Functions
Manipulation and Creation of States and Operators
Quantum States
 basis(dimensions, n=None, offset=None, *, dtype=None)[source]
Generates the vector representation of a Fock state.
 Parameters:
 dimensionsint or list of ints, Space
Number of basis states in Hilbert space. If a list, then the resultant object will be a tensor product over spaces with those dimensions.
 nint or list of ints, optional (default 0 for all dimensions)
Integer corresponding to desired number state, defaults to 0 for all dimensions if omitted. The shape must match
dimensions
, e.g. ifdimensions
is a list, thenn
must either be omitted or a list of equal length. offsetint or list of ints, optional (default 0 for all dimensions)
The lowest number state that is included in the finite number state representation of the state in the relevant dimension.
 dtypetype or str, optional
storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 state
Qobj
Qobj representing the requested number state
n>
.
 state
Notes
A subtle incompatibility with the quantum optics toolbox: In QuTiP:
basis(N, 0) = ground state
but in the qotoolbox:
basis(N, 1) = ground state
Examples
>>> basis(5,2) Quantum object: dims = [[5], [1]], shape = (5, 1), type = ket Qobj data = [[ 0.+0.j] [ 0.+0.j] [ 1.+0.j] [ 0.+0.j] [ 0.+0.j]] >>> basis([2,2,2], [0,1,0]) Quantum object: dims = [[2, 2, 2], [1, 1, 1]], shape = (8, 1), type = ket Qobj data = [[0.] [0.] [1.] [0.] [0.] [0.] [0.] [0.]]
 bell_state(state='00', *, dtype=None)[source]
Returns the selected Bell state:
\[\begin{split}\begin{aligned} \lvert B_{00}\rangle &= \frac1{\sqrt2}(\lvert00\rangle+\lvert11\rangle)\\ \lvert B_{01}\rangle &= \frac1{\sqrt2}(\lvert00\rangle\lvert11\rangle)\\ \lvert B_{10}\rangle &= \frac1{\sqrt2}(\lvert01\rangle+\lvert10\rangle)\\ \lvert B_{11}\rangle &= \frac1{\sqrt2}(\lvert01\rangle\lvert10\rangle)\\ \end{aligned}\end{split}\] Parameters:
 statestr [‘00’, ‘01’, ‘10’, ‘11’]
Which bell state to return
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 Bell_stateqobj
Bell state
 bra(seq, dim=2, *, dtype=None)[source]
Produces a multiparticle bra state for a list or string, where each element stands for state of the respective particle.
 Parameters:
 seqstr / list of ints or characters
Each element defines state of the respective particle. (e.g. [1,1,0,1] or a string “1101”). For qubits it is also possible to use the following conventions:
‘g’/’e’ (ground and excited state)
‘u’/’d’ (spin up and down)
‘H’/’V’ (horizontal and vertical polarization)
Note: for dimension > 9 you need to use a list.
 dimint (default: 2) / list of ints
Space dimension for each particle: int if there are the same, list if they are different.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 braqobj
Examples
>>> bra("10") Quantum object: dims = [[1, 1], [2, 2]], shape = [1, 4], type = bra Qobj data = [[ 0. 0. 1. 0.]]
>>> bra("Hue") Quantum object: dims = [[1, 1, 1], [2, 2, 2]], shape = [1, 8], type = bra Qobj data = [[ 0. 1. 0. 0. 0. 0. 0. 0.]]
>>> bra("12", 3) Quantum object: dims = [[1, 1], [3, 3]], shape = [1, 9], type = bra Qobj data = [[ 0. 0. 0. 0. 0. 1. 0. 0. 0.]]
>>> bra("31", [5, 2]) Quantum object: dims = [[1, 1], [5, 2]], shape = [1, 10], type = bra Qobj data = [[ 0. 0. 0. 0. 0. 0. 0. 1. 0. 0.]]
 coherent(N, alpha, offset=0, method=None, *, dtype=None)[source]
Generates a coherent state with eigenvalue alpha.
Constructed using displacement operator on vacuum state.
 Parameters:
 Nint
Number of Fock states in Hilbert space.
 alphafloat/complex
Eigenvalue of coherent state.
 offsetint, default: 0
The lowest number state that is included in the finite number state representation of the state. Using a nonzero offset will make the default method ‘analytic’.
 methodstring {‘operator’, ‘analytic’}, optional
Method for generating coherent state.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 stateqobj
Qobj quantum object for coherent state
Notes
Select method ‘operator’ (default) or ‘analytic’. With the ‘operator’ method, the coherent state is generated by displacing the vacuum state using the displacement operator defined in the truncated Hilbert space of size ‘N’. This method guarantees that the resulting state is normalized. With ‘analytic’ method the coherent state is generated using the analytical formula for the coherent state coefficients in the Fock basis. This method does not guarantee that the state is normalized if truncated to a small number of Fock states, but would in that case give more accurate coefficients.
Examples
>>> coherent(5,0.25j) Quantum object: dims = [[5], [1]], shape = [5, 1], type = ket Qobj data = [[ 9.69233235e01+0.j ] [ 0.00000000e+00+0.24230831j] [ 4.28344935e02+0.j ] [ 0.00000000e+000.00618204j] [ 7.80904967e04+0.j ]]
 coherent_dm(N, alpha, offset=0, method='operator', *, dtype=None)[source]
Density matrix representation of a coherent state.
Constructed via outer product of
coherent
 Parameters:
 Nint
Number of basis states in Hilbert space.
 alphafloat/complex
Eigenvalue for coherent state.
 offsetint, default: 0
The lowest number state that is included in the finite number state representation of the state.
 methodstring {‘operator’, ‘analytic’}, optional
Method for generating coherent density matrix.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 dmqobj
Density matrix representation of coherent state.
Notes
Select method ‘operator’ (default) or ‘analytic’. With the ‘operator’ method, the coherent density matrix is generated by displacing the vacuum state using the displacement operator defined in the truncated Hilbert space of size ‘N’. This method guarantees that the resulting density matrix is normalized. With ‘analytic’ method the coherent density matrix is generated using the analytical formula for the coherent state coefficients in the Fock basis. This method does not guarantee that the state is normalized if truncated to a small number of Fock states, but would in that case give more accurate coefficients.
Examples
>>> coherent_dm(3,0.25j) Quantum object: dims = [[3], [3]], shape = [3, 3], type = oper, isHerm = True Qobj data = [[ 0.93941695+0.j 0.000000000.23480733j 0.04216943+0.j ] [ 0.00000000+0.23480733j 0.05869011+0.j 0.000000000.01054025j] [0.04216943+0.j 0.00000000+0.01054025j 0.00189294+0.j ]]
 fock(dimensions, n=None, offset=None, *, dtype=None)[source]
Bosonic Fock (number) state.
Same as
basis
. Parameters:
 dimensionsint or list of ints, Space
Number of basis states in Hilbert space. If a list, then the resultant object will be a tensor product over spaces with those dimensions.
 nint or list of ints, default: 0 for all dimensions
Integer corresponding to desired number state, defaults to 0 for all dimensions if omitted. The shape must match
dimensions
, e.g. ifdimensions
is a list, thenn
must either be omitted or a list of equal length. offsetint or list of ints, default: 0 for all dimensions
The lowest number state that is included in the finite number state representation of the state in the relevant dimension.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 Requested number state \(\leftn\right>\).
Examples
>>> fock(4,3) Quantum object: dims = [[4], [1]], shape = [4, 1], type = ket Qobj data = [[ 0.+0.j] [ 0.+0.j] [ 0.+0.j] [ 1.+0.j]]
 fock_dm(dimensions, n=None, offset=None, *, dtype=None)[source]
Density matrix representation of a Fock state
Constructed via outer product of
basis
. Parameters:
 dimensionsint or list of ints, Space
Number of basis states in Hilbert space. If a list, then the resultant object will be a tensor product over spaces with those dimensions.
 nint or list of ints, default: 0 for all dimensions
Integer corresponding to desired number state, defaults to 0 for all dimensions if omitted. The shape must match
dimensions
, e.g. ifdimensions
is a list, thenn
must either be omitted or a list of equal length. offsetint or list of ints, default: 0 for all dimensions
The lowest number state that is included in the finite number state representation of the state in the relevant dimension.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 dmqobj
Density matrix representation of Fock state.
Examples
>>> fock_dm(3,1) Quantum object: dims = [[3], [3]], shape = [3, 3], type = oper, isHerm = True Qobj data = [[ 0.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 1.+0.j 0.+0.j] [ 0.+0.j 0.+0.j 0.+0.j]]
 ghz_state(N_qubit, *, dtype=None)[source]
 Returns the Nqubit GHZstate:
[ 00...00> + 11...11> ] / sqrt(2)
 Parameters:
 N_qubitint
Number of qubits in state
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 Gqobj
Nqubit GHZstate
 ket(seq, dim=2, *, dtype=None)[source]
Produces a multiparticle ket state for a list or string, where each element stands for state of the respective particle.
 Parameters:
 seqstr / list of ints or characters
Each element defines state of the respective particle. (e.g. [1,1,0,1] or a string “1101”). For qubits it is also possible to use the following conventions:  ‘g’/’e’ (ground and excited state)  ‘u’/’d’ (spin up and down)  ‘H’/’V’ (horizontal and vertical polarization) Note: for dimension > 9 you need to use a list.
 dimint or list of ints, default: 2
Space dimension for each particle: int if there are the same, list if they are different.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 ketqobj
Examples
>>> ket("10") Quantum object: dims = [[2, 2], [1, 1]], shape = [4, 1], type = ket Qobj data = [[ 0.] [ 0.] [ 1.] [ 0.]]
>>> ket("Hue") Quantum object: dims = [[2, 2, 2], [1, 1, 1]], shape = [8, 1], type = ket Qobj data = [[ 0.] [ 1.] [ 0.] [ 0.] [ 0.] [ 0.] [ 0.] [ 0.]]
>>> ket("12", 3) Quantum object: dims = [[3, 3], [1, 1]], shape = [9, 1], type = ket Qobj data = [[ 0.] [ 0.] [ 0.] [ 0.] [ 0.] [ 1.] [ 0.] [ 0.] [ 0.]]
>>> ket("31", [5, 2]) Quantum object: dims = [[5, 2], [1, 1]], shape = [10, 1], type = ket Qobj data = [[ 0.] [ 0.] [ 0.] [ 0.] [ 0.] [ 0.] [ 0.] [ 1.] [ 0.] [ 0.]]
 ket2dm(Q)[source]
Takes input ket or bra vector and returns density matrix formed by outer product. This is completely identical to calling
Q.proj()
. Parameters:
 Q
Qobj
Ket or bra type quantum object.
 Q
 Returns:
 dm
Qobj
Density matrix formed by outer product of Q.
 dm
Examples
>>> x=basis(3,2) >>> ket2dm(x) Quantum object: dims = [[3], [3]], shape = [3, 3], type = oper, isHerm = True Qobj data = [[ 0.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 0.+0.j 0.+0.j] [ 0.+0.j 0.+0.j 1.+0.j]]
 maximally_mixed_dm(dimensions, *, dtype=None)[source]
Returns the maximally mixed density matrix for a Hilbert space of dimension N.
 Parameters:
 dimensionsint or list of ints, Space
Number of basis states in Hilbert space. If a list, then the resultant object will be a tensor product over spaces with those dimensions.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 dm
Qobj
Thermal state density matrix.
 dm
 phase_basis(N, m, phi0=0, *, dtype=None)[source]
Basis vector for the mth phase of the PeggBarnett phase operator.
 Parameters:
 Nint
Number of basis states in Hilbert space.
 mint
Integer corresponding to the mth discrete phase
phi_m = phi0 + 2 * pi * m / N
 phi0float, default: 0
Reference phase angle.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 stateqobj
Ket vector for mth PeggBarnett phase operator basis state.
Notes
The PeggBarnett basis states form a complete set over the truncated Hilbert space.
 projection(dimensions, n, m, offset=None, *, dtype=None)[source]
The projection operator that projects state \(\lvert m\rangle\) on state \(\lvert n\rangle\).
 Parameters:
 dimensionsint or list of ints, Space
Number of basis states in Hilbert space. If a list, then the resultant object will be a tensor product over spaces with those dimensions.
 n, mint
The number states in the projection.
 offsetint, default: 0
The lowest number state that is included in the finite number state representation of the projector.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 operqobj
Requested projection operator.
 qutrit_basis(*, dtype=None)[source]
Basis states for a three level system (qutrit)
 dtypetype or str, optional
storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 qstatesarray
Array of qutrit basis vectors
 singlet_state(*, dtype=None)[source]
Returns the two particle singletstate:
\[\lvert S\rangle = \frac1{\sqrt2}(\lvert01\rangle\lvert10\rangle)\]that is identical to the fourth bell state.
 Parameters:
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 Bell_stateqobj
\(\lvert B_{11}\rangle\) Bell state
 spin_coherent(j, theta, phi, type='ket', *, dtype=None)[source]
Generate the coherent spin state \(\lvert \theta, \phi\rangle\).
 Parameters:
 jfloat
The spin of the state.
 thetafloat
Angle from z axis.
 phifloat
Angle from x axis.
 typestring {‘ket’, ‘bra’, ‘dm’}, default: ‘ket’
Type of state to generate.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 stateqobj
Qobj quantum object for spin coherent state
 spin_state(j, m, type='ket', *, dtype=None)[source]
Generates the spin state \(\lvert j, m\rangle\), i.e. the eigenstate of the spinj Sz operator with eigenvalue m.
 Parameters:
 jfloat
The spin of the state ().
 mint
Eigenvalue of the spinj Sz operator.
 typestring {‘ket’, ‘bra’, ‘dm’}, default: ‘ket’
Type of state to generate.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 stateqobj
Qobj quantum object for spin state
 state_index_number(dims, index)[source]
Return a quantum number representation given a state index, for a system of composite structure defined by dims.
Example
>>> state_index_number([2, 2, 2], 6) [1, 1, 0]
 Parameters:
 dimslist or array
The quantum state dimensions array, as it would appear in a Qobj.
 indexinteger
The index of the state in standard enumeration ordering.
 Returns:
 statetuple
The state number tuple corresponding to index index in standard enumeration ordering.
 state_number_enumerate(dims, excitations=None)[source]
An iterator that enumerates all the state number tuples (quantum numbers of the form (n1, n2, n3, …)) for a system with dimensions given by dims.
Example
>>> for state in state_number_enumerate([2,2]): >>> print(state) ( 0 0 ) ( 0 1 ) ( 1 0 ) ( 1 1 )
 Parameters:
 dimslist or array
The quantum state dimensions array, as it would appear in a Qobj.
 excitationsinteger, optional
Restrict state space to states with excitation numbers below or equal to this value.
 Returns:
 state_numbertuple
Successive state number tuples that can be used in loops and other iterations, using standard state enumeration by definition.
 state_number_index(dims, state)[source]
Return the index of a quantum state corresponding to state, given a system with dimensions given by dims.
Example
>>> state_number_index([2, 2, 2], [1, 1, 0]) 6
 Parameters:
 dimslist or array
The quantum state dimensions array, as it would appear in a Qobj.
 statelist
State number array.
 Returns:
 idxint
The index of the state given by state in standard enumeration ordering.
 state_number_qobj(dims, state, *, dtype=None)[source]
Return a Qobj representation of a quantum state specified by the state array state.
Note
Deprecated in QuTiP 5.0, use
basis
instead.Example
>>> state_number_qobj([2, 2, 2], [1, 0, 1]) Quantum object: dims = [[2, 2, 2], [1, 1, 1]], shape = [8, 1], type = ket Qobj data = [[ 0.] [ 0.] [ 0.] [ 0.] [ 0.] [ 1.] [ 0.] [ 0.]]
 thermal_dm(N, n, method='operator', *, dtype=None)[source]
Density matrix for a thermal state of n particles
 Parameters:
 Nint
Number of basis states in Hilbert space.
 nfloat
Expectation value for number of particles in thermal state.
 methodstring {‘operator’, ‘analytic’}, default: ‘operator’
string
that sets the method used to generate the thermal state probabilities dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 dmqobj
Thermal state density matrix.
Notes
The ‘operator’ method (default) generates the thermal state using the truncated number operator
num(N)
. This is the method that should be used in computations. The ‘analytic’ method uses the analytic coefficients derived in an infinite Hilbert space. The analytic form is not necessarily normalized, if truncated too aggressively.Examples
>>> thermal_dm(5, 1) Quantum object: dims = [[5], [5]], shape = [5, 5], type = oper, isHerm = True Qobj data = [[ 0.51612903 0. 0. 0. 0. ] [ 0. 0.25806452 0. 0. 0. ] [ 0. 0. 0.12903226 0. 0. ] [ 0. 0. 0. 0.06451613 0. ] [ 0. 0. 0. 0. 0.03225806]]
>>> thermal_dm(5, 1, 'analytic') Quantum object: dims = [[5], [5]], shape = [5, 5], type = oper, isHerm = True Qobj data = [[ 0.5 0. 0. 0. 0. ] [ 0. 0.25 0. 0. 0. ] [ 0. 0. 0.125 0. 0. ] [ 0. 0. 0. 0.0625 0. ] [ 0. 0. 0. 0. 0.03125]]
 triplet_states(*, dtype=None)[source]
Returns a list of the two particle tripletstates:
\[\lvert T_1\rangle = \lvert11\rangle \lvert T_2\rangle = \frac1{\sqrt2}(\lvert01\rangle + \lvert10\rangle) \lvert T_3\rangle = \lvert00\rangle\] Parameters:
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 trip_stateslist
2 particle triplet states
 w_state(N_qubit, *, dtype=None)[source]
 Returns the Nqubit Wstate:
[ 100..0> + 010..0> + 001..0> + ... 000..1> ] / sqrt(n)
 Parameters:
 N_qubitint
Number of qubits in state
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 W
Qobj
Nqubit Wstate
 W
 zero_ket(dimensions, *, dtype=None)[source]
Creates the zero ket vector with shape Nx1 and dimensions dims.
 Parameters:
 dimensionsint or list of ints, Space
Number of basis states in Hilbert space. If a list, then the resultant object will be a tensor product over spaces with those dimensions.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 zero_ketqobj
Zero ket on given Hilbert space.
Quantum Operators
This module contains functions for generating Qobj representation of a variety of commonly occuring quantum operators.
 charge(Nmax, Nmin=None, frac=1, *, dtype=None)[source]
Generate the diagonal charge operator over charge states from Nmin to Nmax.
 Parameters:
 Nmaxint
Maximum charge state to consider.
 Nminint, default: Nmax
Lowest charge state to consider.
 fracfloat, default: 1
Specify fractional charge if needed.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 CQobj
Charge operator over [Nmin, Nmax].
Notes
New in version 3.2.
 commutator(A, B, kind='normal')[source]
Return the commutator of kind kind (normal, anti) of the two operators A and B.
 Parameters:
 A, B
Qobj
,QobjEvo
The operators to compute the commutator of.
 kind: str {“normal”, “anti”}, default: “anti”
Which kind of commutator to compute.
 A, B
 create(N, offset=0, *, dtype=None)[source]
Creation (raising) operator.
 Parameters:
 Nint
Number of basis states in the Hilbert space.
 offsetint, default: 0
The lowest number state that is included in the finite number state representation of the operator.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 operqobj
Qobj for raising operator.
Examples
>>> create(4) Quantum object: dims=[[4], [4]], shape=(4, 4), type='oper', isherm=False Qobj data = [[ 0.00000000+0.j 0.00000000+0.j 0.00000000+0.j 0.00000000+0.j] [ 1.00000000+0.j 0.00000000+0.j 0.00000000+0.j 0.00000000+0.j] [ 0.00000000+0.j 1.41421356+0.j 0.00000000+0.j 0.00000000+0.j] [ 0.00000000+0.j 0.00000000+0.j 1.73205081+0.j 0.00000000+0.j]]
 destroy(N, offset=0, *, dtype=None)[source]
Destruction (lowering) operator.
 Parameters:
 Nint
Number of basis states in the Hilbert space.
 offsetint, default: 0
The lowest number state that is included in the finite number state representation of the operator.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 operqobj
Qobj for lowering operator.
Examples
>>> destroy(4) Quantum object: dims=[[4], [4]], shape=(4, 4), type='oper', isherm=False Qobj data = [[ 0.00000000+0.j 1.00000000+0.j 0.00000000+0.j 0.00000000+0.j] [ 0.00000000+0.j 0.00000000+0.j 1.41421356+0.j 0.00000000+0.j] [ 0.00000000+0.j 0.00000000+0.j 0.00000000+0.j 1.73205081+0.j] [ 0.00000000+0.j 0.00000000+0.j 0.00000000+0.j 0.00000000+0.j]]
 displace(N, alpha, offset=0, *, dtype=None)[source]
Singlemode displacement operator.
 Parameters:
 Nint
Number of basis states in the Hilbert space.
 alphafloat/complex
Displacement amplitude.
 offsetint, default: 0
The lowest number state that is included in the finite number state representation of the operator.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 operqobj
Displacement operator.
Examples
>>> displace(4,0.25) Quantum object: dims = [[4], [4]], shape = [4, 4], type = oper, isHerm = False Qobj data = [[ 0.96923323+0.j 0.24230859+0.j 0.04282883+0.j 0.00626025+0.j] [ 0.24230859+0.j 0.90866411+0.j 0.33183303+0.j 0.07418172+0.j] [ 0.04282883+0.j 0.33183303+0.j 0.84809499+0.j 0.41083747+0.j] [ 0.00626025+0.j 0.07418172+0.j 0.41083747+0.j 0.90866411+0.j]]
 fcreate(n_sites, site, dtype=None)[source]
Fermionic creation operator. We use the JordanWigner transformation, making use of the JordanWigner ZZ..Z strings, to construct this as follows:
\[a_j = \sigma_z^{\otimes j} \otimes (\frac{\sigma_x  i \sigma_y}{2}) \otimes I^{\otimes Nj1}\] Parameters:
 n_sitesint
Number of sites in Fock space.
 siteint
The site in Fock space to add a fermion to. Corresponds to j in the above JW transform.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 operqobj
Qobj for raising operator.
Examples
>>> fcreate(2) Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = False Qobj data = [[0. 0. 0. 0.] [0. 0. 0. 0.] [1. 0. 0. 0.] [0. 1. 0. 0.]]
 fdestroy(n_sites, site, dtype=None)[source]
Fermionic destruction operator. We use the JordanWigner transformation, making use of the JordanWigner ZZ..Z strings, to construct this as follows:
\[a_j = \sigma_z^{\otimes j} \otimes (\frac{\sigma_x + i \sigma_y}{2}) \otimes I^{\otimes Nj1}\] Parameters:
 n_sitesint
Number of sites in Fock space.
 siteint, default: 0
The site in Fock space to add a fermion to. Corresponds to j in the above JW transform.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 operqobj
Qobj for destruction operator.
Examples
>>> fdestroy(2) Quantum object: dims=[[2 2], [2 2]], shape=(4, 4), type='oper', isherm=False Qobj data = [[0. 0. 1. 0.] [0. 0. 0. 1.] [0. 0. 0. 0.] [0. 0. 0. 0.]]
 identity(dimensions, *, dtype=None)
Identity operator.
 Parameters:
 dimensions(int) or (list of int) or (list of list of int), Space
Number of basis states in the Hilbert space. If provided as a list of ints, then the dimension is the product over this list, but the
dims
property of the new Qobj are set to this list. This can produce either oper or super depending on the passed dimensions. dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 operqobj
Identity operator Qobj.
Examples
>>> qeye(3) Quantum object: dims = [[3], [3]], shape = (3, 3), type = oper, isherm = True Qobj data = [[ 1. 0. 0.] [ 0. 1. 0.] [ 0. 0. 1.]] >>> qeye([2,2]) Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True Qobj data = [[1. 0. 0. 0.] [0. 1. 0. 0.] [0. 0. 1. 0.] [0. 0. 0. 1.]]
 jmat(j, which=None, *, dtype=None)[source]
Higherorder spin operators:
 Parameters:
 jfloat
Spin of operator
 whichstr, optional
Which operator to return ‘x’,’y’,’z’,’+’,’‘. If not given, then output is [‘x’,’y’,’z’]
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 jmatQobj or tuple of Qobj
qobj
for requested spin operator(s).
Examples
>>> jmat(1) [ Quantum object: dims = [[3], [3]], shape = [3, 3], type = oper, isHerm = True Qobj data = [[ 0. 0.70710678 0. ] [ 0.70710678 0. 0.70710678] [ 0. 0.70710678 0. ]] Quantum object: dims = [[3], [3]], shape = [3, 3], type = oper, isHerm = True Qobj data = [[ 0.+0.j 0.0.70710678j 0.+0.j ] [ 0.+0.70710678j 0.+0.j 0.0.70710678j] [ 0.+0.j 0.+0.70710678j 0.+0.j ]] Quantum object: dims = [[3], [3]], shape = [3, 3], type = oper, isHerm = True Qobj data = [[ 1. 0. 0.] [ 0. 0. 0.] [ 0. 0. 1.]]]
 momentum(N, offset=0, *, dtype=None)[source]
Momentum operator p=1j/sqrt(2)*(aa.dag())
 Parameters:
 Nint
Number of basis states in the Hilbert space.
 offsetint, default: 0
The lowest number state that is included in the finite number state representation of the operator.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 operqobj
Momentum operator as Qobj.
 num(N, offset=0, *, dtype=None)[source]
Quantum object for number operator.
 Parameters:
 Nint
Number of basis states in the Hilbert space.
 offsetint, default: 0
The lowest number state that is included in the finite number state representation of the operator.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 oper: qobj
Qobj for number operator.
Examples
>>> num(4) Quantum object: dims=[[4], [4]], shape=(4, 4), type='oper', isherm=True Qobj data = [[0 0 0 0] [0 1 0 0] [0 0 2 0] [0 0 0 3]]
 phase(N, phi0=0, *, dtype=None)[source]
Singlemode PeggBarnett phase operator.
 Parameters:
 Nint
Number of basis states in the Hilbert space.
 phi0float, default: 0
Reference phase.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 operqobj
Phase operator with respect to reference phase.
Notes
The PeggBarnett phase operator is Hermitian on a truncated Hilbert space.
 position(N, offset=0, *, dtype=None)[source]
Position operator \(x = 1 / sqrt(2) * (a + a.dag())\)
 Parameters:
 Nint
Number of basis states in the Hilbert space.
 offsetint, default: 0
The lowest number state that is included in the finite number state representation of the operator.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 operqobj
Position operator as Qobj.
 qdiags(diagonals, offsets=None, dims=None, shape=None, *, dtype=None)[source]
Constructs an operator from an array of diagonals.
 Parameters:
 diagonalssequence of array_like
Array of elements to place along the selected diagonals.
 offsetssequence of ints, optional
 Sequence for diagonals to be set:
k=0 main diagonal
k>0 kth upper diagonal
k<0 kth lower diagonal
 dimslist, optional
Dimensions for operator
 shapelist, tuple, optional
Shape of operator. If omitted, a square operator large enough to contain the diagonals is generated.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
Examples
>>> qdiags(sqrt(range(1, 4)), 1) Quantum object: dims = [[4], [4]], shape = [4, 4], type = oper, isherm = False Qobj data = [[ 0. 1. 0. 0. ] [ 0. 0. 1.41421356 0. ] [ 0. 0. 0. 1.73205081] [ 0. 0. 0. 0. ]]
 qeye(dimensions, *, dtype=None)[source]
Identity operator.
 Parameters:
 dimensions(int) or (list of int) or (list of list of int), Space
Number of basis states in the Hilbert space. If provided as a list of ints, then the dimension is the product over this list, but the
dims
property of the new Qobj are set to this list. This can produce either oper or super depending on the passed dimensions. dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 operqobj
Identity operator Qobj.
Examples
>>> qeye(3) Quantum object: dims = [[3], [3]], shape = (3, 3), type = oper, isherm = True Qobj data = [[ 1. 0. 0.] [ 0. 1. 0.] [ 0. 0. 1.]] >>> qeye([2,2]) Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True Qobj data = [[1. 0. 0. 0.] [0. 1. 0. 0.] [0. 0. 1. 0.] [0. 0. 0. 1.]]
 qeye_like(qobj)[source]
Identity operator with the same dims and type as the reference quantum object.
 Parameters:
 qobjQobj, QobjEvo
Reference quantum object to copy the dims from.
 Returns:
 operqobj
Identity operator Qobj.
 qutrit_ops(*, dtype=None)[source]
Operators for a three level system (qutrit).
 Parameters:
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 opers: array
array of qutrit operators.
 qzero(dimensions, dims_right=None, *, dtype=None)[source]
Zero operator.
 Parameters:
 dimensionsint, list of int, list of list of int, Space
Number of basis states in the Hilbert space. If provided as a list of ints, then the dimension is the product over this list, but the
dims
property of the new Qobj are set to this list. This can produce either oper or super depending on the passed dimensions. dims_rightint, list of int, list of list of int, Space, optional
Number of basis states in the right Hilbert space when the operator is rectangular.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 qzeroqobj
Zero operator Qobj.
 qzero_like(qobj)[source]
Zero operator of the same dims and type as the reference.
 Parameters:
 qobjQobj, QobjEvo
Reference quantum object to copy the dims from.
 Returns:
 qzeroqobj
Zero operator Qobj.
 sigmam()[source]
Annihilation operator for Pauli spins.
Examples
>>> sigmam() Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isHerm = False Qobj data = [[ 0. 0.] [ 1. 0.]]
 sigmap()[source]
Creation operator for Pauli spins.
Examples
>>> sigmap() Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isHerm = False Qobj data = [[ 0. 1.] [ 0. 0.]]
 sigmax()[source]
Pauli spin 1/2 sigmax operator
Examples
>>> sigmax() Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isHerm = False Qobj data = [[ 0. 1.] [ 1. 0.]]
 sigmay()[source]
Pauli spin 1/2 sigmay operator.
Examples
>>> sigmay() Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isHerm = True Qobj data = [[ 0.+0.j 0.1.j] [ 0.+1.j 0.+0.j]]
 sigmaz()[source]
Pauli spin 1/2 sigmaz operator.
Examples
>>> sigmaz() Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isHerm = True Qobj data = [[ 1. 0.] [ 0. 1.]]
 spin_Jm(j, *, dtype=None)[source]
Spinj annihilation operator
 Parameters:
 jfloat
Spin of operator
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 opQobj
qobj
representation of the operator.
 spin_Jp(j, *, dtype=None)[source]
Spinj creation operator
 Parameters:
 jfloat
Spin of operator
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 opQobj
qobj
representation of the operator.
 spin_Jx(j, *, dtype=None)[source]
Spinj x operator
 Parameters:
 jfloat
Spin of operator
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 opQobj
qobj
representation of the operator.
 spin_Jy(j, *, dtype=None)[source]
Spinj y operator
 Parameters:
 jfloat
Spin of operator
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 opQobj
qobj
representation of the operator.
 spin_Jz(j, *, dtype=None)[source]
Spinj z operator
 Parameters:
 jfloat
Spin of operator
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 opQobj
qobj
representation of the operator.
 squeeze(N, z, offset=0, *, dtype=None)[source]
Singlemode squeezing operator.
 Parameters:
 Nint
Dimension of hilbert space.
 zfloat/complex
Squeezing parameter.
 offsetint, default: 0
The lowest number state that is included in the finite number state representation of the operator.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 oper
Qobj
Squeezing operator.
 oper
Examples
>>> squeeze(4, 0.25) Quantum object: dims = [[4], [4]], shape = [4, 4], type = oper, isHerm = False Qobj data = [[ 0.98441565+0.j 0.00000000+0.j 0.17585742+0.j 0.00000000+0.j] [ 0.00000000+0.j 0.95349007+0.j 0.00000000+0.j 0.30142443+0.j] [0.17585742+0.j 0.00000000+0.j 0.98441565+0.j 0.00000000+0.j] [ 0.00000000+0.j 0.30142443+0.j 0.00000000+0.j 0.95349007+0.j]]
 squeezing(a1, a2, z)[source]
Generalized squeezing operator.
\[S(z) = \exp\left(\frac{1}{2}\left(z^*a_1a_2  za_1^\dagger a_2^\dagger\right)\right)\]
 tunneling(N, m=1, *, dtype=None)[source]
Tunneling operator with elements of the form \(\\sum N><N+m + N+m><N\).
 Parameters:
 Nint
Number of basis states in the Hilbert space.
 mint, default: 1
Number of excitations in tunneling event.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 TQobj
Tunneling operator.
Energy Restricted Operators
 enr_destroy(dims, excitations, *, dtype=None)[source]
Generate annilation operators for modes in a excitationnumberrestricted state space. For example, consider a system consisting of 4 modes, each with 5 states. The total hilbert space size is 5**4 = 625. If we are only interested in states that contain up to 2 excitations, we only need to include states such as
(0, 0, 0, 0) (0, 0, 0, 1) (0, 0, 0, 2) (0, 0, 1, 0) (0, 0, 1, 1) (0, 0, 2, 0) …
This function creates annihilation operators for the 4 modes that act within this state space:
a1, a2, a3, a4 = enr_destroy([5, 5, 5, 5], excitations=2)
From this point onwards, the annihiltion operators a1, …, a4 can be used to setup a Hamiltonian, collapse operators and expectationvalue operators, etc., following the usual pattern.
 Parameters:
 dimslist
A list of the dimensions of each subsystem of a composite quantum system.
 excitationsinteger
The maximum number of excitations that are to be included in the state space.
 dtypetype or str, optional
Storage representation. Any datalayer known to qutip.data.to is accepted.
 Returns:
 a_opslist of qobj
A list of annihilation operators for each mode in the composite quantum system described by dims.
 enr_fock(dims, excitations, state, *, dtype=None)[source]
Generate the Fock state representation in a excitationnumber restricted state space. The dims argument is a list of integers that define the number of quantums states of each component of a composite quantum system, and the excitations specifies the maximum number of excitations for the basis states that are to be included in the state space. The state argument is a tuple of integers that specifies the state (in the number basis representation) for which to generate the Fock state representation.
 Parameters:
 dimslist
A list of the dimensions of each subsystem of a composite quantum system.
 excitationsinteger
The maximum number of excitations that are to be included in the state space.
 statelist of integers
The state in the number basis representation.
 dtypetype or str, optional
Storage representation. Any datalayer known to qutip.data.to is accepted.
 Returns:
 ketQobj
A Qobj instance that represent a Fock state in the exicationnumber restricted state space defined by dims and exciations.
 enr_identity(dims, excitations, *, dtype=None)[source]
Generate the identity operator for the excitationnumber restricted state space defined by the dims and exciations arguments. See the docstring for enr_fock for a more detailed description of these arguments.
 Parameters:
 dimslist
A list of the dimensions of each subsystem of a composite quantum system.
 excitationsinteger
The maximum number of excitations that are to be included in the state space.
 dtypetype or str, optional
Storage representation. Any datalayer known to qutip.data.to is accepted.
 Returns:
 opQobj
A Qobj instance that represent the identity operator in the exicationnumberrestricted state space defined by dims and exciations.
 enr_state_dictionaries(dims, excitations)[source]
Return the number of states, and lookupdictionaries for translating a state tuple to a state index, and vice versa, for a system with a given number of components and maximum number of excitations.
 Parameters:
 dims: list
A list with the number of states in each subsystem.
 excitationsinteger
The maximum numbers of dimension
 Returns:
 nstates, state2idx, idx2state: integer, dict, dict
The number of states nstates, a dictionary for looking up state indices from a state tuple, and a dictionary for looking up state state tuples from state indices. state2idx and idx2state are reverses of each other, i.e.,
state2idx[idx2state[idx]] = idx
andidx2state[state2idx[state]] = state
.
 enr_thermal_dm(dims, excitations, n, *, dtype=None)[source]
Generate the density operator for a thermal state in the excitationnumber restricted state space defined by the dims and exciations arguments. See the documentation for enr_fock for a more detailed description of these arguments. The temperature of each mode in dims is specified by the average number of excitatons n.
 Parameters:
 dimslist
A list of the dimensions of each subsystem of a composite quantum system.
 excitationsinteger
The maximum number of excitations that are to be included in the state space.
 ninteger
The average number of exciations in the thermal state. n can be a float (which then applies to each mode), or a list/array of the same length as dims, in which each element corresponds specifies the temperature of the corresponding mode.
 dtypetype or str, optional
Storage representation. Any datalayer known to qutip.data.to is accepted.
 Returns:
 dmQobj
Thermal state density matrix.
Quantum Objects
The Quantum Object (Qobj) class, for representing quantum states and operators, and related functions.
 ptrace(Q, sel)[source]
Partial trace of the Qobj with selected components remaining.
 Parameters:
 Q
Qobj
Composite quantum object.
 selint/list
An
int
orlist
of components to keep after partial trace.
 Q
 Returns:
 oper
Qobj
Quantum object representing partial trace with selected components remaining.
 oper
Notes
This function is for legacy compatibility only. It is recommended to use the
ptrace()
Qobj method.
Random Operators and States
This module is a collection of random state and operator generators.
 rand_dm(dimensions, density=0.75, distribution='ginibre', *, eigenvalues=(), rank=None, seed=None, dtype=None)[source]
Creates a random density matrix of the desired dimensions.
 Parameters:
 dimensions(int) or (list of int) or (list of list of int)
Dimension of Hilbert space. If provided as a list of ints, then the dimension is the product over this list, but the
dims
property of the new Qobj are set to this list. This can produce eitheroper
orsuper
depending on the passeddimensions
. densityfloat, default: 0.75
Density between [0,1] of output density matrix. Used by the “pure”, “eigen” and “herm”.
distribution : str {“ginibre”, “hs”, “pure”, “eigen”, “uniform”},
 default: “ginibre”
Method used to obtain the density matrices.
“ginibre” : Ginibre random density operator of rank
rank
by using the algorithm of [BCSZ08].“hs” : HilbertSchmidt ensemble, equivalent to a full rank ginibre operator.
“pure” : Density matrix created from a random ket.
“eigen” : A density matrix with the given
eigenvalues
.“herm” : Build from a random hermitian matrix using
rand_herm
.
 eigenvaluesarray_like, optional
Eigenvalues of the output Hermitian matrix. The len must match the shape of the matrix.
 rankint, optional
When using the “ginibre” distribution, rank of the density matrix. Will default to a full rank operator when not provided.
 seedint, SeedSequence, Generator, optional
Seed to create the random number generator or a pre prepared generator. When none is suplied, a default generator is used.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 operqobj
Density matrix quantum operator.
 rand_herm(dimensions, density=0.3, distribution='fill', *, eigenvalues=(), seed=None, dtype=None)[source]
Creates a random sparse Hermitian quantum object.
 Parameters:
 dimensions(int) or (list of int) or (list of list of int)
Dimension of Hilbert space. If provided as a list of ints, then the dimension is the product over this list, but the
dims
property of the new Qobj are set to this list. This can produce either oper or super depending on the passed dimensions. densityfloat, default: 0.30
Density between [0,1] of output Hermitian operator.
 distributionstr {“fill”, “pos_def”, “eigen”}, default: “fill”
Method used to obtain the density matrices.
“fill” : Uses \(H=0.5*(X+X^{+})\) where \(X\) is a randomly generated quantum operator with elements uniformly distributed between
[1, 1] + [1j, 1j]
.“eigen” : A density matrix with the given
eigenvalues
. It uses random complex Jacobi rotations to shuffle the operator.“pos_def” : Return a positive semidefinite matrix by diagonal dominance.
 eigenvaluesarray_like, optional
Eigenvalues of the output Hermitian matrix. The len must match the shape of the matrix.
 seedint, SeedSequence, Generator, optional
Seed to create the random number generator or a pre prepared generator. When none is suplied, a default generator is used.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 oper
Qobj
Hermitian quantum operator.
 oper
Notes
If given a list of eigenvalues the object is created using complex Jacobi rotations. While this method is fast for small matrices, it should not be repeatedly used for generating matrices larger than ~1000x1000.
 rand_ket(dimensions, density=1, distribution='haar', *, seed=None, dtype=None)[source]
Creates a random ket vector.
 Parameters:
 dimensions(int) or (list of int) or (list of list of int)
Dimension of Hilbert space. If provided as a list of ints, then the dimension is the product over this list, but the
dims
property of the new Qobj are set to this list. This can produce either oper or super depending on the passed dimensions. densityfloat, default: 1
Density between [0,1] of output ket state when using the
fill
method. distributionstr {“haar”, “fill”}, default: “haar”
Method used to obtain the kets.
haar : Haar random pure state obtained by applying a Haar random unitary to a fixed pure state.
fill : Fill the ket with uniformly distributed random complex number.
 seedint, SeedSequence, Generator, optional
Seed to create the random number generator or a pre prepared generator. When none is suplied, a default generator is used.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 operqobj
Ket quantum state vector.
 rand_kraus_map(dimensions, *, seed=None, dtype=None)[source]
Creates a random CPTP map on an Ndimensional Hilbert space in Kraus form.
 Parameters:
 dimensions(int) or (list of int) or (list of list of int)
Dimension of Hilbert space. If provided as a list of ints, then the dimension is the product over this list, but the
dims
property of the new Qobj are set to this list. This can produce either oper or super depending on the passed dimensions. seedint, SeedSequence, Generator, optional
Seed to create the random number generator or a pre prepared generator. When none is suplied, a default generator is used.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 oper_listlist of qobj
N^2 x N x N qobj operators.
 rand_stochastic(dimensions, density=0.75, kind='left', *, seed=None, dtype=None)[source]
Generates a random stochastic matrix.
 Parameters:
 dimensions(int) or (list of int) or (list of list of int)
Dimension of Hilbert space. If provided as a list of ints, then the dimension is the product over this list, but the
dims
property of the new Qobj are set to this list. This can produce either oper or super depending on the passed dimensions. densityfloat, default: 0.75
Density between [0,1] of output density matrix.
 kindstr {“left”, “right”}, default: “left”
Generate ‘left’ or ‘right’ stochastic matrix.
 seedint, SeedSequence, Generator, optional
Seed to create the random number generator or a pre prepared generator. When none is suplied, a default generator is used.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 operqobj
Quantum operator form of stochastic matrix.
 rand_super(dimensions, *, superrep='super', seed=None, dtype=None)[source]
Returns a randomly drawn superoperator acting on operators acting on N dimensions.
 Parameters:
 dimensions(int) or (list of int) or (list of list of int)
Dimension of Hilbert space. If provided as a list of ints, then the dimension is the product over this list, but the
dims
property of the new Qobj are set to this list. This can produce either oper or super depending on the passed dimensions. superropstr, default: “super”
Representation of the super operator
 seedint, SeedSequence, Generator, optional
Seed to create the random number generator or a pre prepared generator. When none is suplied, a default generator is used.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 rand_super_bcsz(dimensions, enforce_tp=True, rank=None, *, superrep='super', seed=None, dtype=None)[source]
Returns a random superoperator drawn from the Bruzda et al ensemble for CPTP maps [BCSZ08]. Note that due to finite numerical precision, for ranks less than fullrank, zero eigenvalues may become slightly negative, such that the returned operator is not actually completely positive.
 Parameters:
 dimensions(int) or (list of int) or (list of list of int)
Dimension of Hilbert space. If an int is provided, it is understood as the Square root of the dimension of the superoperator to be returned, with the corresponding dims as
[[[N],[N]], [[N],[N]]]
. If provided as a list of ints, then the dimensions is understood as the space of density matrices this superoperator is applied to:dimensions=[2,2]
dims=[[[2,2],[2,2]], [[2,2],[2,2]]]
. enforce_tpbool, default: True
If True, the tracepreserving condition of [BCSZ08] is enforced; otherwise only complete positivity is enforced.
 rankint, optional
Rank of the sampled superoperator. If None, a fullrank superoperator is generated.
 seedint, SeedSequence, Generator, optional
Seed to create the random number generator or a pre prepared generator. When none is suplied, a default generator is used.
 superropstr, default: “super”
representation of the
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 rhoQobj
A superoperator acting on vectorized dim × dim density operators, sampled from the BCSZ distribution.
 rand_unitary(dimensions, density=1, distribution='haar', *, seed=None, dtype=None)[source]
Creates a random sparse unitary quantum object.
 Parameters:
 dimensions(int) or (list of int) or (list of list of int)
Dimension of Hilbert space. If provided as a list of ints, then the dimension is the product over this list, but the
dims
property of the new Qobj are set to this list. This can produce either oper or super depending on the passed dimensions. densityfloat, default: 1
Density between [0,1] of output unitary operator.
 distributionstr {“haar”, “exp”}, default: “haar”
Method used to obtain the unitary matrices.
haar : Haar random unitary matrix using the algorithm of [Mez07].
exp : Uses \(\exp(iH)\), where H is a randomly generated Hermitian operator.
 seedint, SeedSequence, Generator, optional
Seed to create the random number generator or a pre prepared generator. When none is suplied, a default generator is used.
 dtypetype or str, optional
Storage representation. Any datalayer known to
qutip.data.to
is accepted.
 Returns:
 operqobj
Unitary quantum operator.
Superoperators and Liouvillians
 lindblad_dissipator(a, b=None, data_only=False, chi=None)[source]
Lindblad dissipator (generalized) for a single pair of collapse operators (a, b), or for a single collapse operator (a) when b is not specified:
\[\mathcal{D}[a,b]\rho = a \rho b^\dagger  \frac{1}{2}a^\dagger b\rho  \frac{1}{2}\rho a^\dagger b\] Parameters:
 aQobj or QobjEvo
Left part of collapse operator.
 bQobj or QobjEvo, optional
Right part of collapse operator. If not specified, b defaults to a.
 chifloat, optional
In some systems it is possible to determine the statistical moments (mean, variance, etc) of the probability distribution of the occupation numbers of states by numerically evaluating the derivatives of the steady state occupation probability as a function of an artificial phase parameter
chi
which multiplies thea \rho a^dagger
term of the dissipator bye ^ (i * chi)
. The factore ^ (i * chi)
is introduced via the generating function of the statistical moments. For examples of the technique, see Full counting statistics of nanoelectromechanical systems and Photonmediated electron transport in hybrid circuitQED. This parameter is deprecated and may be removed in QuTiP 5. data_onlybool, default: False
Return the data object instead of a Qobj
 Returns:
 Dqobj, QobjEvo
Lindblad dissipator superoperator.
 liouvillian(H=None, c_ops=None, data_only=False, chi=None)[source]
Assembles the Liouvillian superoperator from a Hamiltonian and a
list
of collapse operators. Parameters:
 HQobj or QobjEvo, optional
System Hamiltonian or Hamiltonian component of a Liouvillian. Considered 0 if not given.
 c_opsarray_like of Qobj or QobjEvo, optional
A
list
orarray
of collapse operators. data_onlybool, default: False
Return the data object instead of a Qobj
 chiarray_like of float, optional
In some systems it is possible to determine the statistical moments (mean, variance, etc) of the probability distributions of occupation of various states by numerically evaluating the derivatives of the steady state occupation probability as a function of artificial phase parameters
chi
which are included in thelindblad_dissipator
for each collapse operator. See the documentation oflindblad_dissipator
for references and further details. This parameter is deprecated and may be removed in QuTiP 5.
 Returns:
 LQobj or QobjEvo
Liouvillian superoperator.
 operator_to_vector(op)[source]
Create a vector representation given a quantum operator in matrix form. The passed object should have a
Qobj.type
of ‘oper’ or ‘super’; this function is not designed for generalpurpose matrix reshaping. Parameters:
 opQobj or QobjEvo
Quantum operator in matrix form. This must have a type of ‘oper’ or ‘super’.
 Returns:
 Qobj or QobjEvo
The same object, but recast into a columnstackedvector form of type ‘operatorket’. The output is the same type as the passed object.
 spost(A)[source]
Superoperator formed from postmultiplication by operator A
 Parameters:
 AQobj or QobjEvo
Quantum operator for post multiplication.
 Returns:
 superQobj or QobjEvo
Superoperator formed from input qauntum object.
 spre(A)[source]
Superoperator formed from premultiplication by operator A.
 Parameters:
 AQobj or QobjEvo
Quantum operator for premultiplication.
 Returns:
 super :Qobj or QobjEvo
Superoperator formed from input quantum object.
 sprepost(A, B)[source]
Superoperator formed from premultiplication by A and postmultiplication by B.
 Parameters:
 AQobj or QobjEvo
Quantum operator for premultiplication.
 BQobj or QobjEvo
Quantum operator for postmultiplication.
 Returns:
 superQobj or QobjEvo
Superoperator formed from input quantum objects.
 vector_to_operator(op)[source]
Create a matrix representation given a quantum operator in vector form. The passed object should have a
Qobj.type
of ‘operatorket’; this function is not designed for generalpurpose matrix reshaping. Parameters:
 opQobj or QobjEvo
Quantum operator in columnstackedvector form. This must have a type of ‘operatorket’.
 Returns:
 Qobj or QobjEvo
The same object, but recast into “standard” operator form. The output is the same type as the passed object.
Superoperator Representations
This module implements transformations between superoperator representations, including supermatrix, Kraus, Choi and Chi (process) matrix formalisms.
 kraus_to_choi(kraus_ops)[source]
Convert a list of Kraus operators into Choi representation of the channel.
Essentially, kraus operators are a decomposition of a Choi matrix, and its reconstruction from them should go as \(E = \sum_{i} K_i\rangle\rangle \langle\langle K_i\), where we use vector representation of Kraus operators.
 Parameters:
 kraus_opslist[Qobj]
The list of Kraus operators to be converted to Choi representation.
 Returns:
 choiQobj
A quantum object representing the same map as
kraus_ops
, such thatchoi.superrep == "choi"
.
 kraus_to_super(kraus_list)[source]
Convert a list of Kraus operators to a superoperator.
 Parameters:
 kraus_listlist of Qobj
The list of Kraus super operators to convert.
 to_chi(q_oper)[source]
Converts a Qobj representing a quantum map to a representation as a chi (process) matrix in the Pauli basis, such that the trace of the returned operator is equal to the dimension of the system.
 Parameters:
 q_operQobj
Superoperator to be converted to Chi representation. If
q_oper
istype="oper"
, then it is taken to act by conjugation, such thatto_chi(A) == to_chi(sprepost(A, A.dag()))
.
 Returns:
 chiQobj
A quantum object representing the same map as
q_oper
, such thatchi.superrep == "chi"
.
 Raises:
 TypeError:
If the given quantum object is not a map, or cannot be converted to Chi representation.
 to_choi(q_oper)[source]
Converts a Qobj representing a quantum map to the Choi representation, such that the trace of the returned operator is equal to the dimension of the system.
 Parameters:
 q_operQobj
Superoperator to be converted to Choi representation. If
q_oper
istype="oper"
, then it is taken to act by conjugation, such thatto_choi(A) == to_choi(sprepost(A, A.dag()))
.
 Returns:
 choiQobj
A quantum object representing the same map as
q_oper
, such thatchoi.superrep == "choi"
.
 Raises:
 TypeError:
If the given quantum object is not a map, or cannot be converted to Choi representation.
 to_kraus(q_oper, tol=1e09)[source]
Converts a Qobj representing a quantum map to a list of quantum objects, each representing an operator in the Kraus decomposition of the given map.
 Parameters:
 q_operQobj
Superoperator to be converted to Kraus representation. If
q_oper
istype="oper"
, then it is taken to act by conjugation, such thatto_kraus(A) == to_kraus(sprepost(A, A.dag())) == [A]
. tolFloat, default: 1e9
Optional threshold parameter for eigenvalues/Kraus ops to be discarded.
 Returns:
 kraus_opslist of Qobj
A list of quantum objects, each representing a Kraus operator in the decomposition of
q_oper
.
 Raises:
 TypeError: if the given quantum object is not a map, or cannot be
decomposed into Kraus operators.
 to_stinespring(q_oper, threshold=1e10)[source]
Converts a Qobj representing a quantum map \(\Lambda\) to a pair of partial isometries
A
andB
such that \(\Lambda(X) = \Tr_2(A X B^\dagger)\) for all inputsX
, where the partial trace is taken over a a new index on the output dimensions ofA
andB
.For completely positive inputs,
A
will always equalB
up to precision errors. Parameters:
 q_operQobj
Superoperator to be converted to a Stinespring pair.
 thresholdfloat, default: 1e10
Threshold parameter for eigenvalues/Kraus ops to be discarded.
 Returns:
 A, BQobj
Quantum objects representing each of the Stinespring matrices for the input Qobj.
 to_super(q_oper)[source]
Converts a Qobj representing a quantum map to the supermatrix (Liouville) representation.
 Parameters:
 q_operQobj
Superoperator to be converted to supermatrix representation. If
q_oper
istype="oper"
, then it is taken to act by conjugation, such thatto_super(A) == sprepost(A, A.dag())
.
 Returns:
 superopQobj
A quantum object representing the same map as
q_oper
, such thatsuperop.superrep == "super"
.
 Raises:
 TypeError
If the given quantum object is not a map, or cannot be converted to supermatrix representation.
Operators and Superoperator Dimensions
Internal use module for manipulating dims specifications.
 from_tensor_rep(tensorrep, dims)[source]
Reverse operator of
to_tensor_rep
. Create a Qobj From a Ndimensions numpy array and dimensions with N indices. Parameters:
 tensorrep: ndarray
Numpy array with one dimension for each index in dims.
 dims: list of list, Dimensions
Dimensions of the Qobj.
 Returns:
 Qobj
Re constructed Qobj
 to_tensor_rep(q_oper)[source]
Transform a
Qobj
to a numpy array whose shape is the flattened dimensions. Parameters:
 q_oper: Qobj
Object to reshape
 Returns:
 ndarray:
Numpy array with one dimension for each index in dims.
Examples
>>> ket.dims [[2, 3], [1]] >>> to_tensor_rep(ket).shape (2, 3, 1)
>>> oper.dims [[2, 3], [2, 3]] >>> to_tensor_rep(oper).shape (2, 3, 2, 3)
>>> super_oper.dims [[[2, 3], [2, 3]], [[2, 3], [2, 3]]] >>> to_tensor_rep(super_oper).shape (2, 3, 2, 3, 2, 3, 2, 3)
Functions acting on states and operators
Expectation Values
 expect(oper, state)[source]
Calculate the expectation value for operator(s) and state(s). The expectation of state
k
on operatorA
is defined ask.dag() @ A @ k
, and for density matrixR
on operatorA
it istrace(A @ R)
. Parameters:
 operqobj/arraylike
A single or a list of operators for expectation value.
 stateqobj/arraylike
A single or a list of quantum states or density matrices.
 Returns:
 exptfloat/complex/arraylike
Expectation value.
real
ifoper
is Hermitian,complex
otherwise. A (nested) array of expectaction values ifstate
oroper
are arrays.
Examples
>>> expect(num(4), basis(4, 3)) == 3 True
Tensor
Module for the creation of composite quantum objects via the tensor product.
 composite(*args)[source]
Given two or more operators, kets or bras, returns the Qobj corresponding to a composite system over each argument. For ordinary operators and vectors, this is the tensor product, while for superoperators and vectorized operators, this is the columnreshuffled tensor product.
If a mix of Qobjs supported on Hilbert and Liouville spaces are passed in, the former are promoted. Ordinary operators are assumed to be unitaries, and are promoted using
to_super
, while kets and bras are promoted by taking their projectors and usingoperator_to_vector(ket2dm(arg))
.
 super_tensor(*args)[source]
Calculate the tensor product of input superoperators, by tensoring together the underlying Hilbert spaces on which each vectorized operator acts.
 Parameters:
 argsarray_like
list
orarray
of quantum objects withtype="super"
.
 Returns:
 objqobj
A composite quantum object.
 tensor(*args)[source]
Calculates the tensor product of input operators.
 Parameters:
 argsarray_like
list
orarray
of quantum objects for tensor product.
 Returns:
 objqobj
A composite quantum object.
Examples
>>> tensor([sigmax(), sigmax()]) Quantum object: dims = [[2, 2], [2, 2]], shape = [4, 4], type = oper, isHerm = True Qobj data = [[ 0.+0.j 0.+0.j 0.+0.j 1.+0.j] [ 0.+0.j 0.+0.j 1.+0.j 0.+0.j] [ 0.+0.j 1.+0.j 0.+0.j 0.+0.j] [ 1.+0.j 0.+0.j 0.+0.j 0.+0.j]]
 tensor_contract(qobj, *pairs)[source]
Contracts a qobj along one or more index pairs. Note that this uses dense representations and thus should not be used for very large Qobjs.
 Parameters:
 qobj: Qobj
Operator to contract subspaces on.
 pairstuple
One or more tuples
(i, j)
indicating that thei
andj
dimensions of the original qobj should be contracted.
 Returns:
 cqobjQobj
The original Qobj with all named index pairs contracted away.
Partial Transpose
 partial_transpose(rho, mask, method='dense')[source]
Return the partial transpose of a Qobj instance rho, where mask is an array/list with length that equals the number of components of rho (that is, the length of rho.dims[0]), and the values in mask indicates whether or not the corresponding subsystem is to be transposed. The elements in mask can be boolean or integers 0 or 1, where True/1 indicates that the corresponding subsystem should be tranposed.
 Parameters:
 rho
Qobj
A density matrix.
 masklist / array
A mask that selects which subsystems should be transposed.
 methodstr {“dense”, “sparse”}, default: “dense”
Choice of method. The “sparse” implementation can be faster for large and sparse systems (hundreds of quantum states).
 rho
 Returns:
 rho_pr:
Qobj
A density matrix with the selected subsystems transposed.
 rho_pr:
Entropy Functions
 concurrence(rho)[source]
Calculate the concurrence entanglement measure for a twoqubit state.
 Parameters:
 stateqobj
Ket, bra, or density matrix for a twoqubit state.
 Returns:
 concurfloat
Concurrence
References
[1]https://en.wikipedia.org/wiki/Concurrence_(quantum_computing)
 entropy_conditional(rho, selB, base=2.718281828459045, sparse=False)[source]
Calculates the conditional entropy \(S(AB)=S(A,B)S(B)\) of a selected density matrix component.
 Parameters:
 rhoqobj
Density matrix of composite object
 selBint/list
Selected components for density matrix B
 base{e, 2}, default: e
Base of logarithm.
 sparsebool, default: False
Use sparse eigensolver.
 Returns:
 ent_condfloat
Value of conditional entropy
 entropy_linear(rho)[source]
Linear entropy of a density matrix.
 Parameters:
 rhoqobj
sensity matrix or ket/bra vector.
 Returns:
 entropyfloat
Linear entropy of rho.
Examples
>>> rho=0.5*fock_dm(2,0)+0.5*fock_dm(2,1) >>> entropy_linear(rho) 0.5
 entropy_mutual(rho, selA, selB, base=2.718281828459045, sparse=False)[source]
Calculates the mutual information S(A:B) between selection components of a system density matrix.
 Parameters:
 rhoqobj
Density matrix for composite quantum systems
 selAint/list
int or list of first selected density matrix components.
 selBint/list
int or list of second selected density matrix components.
 base{e, 2}, default: e
Base of logarithm.
 sparsebool, default: False
Use sparse eigensolver.
 Returns:
 ent_mutfloat
Mutual information between selected components.
 entropy_relative(rho, sigma, base=2.718281828459045, sparse=False, tol=1e12)[source]
Calculates the relative entropy S(rhosigma) between two density matrices.
 Parameters:
 rho
Qobj
First density matrix (or ket which will be converted to a density matrix).
 sigma
Qobj
Second density matrix (or ket which will be converted to a density matrix).
 base{e, 2}, default: e
Base of logarithm. Defaults to e.
 sparsebool, default: False
Flag to use sparse solver when determining the eigenvectors of the density matrices. Defaults to False.
 tolfloat, default: 1e12
Tolerance to use to detect 0 eigenvalues or dot producted between eigenvectors. Defaults to 1e12.
 rho
 Returns:
 rel_entfloat
Value of relative entropy. Guaranteed to be greater than zero and should equal zero only when rho and sigma are identical.
References
See Nielsen & Chuang, “Quantum Computation and Quantum Information”, Section 11.3.1, pg. 511 for a detailed explanation of quantum relative entropy.
Examples
First we define two density matrices:
>>> rho = qutip.ket2dm(qutip.ket("00")) >>> sigma = rho + qutip.ket2dm(qutip.ket("01")) >>> sigma = sigma.unit()
Then we calculate their relative entropy using base 2 (i.e.
log2
) and base e (i.e.log
).>>> qutip.entropy_relative(rho, sigma, base=2) 1.0 >>> qutip.entropy_relative(rho, sigma) 0.6931471805599453
 entropy_vn(rho, base=2.718281828459045, sparse=False)[source]
VonNeumann entropy of density matrix
 Parameters:
 rhoqobj
Density matrix.
 base{e, 2}, default: e
Base of logarithm.
 sparsebool, default: False
Use sparse eigensolver.
 Returns:
 entropyfloat
VonNeumann entropy of rho.
Examples
>>> rho=0.5*fock_dm(2,0)+0.5*fock_dm(2,1) >>> entropy_vn(rho,2) 1.0
Density Matrix Metrics
This module contains a collection of functions for calculating metrics (distance measures) between states and operators.
 average_gate_fidelity(oper, target=None)[source]
Returns the average gate fidelity of a quantum channel to the target channel, or to the identity channel if no target is given.
 Parameters:
 Returns:
 fidfloat
Average gate fidelity between oper and target, or between oper and identity.
Notes
The average gate fidelity is defined for example in: A. Gilchrist, N.K. Langford, M.A. Nielsen, Phys. Rev. A 71, 062310 (2005). The definition of state fidelity that the average gate fidelity is based on is the one from R. Jozsa, Journal of Modern Optics, 41:12, 2315 (1994). It is the square of the fidelity implemented in
qutip.core.metrics.fidelity
which follows Nielsen & Chuang, “Quantum Computation and Quantum Information”
 bures_angle(A, B)[source]
Returns the Bures Angle between two density matrices A & B.
The Bures angle ranges from 0, for states with unit fidelity, to pi/2.
 Parameters:
 Aqobj
Density matrix or state vector.
 Bqobj
Density matrix or state vector with same dimensions as A.
 Returns:
 anglefloat
Bures angle between density matrices.
 bures_dist(A, B)[source]
Returns the Bures distance between two density matrices A & B.
The Bures distance ranges from 0, for states with unit fidelity, to sqrt(2).
 Parameters:
 Aqobj
Density matrix or state vector.
 Bqobj
Density matrix or state vector with same dimensions as A.
 Returns:
 distfloat
Bures distance between density matrices.
 dnorm(A, B=None, solver='CVXOPT', verbose=False, force_solve=False, sparse=True)[source]
Calculates the diamond norm of the quantum map q_oper, using the simplified semidefinite program of [Wat13].
The diamond norm SDP is solved by using CVXPY.
 Parameters:
 AQobj
Quantum map to take the diamond norm of.
 BQobj or None
If provided, the diamond norm of \(A  B\) is taken instead.
 solverstr {“CVXOPT”, “SCS”}, default: “CVXOPT”
Solver to use with CVXPY. “SCS” tends to be significantly faster, but somewhat less accurate.
 verbosebool, default: False
If True, prints additional information about the solution.
 force_solvebool, default: False
If True, forces dnorm to solve the associated SDP, even if a special case is known for the argument.
 sparsebool, default: True
Whether to use sparse matrices in the convex optimisation problem. Default True.
 Returns:
 dnfloat
Diamond norm of q_oper.
 Raises:
 ImportError
If CVXPY cannot be imported.
 fidelity(A, B)[source]
Calculates the fidelity (pseudometric) between two density matrices.
 Parameters:
 Aqobj
Density matrix or state vector.
 Bqobj
Density matrix or state vector with same dimensions as A.
 Returns:
 fidfloat
Fidelity pseudometric between A and B.
Notes
Uses the definition from Nielsen & Chuang, “Quantum Computation and Quantum Information”. It is the square root of the fidelity defined in R. Jozsa, Journal of Modern Optics, 41:12, 2315 (1994), used in
qutip.core.metrics.process_fidelity
.Examples
>>> x = fock_dm(5,3) >>> y = coherent_dm(5,1) >>> np.testing.assert_almost_equal(fidelity(x,y), 0.24104350624628332)
 hellinger_dist(A, B, sparse=False, tol=0)[source]
Calculates the quantum Hellinger distance between two density matrices.
Formula:
hellinger_dist(A, B) = sqrt(2  2 * tr(sqrt(A) * sqrt(B)))
See: D. Spehner, F. Illuminati, M. Orszag, and W. Roga, “Geometric measures of quantum correlations with Bures and Hellinger distances” arXiv:1611.03449
 Parameters:
 Returns:
 hellinger_distfloat
Quantum Hellinger distance between A and B. Ranges from 0 to sqrt(2).
Examples
>>> x = fock_dm(5,3) >>> y = coherent_dm(5,1) >>> np.allclose(hellinger_dist(x, y), 1.3725145002591095) True
 hilbert_dist(A, B)[source]
Returns the HilbertSchmidt distance between two density matrices A & B.
 Parameters:
 Aqobj
Density matrix or state vector.
 Bqobj
Density matrix or state vector with same dimensions as A.
 Returns:
 distfloat
HilbertSchmidt distance between density matrices.
Notes
See V. Vedral and M. B. Plenio, Phys. Rev. A 57, 1619 (1998).
 process_fidelity(oper, target=None)[source]
Returns the process fidelity of a quantum channel to the target channel, or to the identity channel if no target is given. The process fidelity between two channels is defined as the state fidelity between their normalized Choi matrices.
 Parameters:
 Returns:
 fidfloat
Process fidelity between oper and target, or between oper and identity.
Notes
Since Qutip 5.0, this function computes the process fidelity as defined for example in: A. Gilchrist, N.K. Langford, M.A. Nielsen, Phys. Rev. A 71, 062310 (2005). Previously, it computed a function that is now implemented as
get_fidelity
in qutipqtrl.The definition of state fidelity that the process fidelity is based on is the one from R. Jozsa, Journal of Modern Optics, 41:12, 2315 (1994). It is the square of the one implemented in
qutip.core.metrics.fidelity
which follows Nielsen & Chuang, “Quantum Computation and Quantum Information”
 tracedist(A, B, sparse=False, tol=0)[source]
Calculates the trace distance between two density matrices.. See: Nielsen & Chuang, “Quantum Computation and Quantum Information”
 Parameters:
 Aqobj
Density matrix or state vector.
 Bqobj
Density matrix or state vector with same dimensions as A.
 tolfloat, default: 0
Tolerance used by sparse eigensolver, if used. (0 = Machine precision)
 sparsebool, default: False
Use sparse eigensolver.
 Returns:
 tracedistfloat
Trace distance between A and B.
Examples
>>> x=fock_dm(5,3) >>> y=coherent_dm(5,1) >>> np.testing.assert_almost_equal(tracedist(x,y), 0.9705143161472971)
Continuous Variables
This module contains a collection functions for calculating continuous variable quantities from fockbasis representation of the state of multimode fields.
 correlation_matrix(basis, rho=None)[source]
Given a basis set of operators \(\{a\}_n\), calculate the correlation matrix:
\[C_{mn} = \langle a_m a_n \rangle\] Parameters:
 basislist
List of operators that defines the basis for the correlation matrix.
 rhoQobj, optional
Density matrix for which to calculate the correlation matrix. If rho is None, then a matrix of correlation matrix operators is returned instead of expectation values of those operators.
 Returns:
 corr_matndarray
A 2dimensional array of correlation values or operators.
 correlation_matrix_field(a1, a2, rho=None)[source]
Calculates the correlation matrix for given field operators \(a_1\) and \(a_2\). If a density matrix is given the expectation values are calculated, otherwise a matrix with operators is returned.
 Parameters:
 a1Qobj
Field operator for mode 1.
 a2Qobj
Field operator for mode 2.
 rhoQobj, optional
Density matrix for which to calculate the covariance matrix.
 Returns:
 cov_matndarray
Array of complex numbers or Qobj’s A 2dimensional array of covariance values, or, if rho=0, a matrix of operators.
 correlation_matrix_quadrature(a1, a2, rho=None, g=1.4142135623730951)[source]
Calculate the quadrature correlation matrix with given field operators \(a_1\) and \(a_2\). If a density matrix is given the expectation values are calculated, otherwise a matrix with operators is returned.
 Parameters:
 a1Qobj
Field operator for mode 1.
 a2Qobj
Field operator for mode 2.
 rhoQobj, optional
Density matrix for which to calculate the covariance matrix.
 gfloat, default: sqrt(2)
Scaling factor for
a = 0.5 * g * (x + iy)
, defaultg = sqrt(2)
. The value ofg
is related to the value ofhbar
in the commutation relation[x, y] = i * hbar
viahbar=2/g ** 2
giving the default valuehbar=1
.
 Returns:
 corr_matndarray
Array of complex numbers or Qobj’s A 2dimensional array of covariance values for the field quadratures, or, if rho=0, a matrix of operators.
 covariance_matrix(basis, rho, symmetrized=True)[source]
Given a basis set of operators \(\{a\}_n\), calculate the covariance matrix:
\[V_{mn} = \frac{1}{2}\langle a_m a_n + a_n a_m \rangle  \langle a_m \rangle \langle a_n\rangle\]or, if of the optional argument symmetrized=False,
\[V_{mn} = \langle a_m a_n\rangle  \langle a_m \rangle \langle a_n\rangle\] Parameters:
 basislist
List of operators that defines the basis for the covariance matrix.
 rhoQobj
Density matrix for which to calculate the covariance matrix.
 symmetrizedbool, default: True
Flag indicating whether the symmetrized (default) or nonsymmetrized correlation matrix is to be calculated.
 Returns:
 corr_matndarray
A 2dimensional array of covariance values.
 logarithmic_negativity(V, g=1.4142135623730951)[source]
Calculates the logarithmic negativity given a symmetrized covariance matrix, see
qutip.continuous_variables.covariance_matrix
. Note that the twomode field state that is described by V must be Gaussian for this function to applicable. Parameters:
 Vndarray
The covariance matrix.
 gfloat, default: sqrt(2)
Scaling factor for
a = 0.5 * g * (x + iy)
, defaultg = sqrt(2)
. The value ofg
is related to the value ofhbar
in the commutation relation[x, y] = i * hbar
viahbar=2/g ** 2
giving the default valuehbar=1
.
 Returns:
 Nfloat
The logarithmic negativity for the twomode Gaussian state that is described by the the Wigner covariance matrix V.
 wigner_covariance_matrix(a1=None, a2=None, R=None, rho=None, g=1.4142135623730951)[source]
Calculates the Wigner covariance matrix \(V_{ij} = \frac{1}{2}(R_{ij} + R_{ji})\), given the quadrature correlation matrix \(R_{ij} = \langle R_{i} R_{j}\rangle  \langle R_{i}\rangle \langle R_{j}\rangle\), where \(R = (q_1, p_1, q_2, p_2)^T\) is the vector with quadrature operators for the two modes.
Alternatively, if
R = None
, and if annihilation operatorsa1
anda2
for the two modes are supplied instead, the quadrature correlation matrix is constructed from the annihilation operators before then the covariance matrix is calculated. Parameters:
 a1Qobj, optional
Field operator for mode 1.
 a2Qobj, optional
Field operator for mode 2.
 Rndarray, optional
The quadrature correlation matrix.
 rhoQobj, optional
Density matrix for which to calculate the covariance matrix.
 gfloat, default: sqrt(2)
Scaling factor for
a = 0.5 * g * (x + iy)
, defaultg = sqrt(2)
. The value ofg
is related to the value ofhbar
in the commutation relation[x, y] = i * hbar
viahbar=2/g ** 2
giving the default valuehbar=1
.
 Returns:
 cov_matndarray
A 2dimensional array of covariance values.
Measurement
Measurement of quantum states
Module for measuring quantum objects.
 measure(state, ops, tol=None)[source]
A dispatch method that provides measurement results handling both observable style measurements and projector style measurements (POVMs and PVMs).
For return signatures, please check:
measure_observable
for observable measurements.measure_povm
for POVM measurements.
 measure_observable(state, op, tol=None)[source]
Perform a measurement specified by an operator on the given state.
This function simulates the classic quantum measurement described in many introductory texts on quantum mechanics. The measurement collapses the state to one of the eigenstates of the given operator and the result of the measurement is the corresponding eigenvalue.
 Parameters:
 Returns:
 measured_valuefloat
The result of the measurement (one of the eigenvalues of op).
 state
Qobj
The new state (a ket if a ket was given, otherwise a density matrix).
Examples
Measure the zcomponent of the spin of the spinup basis state:
>>> measure_observable(basis(2, 0), sigmaz()) (1.0, Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket Qobj data = [[1.] [ 0.]])
Since the spinup basis is an eigenstate of sigmaz, this measurement always returns 1 as the measurement result (the eigenvalue of the spinup basis) and the original state (up to a global phase).
Measure the xcomponent of the spin of the spindown basis state:
>>> measure_observable(basis(2, 1), sigmax()) (1.0, Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket Qobj data = [[0.70710678] [ 0.70710678]])
This measurement returns 1 fifty percent of the time and 1 the other fifty percent of the time. The new state returned is the corresponding eigenstate of sigmax.
One may also perform a measurement on a density matrix. Below we perform the same measurement as above, but on the density matrix representing the pure spindown state:
>>> measure_observable(ket2dm(basis(2, 1)), sigmax()) (1.0, Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper Qobj data = [[ 0.5 0.5] [0.5 0.5]])
The measurement result is the same, but the new state is returned as a density matrix.
 measure_povm(state, ops, tol=None)[source]
Perform a measurement specified by list of POVMs.
This function simulates a POVM measurement. The measurement collapses the state to one of the resultant states of the measurement and returns the index of the operator corresponding to the collapsed state as well as the collapsed state.
 Parameters:
 state
Qobj
The ket or density matrix specifying the state to measure.
 opslist of
Qobj
List of measurement operators \(M_i\) or kets. Either:
specifying a POVM s.t. \(E_i = M_i^\dagger M_i\)
projection operators if ops correspond to projectors (s.t. \(E_i = M_i^\dagger = M_i\))
kets (transformed to projectors)
 tolfloat, optional
Smallest value for the probabilities. Default is qutip’s core settings’
atol
.
 state
 Returns:
 indexfloat
The resultant index of the measurement.
 state
Qobj
The new state (a ket if a ket was given, otherwise a density matrix).
 measurement_statistics(state, ops, tol=None)[source]
A dispatch method that provides measurement statistics handling both observable style measurements and projector style measurements(POVMs and PVMs).
For return signatures, please check:
measurement_statistics_observable
for observable measurements.measurement_statistics_povm
for POVM measurements.
 Parameters:
 state
Qobj
The ket or density matrix specifying the state to measure.
 ops
Qobj
or list ofQobj
measurement observable (:class:.Qobj); or
list of measurement operators \(M_i\) or kets (list of
Qobj
) Either:specifying a POVM s.t. \(E_i = M_i^\dagger * M_i\)
projection operators if ops correspond to projectors (s.t. \(E_i = M_i^\dagger = M_i\))
kets (transformed to projectors)
 tolfloat, optional
Smallest value for the probabilities. Default is qutip’s core settings’
atol
.
 state
 measurement_statistics_observable(state, op, tol=None)[source]
Return the measurement eigenvalues, eigenstates (or projectors) and measurement probabilities for the given state and measurement operator.
 Parameters:
 Returns:
 eigenvalues: list of float
The list of eigenvalues of the measurement operator.
 projectors: list of
Qobj
Return the projectors onto the eigenstates.
 probabilities: list of float
The probability of measuring the state as being in the corresponding eigenstate (and the measurement result being the corresponding eigenvalue).
 measurement_statistics_povm(state, ops, tol=None)[source]
Returns measurement statistics (resultant states and probabilities) for a measurement specified by a set of positive operator valued measurements on a specified ket or density matrix.
 Parameters:
 state
Qobj
The ket or density matrix specifying the state to measure.
 opslist of
Qobj
List of measurement operators \(M_i\) or kets. Either:
specifying a POVM s.t. \(E_i = M_i^\dagger M_i\)
projection operators if ops correspond to projectors (s.t. \(E_i = M_i^\dagger = M_i\))
kets (transformed to projectors)
 tolfloat, optional
Smallest value for the probabilities. Smaller probabilities will be rounded to
0
. Default is qutip’s core settings’atol
.
 state
 Returns:
 collapsed_stateslist of
Qobj
The collapsed states obtained after measuring the qubits and obtaining the qubit specified by the target in the state specified by the index.
 probabilitieslist of floats
The probability of measuring a state in a the state specified by the index.
 collapsed_stateslist of
Dynamics and TimeEvolution
Schrödinger Equation
This module provides solvers for the unitary Schrodinger equation.
 sesolve(H, psi0, tlist, e_ops=None, args=None, options=None, **kwargs)[source]
Schrodinger equation evolution of a state vector or unitary matrix for a given Hamiltonian.
Evolve the state vector (
psi0
) using a given Hamiltonian (H
), by integrating the set of ordinary differential equations that define the system. Alternatively evolve a unitary matrix in solving the Schrodinger operator equation.The output is either the state vector or unitary matrix at arbitrary points in time (
tlist
), or the expectation values of the supplied operators (e_ops
). If e_ops is a callback function, it is invoked for each time in tlist with time and the state as arguments, and the function does not use any return values. e_ops cannot be used in conjunction with solving the Schrodinger operator equationTimedependent operators
For timedependent problems,
H
andc_ops
can be aQobjEvo
or object that can be interpreted asQobjEvo
such as a list of (Qobj, Coefficient) pairs or a function. Parameters:
 H
Qobj
,QobjEvo
,QobjEvo
compatible format. System Hamiltonian as a Qobj or QobjEvo for timedependent Hamiltonians. List of [
Qobj
,Coefficient
] or callable that can be made intoQobjEvo
are also accepted. psi0
Qobj
initial state vector (ket) or initial unitary operator psi0 = U
 tlistlist / array
list of times for \(t\).
 e_ops
Qobj
, callable, or list, optional Single operator or list of operators for which to evaluate expectation values or callable or list of callable. Callable signature must be, f(t: float, state: Qobj). See
expect
for more detail of operator expectation. argsdict, optional
dictionary of parameters for timedependent Hamiltonians
 optionsdict, optional
Dictionary of options for the solver.
 store_final_state : boolWhether or not to store the final state of the evolution in the result class.
 store_states : bool, NoneWhether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
 normalize_output : boolNormalize output state to hide ODE numerical errors.
 progress_bar : str {‘text’, ‘enhanced’, ‘tqdm’, ‘’}How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
 progress_kwargs : dictkwargs to pass to the progress_bar. Qutip’s bars use chunk_size.
 method : str [“adams”, “bdf”, “lsoda”, “dop853”, “vern9”, etc.]Which differential equation integration method to use.
 atol, rtol : floatAbsolute and relative tolerance of the ODE integrator.
 nsteps : intMaximum number of (internally defined) steps allowed in one
tlist
step.  max_step : floatMaximum lenght of one internal step. When using pulses, it should be less than half the width of the thinnest pulse.
Other options could be supported depending on the integration method, see Integrator.
 H
 Returns:
 result:
Result
An instance of the class
Result
, which contains a list of arrayresult.expect
of expectation values for the times specified bytlist
, and/or a listresult.states
of state vectors or density matrices corresponding to the times intlist
[ife_ops
is an empty list ofstore_states=True
in options].
 result:
Master Equation
This module provides solvers for the Lindblad master equation and von Neumann equation.
 mesolve(H, rho0, tlist, c_ops=None, e_ops=None, args=None, options=None, **kwargs)[source]
Master equation evolution of a density matrix for a given Hamiltonian and set of collapse operators, or a Liouvillian.
Evolve the state vector or density matrix (
rho0
) using a given Hamiltonian or Liouvillian (H
) and an optional set of collapse operators (c_ops
), by integrating the set of ordinary differential equations that define the system. In the absence of collapse operators the system is evolved according to the unitary evolution of the Hamiltonian.The output is either the state vector at arbitrary points in time (
tlist
), or the expectation values of the supplied operators (e_ops
). If e_ops is a callback function, it is invoked for each time intlist
with time and the state as arguments, and the function does not use any return values.If either
H
or the Qobj elements inc_ops
are superoperators, they will be treated as direct contributions to the total system Liouvillian. This allows the solution of master equations that are not in standard Lindblad form.Timedependent operators
For timedependent problems,
H
andc_ops
can be aQobjEvo
or object that can be interpreted asQobjEvo
such as a list of (Qobj, Coefficient) pairs or a function.Additional options
Additional options to mesolve can be set via the
options
argument. Many ODE integration options can be set this way, and thestore_states
andstore_final_state
options can be used to store states even though expectation values are requested via thee_ops
argument. Parameters:
 H
Qobj
,QobjEvo
,QobjEvo
compatible format. Possibly timedependent system Liouvillian or Hamiltonian as a Qobj or QobjEvo. List of [
Qobj
,Coefficient
] or callable that can be made intoQobjEvo
are also accepted. rho0
Qobj
initial density matrix or state vector (ket).
 tlistlist / array
list of times for \(t\).
 c_opslist of (
QobjEvo
,QobjEvo
compatible format) Single collapse operator, or list of collapse operators, or a list of Liouvillian superoperators. None is equivalent to an empty list.
 e_opslist of
Qobj
/ callback function, optional Single operator or list of operators for which to evaluate expectation values or callable or list of callable. Callable signature must be, f(t: float, state: Qobj). See
expect
for more detail of operator expectation. argsdict, optional
dictionary of parameters for timedependent Hamiltonians and collapse operators.
 optionsdict, optional
Dictionary of options for the solver.
 store_final_state : boolWhether or not to store the final state of the evolution in the result class.
 store_states : bool, NoneWhether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
 normalize_output : boolNormalize output state to hide ODE numerical errors.
 progress_bar : str {‘text’, ‘enhanced’, ‘tqdm’, ‘’}How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
 progress_kwargs : dictkwargs to pass to the progress_bar. Qutip’s bars use chunk_size.
 method : str [“adams”, “bdf”, “lsoda”, “dop853”, “vern9”, etc.]Which differential equation integration method to use.
 atol, rtol : floatAbsolute and relative tolerance of the ODE integrator.
 nsteps : intMaximum number of (internally defined) steps allowed in one
tlist
step.  max_step : floatMaximum lenght of one internal step. When using pulses, it should be less than half the width of the thinnest pulse.
Other options could be supported depending on the integration method, see Integrator.
 H
 Returns:
 result:
Result
An instance of the class
Result
, which contains a list of arrayresult.expect
of expectation values for the times specified bytlist
, and/or a listresult.states
of state vectors or density matrices corresponding to the times intlist
[ife_ops
is an empty list ofstore_states=True
in options].
 result:
Notes
When no collapse operator are given and the H is not a superoperator, it will defer to
sesolve
.
Monte Carlo Evolution
 mcsolve(H, state, tlist, c_ops=(), e_ops=None, ntraj=500, *, args=None, options=None, seeds=None, target_tol=None, timeout=None, **kwargs)[source]
Monte Carlo evolution of a state vector \(\psi \rangle\) for a given Hamiltonian and sets of collapse operators. Options for the underlying ODE solver are given by the Options class.
 Parameters:
 H
Qobj
,QobjEvo
,list
, callable. System Hamiltonian as a Qobj, QobjEvo. It can also be any input type that QobjEvo accepts (see
QobjEvo
’s documentation).H
can also be a superoperator (liouvillian) if some collapse operators are to be treated deterministically. state
Qobj
Initial state vector.
 tlistarray_like
Times at which results are recorded.
 c_opslist
A
list
of collapse operators in any input type that QobjEvo accepts (seeQobjEvo
’s documentation). They must be operators even ifH
is a superoperator. If none are given, the solver will defer tosesolve
ormesolve
. e_opslist, optional
A
list
of operator as Qobj, QobjEvo or callable with signature of (t, state: Qobj) for calculating expectation values. When noe_ops
are given, the solver will default to save the states. ntrajint, default: 500
Maximum number of trajectories to run. Can be cut short if a time limit is passed with the
timeout
keyword or if the target tolerance is reached, seetarget_tol
. argsdict, optional
Arguments for timedependent Hamiltonian and collapse operator terms.
 optionsdict, optional
Dictionary of options for the solver.
 store_final_state : boolWhether or not to store the final state of the evolution in the result class.
 store_states : bool, NoneWhether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
 progress_bar : str {‘text’, ‘enhanced’, ‘tqdm’, ‘’}How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
 progress_kwargs : dictkwargs to pass to the progress_bar. Qutip’s bars use chunk_size.
 method : str [“adams”, “bdf”, “lsoda”, “dop853”, “vern9”, etc.]Which differential equation integration method to use.
 atol, rtol : floatAbsolute and relative tolerance of the ODE integrator.
 nsteps : intMaximum number of (internally defined) steps allowed in one
tlist
step.  max_step : floatMaximum length of one internal step. When using pulses, it should be less than half the width of the thinnest pulse.
 keep_runs_results : bool, [False]Whether to store results from all trajectories or just store the averages.
 map : str {“serial”, “parallel”, “loky”, “mpi”}How to run the trajectories. “parallel” uses the multiprocessing module to run in parallel while “loky” and “mpi” use the “loky” and “mpi4py” modules to do so.
 num_cpus : intNumber of cpus to use when running in parallel.
None
detect the number of available cpus.  norm_t_tol, norm_tol, norm_steps : float, float, intParameters used to find the collapse location.
norm_t_tol
andnorm_tol
are the tolerance in time and norm respectively. An error will be raised if the collapse could not be found withinnorm_steps
tries.  mc_corr_eps : floatSmall number used to detect nonphysical collapse caused by numerical imprecision.
 improved_sampling : BoolWhether to use the improved sampling algorithm from Abdelhafez et al. PRA (2019)
Additional options are listed under options. More options may be available depending on the selected differential equation integration method, see Integrator.
 seedsint, SeedSequence, list, optional
Seed for the random number generator. It can be a single seed used to spawn seeds for each trajectory or a list of seeds, one for each trajectory. Seeds are saved in the result and they can be reused with:
seeds=prev_result.seeds
 target_tolfloat, tuple, list, optional
Target tolerance of the evolution. The evolution will compute trajectories until the error on the expectation values is lower than this tolerance. The maximum number of trajectories employed is given by
ntraj
. The error is computed using jackknife resampling.target_tol
can be an absolute tolerance or a pair of absolute and relative tolerance, in that order. Lastly, it can be a list of pairs of (atol, rtol) for each e_ops. timeoutfloat, optional
Maximum time for the evolution in second. When reached, no more trajectories will be computed.
 H
 Returns:
 results
McResult
Object storing all results from the simulation. Which results is saved depends on the presence of
e_ops
and the options used.collapse
andphotocurrent
is available to Monte Carlo simulation results.
 results
Notes
The simulation will end when the first end condition is reached between
ntraj
,timeout
andtarget_tol
.
 nm_mcsolve(H, state, tlist, ops_and_rates=(), e_ops=None, ntraj=500, *, args=None, options=None, seeds=None, target_tol=None, timeout=None)[source]
MonteCarlo evolution corresponding to a Lindblad equation with “rates” that may be negative. Usage of this function is analogous to
mcsolve
, but thec_ops
parameter is replaced by anops_and_rates
parameter to allow for negative rates. Options for the underlying ODE solver are given by the Options class. Parameters:
 H
Qobj
,QobjEvo
,list
, callable. System Hamiltonian as a Qobj, QobjEvo. It can also be any input type that QobjEvo accepts (see
QobjEvo
’s documentation).H
can also be a superoperator (liouvillian) if some collapse operators are to be treated deterministically. state
Qobj
Initial state vector.
 tlistarray_like
Times at which results are recorded.
 ops_and_rateslist
A
list
of tuples(L, Gamma)
, where the Lindblad operatorL
is aQobj
andGamma
represents the corresponding rate, which is allowed to be negative. The Lindblad operators must be operators even ifH
is a superoperator. If none are given, the solver will defer tosesolve
ormesolve
. Each rateGamma
may be just a number (in the case of a constant rate) or, otherwise, specified using any format accepted bycoefficient
. e_opslist, optional
A
list
of operator as Qobj, QobjEvo or callable with signature of (t, state: Qobj) for calculating expectation values. When noe_ops
are given, the solver will default to save the states. ntrajint, default: 500
Maximum number of trajectories to run. Can be cut short if a time limit is passed with the
timeout
keyword or if the target tolerance is reached, seetarget_tol
. argsdict, optional
Arguments for timedependent Hamiltonian and collapse operator terms.
 optionsdict, optional
Dictionary of options for the solver.
 store_final_state : boolWhether or not to store the final state of the evolution in the result class.
 store_states : bool, NoneWhether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
 progress_bar : str {‘text’, ‘enhanced’, ‘tqdm’, ‘’}How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
 progress_kwargs : dictkwargs to pass to the progress_bar. Qutip’s bars use chunk_size.
 method : str [“adams”, “bdf”, “lsoda”, “dop853”, “vern9”, etc.]Which differential equation integration method to use.
 atol, rtol : floatAbsolute and relative tolerance of the ODE integrator.
 nsteps : intMaximum number of (internally defined) steps allowed in one
tlist
step.  max_step : floatMaximum length of one internal step. When using pulses, it should be less than half the width of the thinnest pulse.
 keep_runs_results : bool, [False]Whether to store results from all trajectories or just store the averages.
 map : str {“serial”, “parallel”, “loky”, “mpi”}How to run the trajectories. “parallel” uses the multiprocessing module to run in parallel while “loky” and “mpi” use the “loky” and “mpi4py” modules to do so.
 num_cpus : intNumber of cpus to use when running in parallel.
None
detect the number of available cpus.  norm_t_tol, norm_tol, norm_steps : float, float, intParameters used to find the collapse location.
norm_t_tol
andnorm_tol
are the tolerance in time and norm respectively. An error will be raised if the collapse could not be found withinnorm_steps
tries.  mc_corr_eps : floatSmall number used to detect nonphysical collapse caused by numerical imprecision.
 completeness_rtol, completeness_atol : float, floatParameters used in determining whether the given Lindblad operators satisfy a certain completeness relation. If they do not, an additional Lindblad operator is added automatically (with zero rate).
 martingale_quad_limit : float or intAn upper bound on the number of subintervals used in the adaptive integration of the martingale.
Note that the ‘improved_sampling’ option is not currently supported. Additional options are listed under options. More options may be available depending on the selected differential equation integration method, see Integrator.
 seedsint, SeedSequence, list, optional
Seed for the random number generator. It can be a single seed used to spawn seeds for each trajectory or a list of seeds, one for each trajectory. Seeds are saved in the result and they can be reused with:
seeds=prev_result.seeds
 target_tolfloat, tuple, list, optional
Target tolerance of the evolution. The evolution will compute trajectories until the error on the expectation values is lower than this tolerance. The maximum number of trajectories employed is given by
ntraj
. The error is computed using jackknife resampling.target_tol
can be an absolute tolerance or a pair of absolute and relative tolerance, in that order. Lastly, it can be a list of pairs of (atol, rtol) for each e_ops. timeoutfloat, optional
Maximum time for the evolution in seconds. When reached, no more trajectories will be computed.
 H
 Returns:
 results
NmmcResult
Object storing all results from the simulation. Compared to a result returned by
mcsolve
, this result contains the additional fieldtrace
(andruns_trace
ifstore_final_state
is set). Note that the states on the individual trajectories are not normalized. This field contains the average of their trace, which will converge to one in the limit of sufficiently many trajectories.
 results
Krylov Subspace Solver
 krylovsolve(H, psi0, tlist, krylov_dim, e_ops=None, args=None, options=None)[source]
Schrodinger equation evolution of a state vector for time independent Hamiltonians using Krylov method.
Evolve the state vector (“psi0”) finding an approximation for the time evolution operator of Hamiltonian (“H”) by obtaining the projection of the time evolution operator on a set of small dimensional Krylov subspaces (m << dim(H)).
The output is either the state vector or unitary matrix at arbitrary points in time (tlist), or the expectation values of the supplied operators (e_ops). If e_ops is a callback function, it is invoked for each time in tlist with time and the state as arguments, and the function does not use any return values. e_ops cannot be used in conjunction with solving the Schrodinger operator equation
 Parameters:
 H
Qobj
,QobjEvo
,QobjEvo
compatible format. System Hamiltonian as a Qobj or QobjEvo for timedependent Hamiltonians. List of [
Qobj
,Coefficient
] or callable that can be made intoQobjEvo
are also accepted. psi0
Qobj
Initial state vector (ket)
 tlistlist / array
list of times for \(t\).
 krylov_dim: int
Dimension of Krylov approximation subspaces used for the time evolution approximation.
 e_ops
Qobj
, callable, or list, optional Single operator or list of operators for which to evaluate expectation values or callable or list of callable. Callable signature must be, f(t: float, state: Qobj). See
expect
for more detail of operator expectation. argsdict, optional
dictionary of parameters for timedependent Hamiltonians
 optionsdict, optional
Dictionary of options for the solver.
 store_final_state : boolWhether or not to store the final state of the evolution in the result class.
 store_states : bool, NoneWhether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
 normalize_output : boolNormalize output state to hide ODE numerical errors.
 progress_bar : str {‘text’, ‘enhanced’, ‘tqdm’, ‘’}How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
 progress_kwargs : dictkwargs to pass to the progress_bar. Qutip’s bars use chunk_size.
 atol: floatAbsolute tolerance of the ODE integrator.
 nsteps : intMaximum number of (internally defined) steps allowed in one
tlist
step.  min_step, max_step : floatMiniumum and maximum lenght of one internal step.
 always_compute_step: boolIf True, the step lenght is computed each time a new Krylov subspace is computed. Otherwise it is computed only once when creating the integrator.
 sub_system_tol: floatTolerance to detect an happy breakdown. An happy breakdown happens when the initial ket is in a subspace of the Hamiltonian smaller than
krylov_dim
.
 H
 Returns:
 result:
Result
An instance of the class
Result
, which contains a list of arrayresult.expect
of expectation values for the times specified bytlist
, and/or a listresult.states
of state vectors or density matrices corresponding to the times intlist
[ife_ops
is an empty list ofstore_states=True
in options].
 result:
BlochRedfield Master Equation
This module provides solvers for the Lindblad master equation and von Neumann equation.
 brmesolve(H, psi0, tlist, a_ops=(), e_ops=(), c_ops=(), args=None, sec_cutoff=0.1, options=None, **kwargs)[source]
Solves for the dynamics of a system using the BlochRedfield master equation, given an input Hamiltonian, Hermitian bathcoupling terms and their associated spectral functions, as well as possible Lindblad collapse operators.
 Parameters:
 H
Qobj
,QobjEvo
Possibly timedependent system Liouvillian or Hamiltonian as a Qobj or QobjEvo. list of [
Qobj
,Coefficient
] or callable that can be made intoQobjEvo
are also accepted. psi0: Qobj
Initial density matrix or state vector (ket).
 tlistarray_like
List of times for evaluating evolution
 a_opslist of (a_op, spectra)
Nested list of system operators that couple to the environment, and the corresponding bath spectra.
 a_op
Qobj
,QobjEvo
The operator coupling to the environment. Must be hermitian.
 spectra
Coefficient
, str, func The corresponding bath spectral responce. Can be a Coefficient using an ‘w’ args, a function of the frequence or a string. Coefficient build from a numpy array are understood as a function of
w
instead oft
. Function are expected to be of the signaturef(w)
orf(t, w, **args)
.The spectra function can depend on
t
if the correspondinga_op
is aQobjEvo
.
Example:
a_ops = [ (a+a.dag(), ('w>0', args={"w": 0})), (QobjEvo(a+a.dag()), 'w > exp(t)'), (QobjEvo([b+b.dag(), lambda t: ...]), lambda w: ...)), (c+c.dag(), SpectraCoefficient(coefficient(array, tlist=ws))), ]
 a_op
 e_opslist of
Qobj
/ callback function, optional Single operator or list of operators for which to evaluate expectation values or callable or list of callable. Callable signature must be, f(t: float, state: Qobj). See
expect
for more detail of operator expectation c_opslist of (
QobjEvo
,QobjEvo
compatible format), optional List of collapse operators.
 argsdict, optional
Dictionary of parameters for timedependent Hamiltonians and collapse operators. The key
w
is reserved for the spectra function. sec_cutofffloat, default: 0.1
Cutoff for secular approximation. Use
1
if secular approximation is not used when evaluating bathcoupling terms. optionsdict, optional
Dictionary of options for the solver.
 store_final_state : boolWhether or not to store the final state of the evolution in the result class.
 store_states : bool, NoneWhether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
 normalize_output : boolNormalize output state to hide ODE numerical errors.
 progress_bar : str {‘text’, ‘enhanced’, ‘tqdm’, ‘’}How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
 progress_kwargs : dictkwargs to pass to the progress_bar. Qutip’s bars use chunk_size.
 tensor_type : str [‘sparse’, ‘dense’, ‘data’]Which data type to use when computing the brtensor. With a cutoff ‘sparse’ is usually the most efficient.
 sparse_eigensolver : bool {False} Whether to use the sparse eigensolver
 method : str [“adams”, “bdf”, “lsoda”, “dop853”, “vern9”, etc.] Which differential equation integration method to use.
 atol, rtol : floatAbsolute and relative tolerance of the ODE integrator.
 nsteps : intMaximum number of (internally defined) steps allowed in one
tlist
step.  max_step : float, 0Maximum lenght of one internal step. When using pulses, it should be less than half the width of the thinnest pulse.
Other options could be supported depending on the integration method, see Integrator.
 H
 Returns:
 result:
Result
An instance of the class
qutip.solver.Result
, which contains either an array of expectation values, for operators given in e_ops, or a list of states for the times specified bytlist
.
 result:
Floquet States and FloquetMarkov Master Equation
 floquet_tensor(H, c_ops, spectra_cb, T=0, w_th=0.0, kmax=5, nT=100)[source]
Construct a tensor that represents the master equation in the floquet basis.
Simplest RWA approximation [Grifoni et al, Phys.Rep. 304 229 (1998)]
 Parameters:
 H
QobjEvo
,FloquetBasis
Periodic Hamiltonian a floquet basis system.
 Tfloat, optional
The period of the timedependence of the hamiltonian. Optional if
H
is aFloquetBasis
object. c_opslist of
Qobj
list of collapse operators.
 spectra_cblist callback functions
List of callback functions that compute the noise power spectrum as a function of frequency for the collapse operators in c_ops.
 w_thfloat, default: 0.0
The temperature in units of frequency.
 kmaxint, default: 5
The truncation of the number of sidebands (default 5).
 nTint, default: 100
The number of integration steps (for calculating X) within one period.
 H
 Returns:
 outputarray
The FloquetMarkov master equation tensor R.
 fmmesolve(H, rho0, tlist, c_ops=None, e_ops=None, spectra_cb=None, T=0, w_th=0.0, args=None, options=None)[source]
Solve the dynamics for the system using the FloquetMarkov master equation.
 Parameters:
 H
Qobj
,QobjEvo
,QobjEvo
compatible format. Periodic system Hamiltonian as
QobjEvo
. List of [Qobj
,Coefficient
] or callable that can be made intoQobjEvo
are also accepted. rho0 / psi0
Qobj
Initial density matrix or state vector (ket).
 tlistlist / array
List of times for \(t\).
 c_opslist of
Qobj
, optional List of collapse operators. Time dependent collapse operators are not supported. Fall back on
fsesolve
if not provided. e_opslist of
Qobj
/ callback function, optional List of operators for which to evaluate expectation values. The states are reverted to the lab basis before applying the
 spectra_cblist callback functions, default:
lambda w: (w > 0)
List of callback functions that compute the noise power spectrum as a function of frequency for the collapse operators in c_ops.
 Tfloat, default=tlist[1]
The period of the timedependence of the hamiltonian. The default value
0
indicates that the ‘tlist’ spans a single period of the driving. w_thfloat, default: 0.0
The temperature of the environment in units of frequency. For example, if the Hamiltonian written in units of 2pi GHz, and the temperature is given in K, use the following conversion:
temperature = 25e3 # unit K h = 6.626e34 kB = 1.38e23 args[‘w_th’] = temperature * (kB / h) * 2 * pi * 1e9
 argsdict, optional
Dictionary of parameters for timedependent Hamiltonian
 optionsdict, optional
Dictionary of options for the solver.
 store_final_state : boolWhether or not to store the final state of the evolution in the result class.
 store_states : bool, NoneWhether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
 store_floquet_states : boolWhether or not to store the density matrices in the floquet basis in
result.floquet_states
.  normalize_output : boolNormalize output state to hide ODE numerical errors.
 progress_bar : str {‘text’, ‘enhanced’, ‘tqdm’, ‘’}How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
 progress_kwargs : dictkwargs to pass to the progress_bar. Qutip’s bars use chunk_size.
 method : str [“adams”, “bdf”, “lsoda”, “dop853”, “vern9”, etc.]Which differential equation integration method to use.
 atol, rtol : floatAbsolute and relative tolerance of the ODE integrator.
 nsteps : intMaximum number of (internally defined) steps allowed in one
tlist
step.  max_step : floatMaximum lenght of one internal step. When using pulses, it should be less than half the width of the thinnest pulse.
Other options could be supported depending on the integration method, see Integrator.
 H
 Returns:
 fsesolve(H, psi0, tlist, e_ops=None, T=0.0, args=None, options=None)[source]
Solve the Schrodinger equation using the Floquet formalism.
 Parameters:
 H
Qobj
,QobjEvo
,QobjEvo
compatible format. Periodic system Hamiltonian as
QobjEvo
. List of [Qobj
,Coefficient
] or callable that can be made intoQobjEvo
are also accepted. psi0
Qobj
Initial state vector (ket). If an operator is provided,
 tlistlist / array
List of times for \(t\).
 e_opslist of
Qobj
/ callback function, optional List of operators for which to evaluate expectation values. If this list is empty, the state vectors for each time in tlist will be returned instead of expectation values.
 Tfloat, default=tlist[1]
The period of the timedependence of the hamiltonian.
 argsdictionary, optional
Dictionary with variables required to evaluate H.
 optionsdict, optional
Options for the results.
 store_final_state : boolWhether or not to store the final state of the evolution in the result class.
 store_states : bool, NoneWhether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
 normalize_output : boolNormalize output state to hide ODE numerical errors.
 H
 Returns:
Stochastic Schrödinger Equation and Master Equation
 smesolve(H, rho0, tlist, c_ops=(), sc_ops=(), heterodyne=False, *, e_ops=(), args={}, ntraj=500, options=None, seeds=None, target_tol=None, timeout=None, **kwargs)[source]
Solve stochastic master equation.
 Parameters:
 H
Qobj
,QobjEvo
,QobjEvo
compatible format. System Hamiltonian as a Qobj or QobjEvo for timedependent Hamiltonians. List of [
Qobj
,Coefficient
] or callable that can be made intoQobjEvo
are also accepted. rho0
Qobj
Initial density matrix or state vector (ket).
 tlistlist / array
List of times for \(t\).
 c_opslist of (
QobjEvo
,QobjEvo
compatible format), optional Deterministic collapse operator which will contribute with a standard Lindblad type of dissipation.
 sc_opslist of (
QobjEvo
,QobjEvo
compatible format) List of stochastic collapse operators.
 e_ops:
qobj
, callable, or list, optional Single operator or list of operators for which to evaluate expectation values or callable or list of callable. Callable signature must be, f(t: float, state: Qobj). See
expect
for more detail of operator expectation. argsdict, optional
Dictionary of parameters for timedependent Hamiltonians and collapse operators.
 ntrajint, default: 500
Number of trajectories to compute.
 heterodynebool, default: False
Whether to use heterodyne or homodyne detection.
 seedsint, SeedSequence, list, optional
Seed for the random number generator. It can be a single seed used to spawn seeds for each trajectory or a list of seeds, one for each trajectory. Seeds are saved in the result and they can be reused with:
seeds=prev_result.seeds
When using a parallel map, the trajectories can be reordered.
 target_tol{float, tuple, list}, optional
Target tolerance of the evolution. The evolution will compute trajectories until the error on the expectation values is lower than this tolerance. The maximum number of trajectories employed is given by
ntraj
. The error is computed using jackknife resampling.target_tol
can be an absolute tolerance or a pair of absolute and relative tolerance, in that order. Lastly, it can be a list of pairs of(atol, rtol)
for each e_ops. timeoutfloat, optional
Maximum time for the evolution in second. When reached, no more trajectories will be computed. Overwrite the option of the same name.
 optionsdict, optional
Dictionary of options for the solver.
 store_final_state : boolWhether or not to store the final state of the evolution in the result class.
 store_states : bool, NoneWhether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
 store_measurement: boolWhether to store the measurement and wiener process for each trajectories.
 keep_runs_results : boolWhether to store results from all trajectories or just store the averages.
 normalize_output : boolNormalize output state to hide ODE numerical errors.
 progress_bar : str {‘text’, ‘enhanced’, ‘tqdm’, ‘’}How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
 progress_kwargs : dictkwargs to pass to the progress_bar. Qutip’s bars use chunk_size.
 method : strWhich stochastic differential equation integration method to use. Main ones are {“euler”, “rouchon”, “platen”, “taylor1.5_imp”}
 map : str {“serial”, “parallel”, “loky”, “mpi”}How to run the trajectories. “parallel” uses the multiprocessing module to run in parallel while “loky” and “mpi” use the “loky” and “mpi4py” modules to do so.
 num_cpus : NoneType, intNumber of cpus to use when running in parallel.
None
detect the number of available cpus.  dt : floatThe finite steps lenght for the Stochastic integration method. Default change depending on the integrator.
Additional options are listed under options. More options may be available depending on the selected differential equation integration method, see SIntegrator.
 H
 Returns:
 ssesolve(H, psi0, tlist, sc_ops=(), heterodyne=False, *, e_ops=(), args={}, ntraj=500, options=None, seeds=None, target_tol=None, timeout=None, **kwargs)[source]
Solve stochastic Schrodinger equation.
 Parameters:
 H
Qobj
,QobjEvo
,QobjEvo
compatible format. System Hamiltonian as a Qobj or QobjEvo for timedependent Hamiltonians. List of [
Qobj
,Coefficient
] or callable that can be made intoQobjEvo
are also accepted. psi0
Qobj
Initial state vector (ket).
 tlistlist / array
List of times for \(t\).
 sc_opslist of (
QobjEvo
,QobjEvo
compatible format) List of stochastic collapse operators.
 e_ops
qobj
, callable, or list, optional Single operator or list of operators for which to evaluate expectation values or callable or list of callable. Callable signature must be, f(t: float, state: Qobj). See
expect
for more detail of operator expectation. argsdict, optional
Dictionary of parameters for timedependent Hamiltonians and collapse operators.
 ntrajint, default: 500
Number of trajectories to compute.
 heterodynebool, default: False
Whether to use heterodyne or homodyne detection.
 seedsint, SeedSequence, list, optional
Seed for the random number generator. It can be a single seed used to spawn seeds for each trajectory or a list of seeds, one for each trajectory. Seeds are saved in the result and they can be reused with:
seeds=prev_result.seeds
 target_tol{float, tuple, list}, optional
Target tolerance of the evolution. The evolution will compute trajectories until the error on the expectation values is lower than this tolerance. The maximum number of trajectories employed is given by
ntraj
. The error is computed using jackknife resampling.target_tol
can be an absolute tolerance or a pair of absolute and relative tolerance, in that order. Lastly, it can be a list of pairs of (atol, rtol) for each e_ops. timeoutfloat, optional
Maximum time for the evolution in second. When reached, no more trajectories will be computed. Overwrite the option of the same name.
 optionsdict, optional
Dictionary of options for the solver.
 store_final_state : boolWhether or not to store the final state of the evolution in the result class.
 store_states : bool, NoneWhether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
 store_measurement: bool Whether to store the measurement and wiener process, or brownian noise for each trajectories.
 keep_runs_results : boolWhether to store results from all trajectories or just store the averages.
 normalize_output : boolNormalize output state to hide ODE numerical errors.
 progress_bar : str {‘text’, ‘enhanced’, ‘tqdm’, ‘’}How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
 progress_kwargs : dictkwargs to pass to the progress_bar. Qutip’s bars use chunk_size.
 method : strWhich stochastic differential equation integration method to use. Main ones are {“euler”, “rouchon”, “platen”, “taylor1.5_imp”}
 map : str {“serial”, “parallel”, “loky”, “mpi”}How to run the trajectories. “parallel” uses the multiprocessing module to run in parallel while “loky” and “mpi” use the “loky” and “mpi4py” modules to do so.
 num_cpus : NoneType, intNumber of cpus to use when running in parallel.
None
detect the number of available cpus.  dt : floatThe finite steps lenght for the Stochastic integration method. Default change depending on the integrator.
Additional options are listed under options. More options may be available depending on the selected differential equation integration method, see SIntegrator.
 H
 Returns:
Constructing time dependent systems
 coefficient(base, *, tlist=None, args={}, args_ctypes={}, order=3, compile_opt=None, function_style=None, boundary_conditions=None, **kwargs)[source]
Build
Coefficient
for time dependent systems:` QobjEvo = Qobj + Qobj * Coefficient + Qobj * Coefficient + ... `
The coefficients can be a function, a string or a numpy array. Other packages may add support for other kind of coefficients.
For function based coefficients, the function signature must be either:
f(t, ...)
where the other arguments are supplied as ordinary “pythonic” arguments (e.g.f(t, w, a=5)
)f(t, args)
where the arguments are supplied in a “dict” namedargs
By default the signature style is controlled by the
qutip.settings.core["function_coefficient_style"]
setting, but it may be overriden here by specifying eitherfunction_style="pythonic"
orfunction_style="dict"
.Examples:
pythonic style function signature:
def f1_t(t, w): return np.exp(1j * t * w) coeff1 = coefficient(f1_t, args={"w": 1.})
dict style function signature:
def f2_t(t, args): return np.exp(1j * t * args["w"]) coeff2 = coefficient(f2_t, args={"w": 1.})
For string based coeffients, the string must be a compilable python code resulting in a complex. The following symbols are defined:
sin, cos, tan, asin, acos, atan, pi, sinh, cosh, tanh, asinh, acosh, atanh, exp, log, log10, erf, zerf, sqrt, real, imag, conj, abs, norm, arg, proj, numpy as np, scipy.special as spe (python interface) and cython_special (scipy cython interface)
Examples:
coeff = coefficient('exp(1j*w1*t)', args={"w1":1.})
‘args’ is needed for string coefficient at compilation. It is a dict of (name:object). The keys must be a valid variables string.
Compilation options can be passed as “compile_opt=CompilationOptions(…)”.
For numpy array format, the array must be an 1d of dtype float or complex. A list of times (float64) at which the coeffients must be given (tlist). The coeffients array must have the same len as the tlist. The time of the tlist do not need to be equidistant, but must be sorted. By default, a cubic spline interpolation will be used to compute the coefficient at time t. The keyword
order
sets the order of the interpolation. Whenorder = 0
, the interpolation is step function that evaluates to the most recent value.Examples:
tlist = np.logspace(5,0,100) H = QobjEvo(np.exp(1j*tlist), tlist=tlist)
scipy.interpolate
’sCubicSpline
,PPoly
andBspline
are also converted to interpolated coefficients (the same kind of coefficient created fromndarray
). Other interpolation methods from scipy are converted to a functionbased coefficient (the same kind of coefficient created from callables). Parameters:
 baseobject
Base object to make into a Coefficient.
 argsdict, optional
Dictionary of arguments to pass to the function or string coefficient.
 orderint, default=3
Order of the spline for array based coefficient.
 tlistiterable, optional
Times for each element of an array based coefficient.
 function_stylestr {“dict”, “pythonic”, None}, optional
Function signature of function based coefficients.
 args_ctypesdict, optional
C type for the args when compiling array based coefficients.
 compile_optCompilationOptions, optional
Sets of options for the compilation of string based coefficients.
 boundary_conditions: 2tupule, str or None, optional
Specify boundary conditions for spline interpolation.
 **kwargs
Extra arguments to pass the the coefficients.
Hierarchical Equations of Motion
This module provides solvers for systembath evoluation using the HEOM (hierarchy equations of motion).
See https://en.wikipedia.org/wiki/Hierarchical_equations_of_motion for a very basic introduction to the technique.
The implementation is derived from the BoFiN library (see https://github.com/tehruhn/bofin) which was itself derived from an earlier implementation in QuTiP itself.
For backwards compatibility with QuTiP 4.6 and below, a new version of HSolverDL (the DrudeLorentz specific HEOM solver) is provided. It is implemented on top of the new HEOMSolver but should largely be a dropin replacement for the old HSolverDL.
 heomsolve(H, bath, max_depth, state0, tlist, *, e_ops=None, args=None, options=None)[source]
Hierarchical Equations of Motion (HEOM) solver that supports multiple baths.
The baths must be all either bosonic or fermionic baths.
If you need to run many evolutions of the same system and bath, consider using
HEOMSolver
directly to avoid having to continually reconstruct the equation hierarchy for every evolution. Parameters:
 H
Qobj
,QobjEvo
Possibly timedependent system Liouvillian or Hamiltonian as a Qobj or QobjEvo. list of [
Qobj
,Coefficient
] or callable that can be made intoQobjEvo
are also accepted. bathBath or list of Bath
A
Bath
containing the exponents of the expansion of the bath correlation funcion and their associated coefficients and coupling operators, or a list of baths.If multiple baths are given, they must all be either fermionic or bosonic baths.
 max_depthint
The maximum depth of the heirarchy (i.e. the maximum number of bath exponent “excitations” to retain).
 state0
Qobj
orHierarchyADOsState
or arraylike If
rho0
is aQobj
the it is the initial state of the system (i.e. aQobj
density matrix).If it is a
HierarchyADOsState
or arraylike, thenrho0
gives the initial state of all ADOs.Usually the state of the ADOs would be determine from a previous call to
.run(...)
with the solver results optionstore_ados
set to True. For example,result = solver.run(...)
could be followed bysolver.run(result.ado_states[1], tlist)
.If a numpy arraylike is passed its shape must be
(number_of_ados, n, n)
where(n, n)
is the system shape (i.e. shape of the system density matrix) and the ADOs must be in the same order as in.ados.labels
. tlistlist
An ordered list of times at which to return the value of the state.
 e_opsQobj / QobjEvo / callable / list / dict / None, optional
A list or dictionary of operators as
Qobj
,QobjEvo
and/or callable functions (they can be mixed) or a single operator or callable function. For an operatorop
, the result will be computed using(state * op).tr()
and the state at each timet
. For callable functions,f
, the result is computed usingf(t, ado_state)
. The values are stored in theexpect
ande_data
attributes of the result (see the return section below). argsdict, optional
Change the
args
of the RHS for the evolution. optionsdict, optional
Generic solver options.
 store_final_state : boolWhether or not to store the final state of the evolution in the result class.
 store_states : bool, NoneWhether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
 store_ados : boolWhether or not to store the HEOM ADOs.
 normalize_output : boolNormalize output state to hide ODE numerical errors.
 progress_bar : str {‘text’, ‘enhanced’, ‘tqdm’, ‘’}How to present the solver progress. ‘tqdm’ uses the python module of the same name and raise an error if not installed. Empty string or False will disable the bar.
 progress_kwargs : dictkwargs to pass to the progress_bar. Qutip’s bars use chunk_size.
 state_data_type: str {‘dense’, ‘CSR’, ‘Dia’, }Name of the data type of the state used during the ODE evolution. Use an empty string to keep the input state type. Many integrator can only work with Dense.
 method : str [“adams”, “bdf”, “lsoda”, “dop853”, “vern9”, etc.]Which differential equation integration method to use.
 atol, rtol : floatAbsolute and relative tolerance of the ODE integrator.
 nsteps : intMaximum number of (internally defined) steps allowed in one
tlist
step.  max_step : float,Maximum lenght of one internal step. When using pulses, it should be less than half the width of the thinnest pulse.
 H
 Returns:
HEOMResult
The results of the simulation run, with the following important attributes:
times
: the timest
(i.e. thetlist
).states
: the system state at each timet
(only available ife_ops
wasNone
or if the solver optionstore_states
was set toTrue
).ado_states
: the full ADO state at each time (only available if the results optionado_return
was set toTrue
). Each element is an instance ofHierarchyADOsState
. The state of a particular ADO may be extracted fromresult.ado_states[i]
by callingextract
.expect
: a list containing the values of eache_ops
at timet
.e_data
: a dictionary containing the values of eache_ops
at tmet
. The keys are those given bye_ops
if it was a dict, otherwise they are the indexes of the suppliede_ops
.
See
HEOMResult
andResult
for the complete list of attributes.
Correlation Functions
 coherence_function_g1(H, state0, taulist, c_ops, a_op, solver='me', args=None, options=None)[source]
Calculate the normalized firstorder quantum coherence function:
\[g^{(1)}(\tau) = \frac{\langle A^\dagger(\tau)A(0)\rangle} {\sqrt{\langle A^\dagger(\tau)A(\tau)\rangle \langle A^\dagger(0)A(0)\rangle}}\]using the quantum regression theorem and the evolution solver indicated by the solver parameter.
 Parameters:
 H
Qobj
,QobjEvo
System Hamiltonian, may be timedependent for solver choice of
me
. state0
Qobj
Initial state density matrix \(\rho(t_0)\) or state vector \(\psi(t_0)\). If ‘state0’ is ‘None’, then the steady state will be used as the initial state. The ‘steadystate’ is only implemented if
c_ops
are provided and the Hamiltonian is constant. taulistarray_like
List of times for \(\tau\). taulist must be positive and contain the element 0.
 c_opslist of {
Qobj
,QobjEvo
} List of collapse operators
 a_op
Qobj
,QobjEvo
Operator A.
 solverstr {‘me’, ‘es’}, default: ‘me’
Choice of solver,
me
for masterequation, andes
for exponential series.es
is equivalent to me withoptions={"method": "diag"}
. argsdict, optional
dictionary of parameters for timedependent Hamiltonians
 optionsdict, optional
Options for the solver.
 H
 Returns:
 g1, G1tuple
The normalized and unnormalized secondorder coherence function.
 coherence_function_g2(H, state0, taulist, c_ops, a_op, solver='me', args=None, options=None)[source]
Calculate the normalized secondorder quantum coherence function:
\[ g^{(2)}(\tau) = \frac{\langle A^\dagger(0)A^\dagger(\tau)A(\tau)A(0)\rangle} {\langle A^\dagger(\tau)A(\tau)\rangle \langle A^\dagger(0)A(0)\rangle}\]using the quantum regression theorem and the evolution solver indicated by the solver parameter.
 Parameters:
 H
Qobj
,QobjEvo
System Hamiltonian, may be timedependent for solver choice of
me
. state0
Qobj
Initial state density matrix \(\rho(t_0)\) or state vector \(\psi(t_0)\). If ‘state0’ is ‘None’, then the steady state will be used as the initial state. The ‘steadystate’ is only implemented if
c_ops
are provided and the Hamiltonian is constant. taulistarray_like
List of times for \(\tau\). taulist must be positive and contain the element 0.
 c_opslist
List of collapse operators, may be timedependent for solver choice of
me
. a_op
Qobj
Operator A.
 argsdict, optional
Dictionary of arguments to be passed to solver.
 solverstr {‘me’, ‘es’}, default: ‘me’
Choice of solver,
me
for masterequation, andes
for exponential series.es
is equivalent tome
withoptions={"method": "diag"}
. optionsdict, optional
Options for the solver.
 H
 Returns:
 g2, G2tuple
The normalized and unnormalized secondorder coherence function.
 correlation_2op_1t(H, state0, taulist, c_ops, a_op, b_op, solver='me', reverse=False, args=None, options=None)[source]
Calculate the twooperator onetime correlation function: \(\left<A(\tau)B(0)\right>\) along one time axis using the quantum regression theorem and the evolution solver indicated by the solver parameter.
 Parameters:
 H
Qobj
,QobjEvo
System Hamiltonian, may be timedependent for solver choice of me.
 state0
Qobj
Initial state density matrix \(\rho(t_0)\) or state vector \(\psi(t_0)\). If ‘state0’ is ‘None’, then the steady state will be used as the initial state. The ‘steadystate’ is only implemented if
c_ops
are provided and the Hamiltonian is constant. taulistarray_like
List of times for \(\tau\). taulist must be positive and contain the element 0.
 c_opslist of {
Qobj
,QobjEvo
} List of collapse operators
 a_op
Qobj
,QobjEvo
Operator A.
 b_op
Qobj
,QobjEvo
Operator B.
 reversebool, default: False
If
True
, calculate \(\left<A(t)B(t+\tau)\right>\) instead of \(\left<A(t+\tau)B(t)\right>\). solverstr {‘me’, ‘es’}, default: ‘me’
Choice of solver,
me
for masterequation, andes
for exponential series.es
is equivalent to me withoptions={"method": "diag"}
. optionsdict, optional
Options for the solver.
 H
 Returns:
 corr_vecndarray
An array of correlation values for the times specified by
taulist
.
See also
correlation_3op
Similar function supporting various solver types.
References
See, Gardiner, Quantum Noise, Section 5.2.
 correlation_2op_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, solver='me', reverse=False, args=None, options=None)[source]
Calculate the twooperator twotime correlation function: \(\left<A(t+\tau)B(t)\right>\) along two time axes using the quantum regression theorem and the evolution solver indicated by the
solver
parameter. Parameters:
 H
Qobj
,QobjEvo
System Hamiltonian, may be timedependent for solver choice of me.
 state0
Qobj
Initial state density matrix \(\rho(t_0)\) or state vector \(\psi(t_0)\). If ‘state0’ is ‘None’, then the steady state will be used as the initial state. The ‘steadystate’ is only implemented if
c_ops
are provided and the Hamiltonian is constant. tlistarray_like
List of times for \(t\). tlist must be positive and contain the element 0. When taking steadysteady correlations only one
tlist
value is necessary, i.e. when \(t \rightarrow \infty\). Iftlist
isNone
,tlist=[0]
is assumed. taulistarray_like
List of times for \(\tau\). taulist must be positive and contain the element 0.
 c_opslist of {
Qobj
,QobjEvo
} List of collapse operators
 a_op
Qobj
,QobjEvo
Operator A.
 b_op
Qobj
,QobjEvo
Operator B.
 reversebool, default: False
If
True
, calculate \(\left<A(t)B(t+\tau)\right>\) instead of \(\left<A(t+\tau)B(t)\right>\). solverstr {‘me’, ‘es’}, default: ‘me’
Choice of solver,
me
for masterequation, andes
for exponential series.es
is equivalent to me withoptions={"method": "diag"}
. optionsdict, optional
Options for the solver.
 H
 Returns:
 corr_matndarray
An 2dimensional array (matrix) of correlation values for the times specified by
tlist
(first index) andtaulist
(second index).
See also
correlation_3op
Similar function supporting various solver types.
References
See, Gardiner, Quantum Noise, Section 5.2.
 correlation_3op(solver, state0, tlist, taulist, A=None, B=None, C=None)[source]
Calculate the threeoperator twotime correlation function:
\(\left<A(t)B(t+\tau)C(t)\right>\).
from a open system
Solver
.Note: it is not possible to calculate a physically meaningful correlation where \(\tau<0\).
 Parameters:
 solver
MESolver
,BRSolver
Qutip solver for an open system.
 state0
Qobj
Initial state density matrix \(\rho(t_0)\) or state vector \(\psi(t_0)\).
 tlistarray_like
List of times for \(t\). tlist must be positive and contain the element 0.
 taulistarray_like
List of times for \(\tau\). taulist must be positive and contain the element 0.
 A, B, C
Qobj
,QobjEvo
, optional, default=None Operators
A
,B
,C
from the equation<A(t)B(t+\tau)C(t)>
in the Schrodinger picture. They do not need to be all provided. For exemple, ifA
is not provided,<B(t+\tau)C(t)>
is computed.
 solver
 Returns:
 corr_matarray
An 2dimensional array (matrix) of correlation values for the times specified by
tlist
(first index) and taulist (second index). Iftlist
isNone
, then a 1dimensional array of correlation values is returned instead.
 correlation_3op_1t(H, state0, taulist, c_ops, a_op, b_op, c_op, solver='me', args=None, options=None)[source]
Calculate the threeoperator twotime correlation function: \(\left<A(0)B(\tau)C(0)\right>\) along one time axis using the quantum regression theorem and the evolution solver indicated by the solver parameter.
Note: it is not possibly to calculate a physically meaningful correlation of this form where \(\tau<0\).
 Parameters:
 H
Qobj
,QobjEvo
System Hamiltonian, may be timedependent for solver choice of
me
. state0
Qobj
Initial state density matrix \(\rho(t_0)\) or state vector \(\psi(t_0)\). If ‘state0’ is ‘None’, then the steady state will be used as the initial state. The ‘steadystate’ is only implemented if
c_ops
are provided and the Hamiltonian is constant. taulistarray_like
List of times for \(\tau\). taulist must be positive and contain the element 0.
 c_opslist of {
Qobj
,QobjEvo
} List of collapse operators
 a_op
Qobj
,QobjEvo
Operator A.
 b_op
Qobj
,QobjEvo
Operator B.
 c_op
Qobj
,QobjEvo
Operator C.
 solverstr {‘me’, ‘es’}, default: ‘me’
Choice of solver,
me
for masterequation, andes
for exponential series.es
is equivalent to me withoptions={"method": "diag"}
. optionsdict, optional
Options for the solver.
 H
 Returns:
 corr_vecarray
An array of correlation values for the times specified by
taulist
.
See also
correlation_3op
Similar function supporting various solver types.
References
See, Gardiner, Quantum Noise, Section 5.2.
 correlation_3op_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, c_op, solver='me', args=None, options=None)[source]
Calculate the threeoperator twotime correlation function: \(\left<A(t)B(t+\tau)C(t)\right>\) along two time axes using the quantum regression theorem and the evolution solver indicated by the solver parameter.
Note: it is not possibly to calculate a physically meaningful correlation of this form where \(\tau<0\).
 Parameters:
 H
Qobj
,QobjEvo
System Hamiltonian, may be timedependent for solver choice of
me
. state0
Qobj
Initial state density matrix \(\rho(t_0)\) or state vector \(\psi(t_0)\). If ‘state0’ is ‘None’, then the steady state will be used as the initial state. The ‘steadystate’ is only implemented if
c_ops
are provided and the Hamiltonian is constant. tlistarray_like
List of times for \(t\). tlist must be positive and contain the element 0. When taking steadysteady correlations only one tlist value is necessary, i.e. when \(t \rightarrow \infty\). If
tlist
isNone
,tlist=[0]
is assumed. taulistarray_like
List of times for \(\tau\). taulist must be positive and contain the element 0.
 c_opslist of {
Qobj
,QobjEvo
} List of collapse operators
 a_op
Qobj
,QobjEvo
Operator A.
 b_op
Qobj
,QobjEvo
Operator B.
 c_op
Qobj
,QobjEvo
Operator C.
 solverstr {‘me’, ‘es’}, default: ‘me’
Choice of solver,
me
for masterequation, andes
for exponential series.es
is equivalent to me withoptions={"method": "diag"}
. optionsdict, optional
Options for the solver. Only used with
me
solver.
 H
 Returns:
 corr_matarray
An 2dimensional array (matrix) of correlation values for the times specified by
tlist
(first index) andtaulist
(second index).
See also
correlation_3op
Similar function supporting various solver types.
References
See, Gardiner, Quantum Noise, Section 5.2.
 spectrum(H, wlist, c_ops, a_op, b_op, solver='es')[source]
Calculate the spectrum of the correlation function \(\lim_{t \to \infty} \left<A(t+\tau)B(t)\right>\), i.e., the Fourier transform of the correlation function:
\[S(\omega) = \int_{\infty}^{\infty} \lim_{t \to \infty} \left<A(t+\tau)B(t)\right> e^{i\omega\tau} d\tau.\]using the solver indicated by the
solver
parameter. Note: this spectrum is only defined for stationary statistics (uses steady state rho0) Parameters:
 Returns:
 spectrumarray
An array with spectrum \(S(\omega)\) for the frequencies specified in wlist.
 spectrum_correlation_fft(tlist, y, inverse=False)[source]
Calculate the power spectrum corresponding to a twotime correlation function using FFT.
 Parameters:
 tlistarray_like
list/array of times \(t\) which the correlation function is given.
 yarray_like
list/array of correlations corresponding to time delays \(t\).
 inverse: bool, default: False
boolean parameter for using a positive exponent in the Fourier Transform instead. Default is False.
 Returns:
 w, Stuple
Returns an array of angular frequencies ‘w’ and the corresponding twosided power spectrum ‘S(w)’.
Steadystate Solvers
 pseudo_inverse(L, rhoss=None, w=None, method='splu', *, use_rcm=False, **kwargs)[source]
Compute the pseudo inverse for a Liouvillian superoperator, optionally given its steady state density matrix (which will be computed if not given).
 Parameters:
 LQobj
A Liouvillian superoperator for which to compute the pseudo inverse.
 rhossQobj, optional
A steadystate density matrix as Qobj instance, for the Liouvillian superoperator L.
 wdouble, optional
frequency at which to evaluate pseudoinverse. Can be zero for dense systems and large sparse systems. Small sparse systems can fail for zero frequencies.
 sparsebool, optional
Flag that indicate whether to use sparse or dense matrix methods when computing the pseudo inverse.
 methodstr, optional
Method used to compte matrix inverse. Choice are ‘pinv’ to use scipy’s function of the same name, or a linear system solver. Default supported solver are:
“solve”, “lstsq” dense solver from numpy.linalg
“spsolve”, “gmres”, “lgmres”, “bicgstab”, “splu” sparse solver from scipy.sparse.linalg
“mkl_spsolve”, sparse solver by mkl.
Extension to qutip, such as qutiptensorflow, can use come with their own solver. When
L
use these data backends, see the corresponding librarieslinalg
for available solver. use_rcmbool, default: False
Use reverse CuthillMckee reordering to minimize fillin in the LU factorization of the Liouvillian.
 kwargsdictionary
Additional keyword arguments for setting parameters for solver methods.
 Returns:
 RQobj
Returns a Qobj instance representing the pseudo inverse of L.
Notes
In general the inverse of a sparse matrix will be dense. If you are applying the inverse to a density matrix then it is better to cast the problem as an Ax=b type problem where the explicit calculation of the inverse is not required. See page 67 of “Electrons in nanostructures” C. Flindt, PhD Thesis available online: https://orbit.dtu.dk/en/publications/electronsinnanostructurescoherentmanipulationandcountingst
Note also that the definition of the pseudoinverse herein is different from numpys pinv() alone, as it includes pre and post projection onto the subspace defined by the projector Q.
 steadystate(A, c_ops=[], *, method='direct', solver=None, **kwargs)[source]
Calculates the steady state for quantum evolution subject to the supplied Hamiltonian or Liouvillian operator and (if given a Hamiltonian) a list of collapse operators.
If the user passes a Hamiltonian then it, along with the list of collapse operators, will be converted into a Liouvillian operator in Lindblad form.
 Parameters:
 A
Qobj
A Hamiltonian or Liouvillian operator.
 c_op_listlist
A list of collapse operators.
 methodstr, {“direct”, “eigen”, “svd”, “power”}, default: “direct”
The allowed methods are composed of 2 parts, the steadystate method:  “direct”: Solving
L(rho_ss) = 0
 “eigen” : Eigenvalue problem  “svd” : Singular value decomposition  “power” : Inversepower method solverstr, optional
‘direct’ and ‘power’ methods only. Solver to use when solving the
L(rho_ss) = 0
equation. Default supported solver are:“solve”, “lstsq” dense solver from numpy.linalg
“spsolve”, “gmres”, “lgmres”, “bicgstab” sparse solver from scipy.sparse.linalg
“mkl_spsolve” sparse solver by mkl.
Extension to qutip, such as qutiptensorflow, can use come with their own solver. When
A
andc_ops
use these data backends, see the corresponding librarieslinalg
for available solver.Extra options for these solver can be passed in
**kw
. use_rcmbool, default: False
Use reverse CuthillMckee reordering to minimize fillin in the LU factorization of the Liouvillian. Used with ‘direct’ or ‘power’ method.
 use_wbmbool, default: False
Use Weighted Bipartite Matching reordering to make the Liouvillian diagonally dominant. This is useful for iterative preconditioners only. Used with ‘direct’ or ‘power’ method.
 weightfloat, optional
Sets the size of the elements used for adding the unity trace condition to the linear solvers. This is set to the average abs value of the Liouvillian elements if not specified by the user. Used with ‘direct’ method.
 power_tolfloat, default: 1e12
Tolerance for the solution when using the ‘power’ method.
 power_maxiterint, default: 10
Maximum number of iteration to use when looking for a solution when using the ‘power’ method.
 power_eps: double, default: 1e15
Small weight used in the “power” method.
 sparse: bool, default: True
Whether to use the sparse eigen solver with the “eigen” method (default sparse). With “direct” and “power” method, when the solver is not specified, it is used to set whether “solve” or “spsolve” is used as default solver.
 **kwargs
Extra options to pass to the linear system solver. See the documentation of the used solver in
numpy.linalg
orscipy.sparse.linalg
to see what extra arguments are supported.
 A
 Returns:
 dmqobj
Steady state density matrix.
 infodict, optional
Dictionary containing solverspecific information about the solution.
Notes
The SVD method works only for dense operators (i.e. small systems).
 steadystate_floquet(H_0, c_ops, Op_t, w_d=1.0, n_it=3, sparse=False, solver=None, **kwargs)[source]
 Calculates the effective steady state for a driven
system with a timedependent cosinusoidal term:
\[\mathcal{\hat{H}}(t) = \hat{H}_0 + \mathcal{\hat{O}} \cos(\omega_d t)\] Parameters:
 H_0
Qobj
A Hamiltonian or Liouvillian operator.
 c_opslist
A list of collapse operators.
 Op_t
Qobj
The the interaction operator which is multiplied by the cosine
 w_dfloat, default: 1.0
The frequency of the drive
 n_itint, default: 3
The number of iterations for the solver
 sparsebool, default: False
Solve for the steady state using sparse algorithms.
 solverstr, optional
Solver to use when solving the linear system. Default supported solver are:
“solve”, “lstsq” dense solver from numpy.linalg
“spsolve”, “gmres”, “lgmres”, “bicgstab” sparse solver from scipy.sparse.linalg
“mkl_spsolve” sparse solver by mkl.
Extensions to qutip, such as qutiptensorflow, may provide their own solvers. When
H_0
andc_ops
use these data backends, see their documentation for the names and details of additional solvers they may provide. **kwargs:
Extra options to pass to the linear system solver. See the documentation of the used solver in
numpy.linalg
orscipy.sparse.linalg
to see what extra arguments are supported.
 H_0
 Returns:
 dmqobj
Steady state density matrix.
Notes
See: Sze Meng Tan, https://copilot.caltech.edu/documents/16743/qousersguide.pdf, Section (10.16)
Propagators
 propagator(H, t, c_ops=(), args=None, options=None, **kwargs)[source]
Calculate the propagator U(t) for the density matrix or wave function such that \(\psi(t) = U(t)\psi(0)\) or \(\rho_{\mathrm vec}(t) = U(t) \rho_{\mathrm vec}(0)\) where \(\rho_{\mathrm vec}\) is the vector representation of the density matrix.
 Parameters:
 H
Qobj
,QobjEvo
,QobjEvo
compatible format Possibly timedependent system Liouvillian or Hamiltonian as a Qobj or QobjEvo.
list
of [Qobj
,Coefficient
] or callable that can be made intoQobjEvo
are also accepted. tfloat or arraylike
Time or list of times for which to evaluate the propagator.
 c_opslist, optional
List of Qobj or QobjEvo collapse operators.
 argsdictionary, optional
Parameters to callback functions for timedependent Hamiltonians and collapse operators.
 optionsdict, optional
Options for the solver.
 **kwargs
Extra parameters to use when creating the
QobjEvo
from a list formatH
.
 H
 Returns:
 U
Qobj
, list Instance representing the propagator(s) \(U(t)\). Return a single Qobj when
t
is a number or a list whent
is a list.
 U
Scattering in Quantum Optical Systems
Photon scattering in quantum optical systems
This module includes a collection of functions for numerically computing photon scattering in driven arbitrary systems coupled to some configuration of output waveguides. The implementation of these functions closely follows the mathematical treatment given in K.A. Fischer, et. al., Scattering of Coherent Pulses from Quantum Optical Systems (2017, arXiv:1710.02875).
 scattering_probability(H, psi0, n_emissions, c_ops, tlist, system_zero_state=None, construct_effective_hamiltonian=True)[source]
Compute the integrated probability of scattering n photons in an arbitrary system. This function accepts a nonlinearly spaced array of times.
 Parameters:
 H
Qobj
or list Systemwaveguide(s) Hamiltonian or effective Hamiltonian in Qobj or listcallback format. If construct_effective_hamiltonian is not specified, an effective Hamiltonian is constructed from H and c_ops.
 psi0
Qobj
Initial state density matrix \(\rho(t_0)\) or state vector \(\psi(t_0)\).
 n_emissionsint
Number of photons emitted by the system (into any combination of waveguides).
 c_opslist
List of collapse operators for each waveguide; these are assumed to include spontaneous decay rates, e.g. \(\sigma = \sqrt \gamma \cdot a\).
 tlistarray_like
List of times for \(\tau_i\). tlist should contain 0 and exceed the pulse duration / temporal region of interest; tlist need not be linearly spaced.
 system_zero_state
Qobj
, optional State representing zero excitations in the system. Defaults to basis(systemDims, 0).
 construct_effective_hamiltonianbool, default: True
Whether an effective Hamiltonian should be constructed from H and c_ops: \(H_{eff} = H  \frac{i}{2} \sum_n \sigma_n^\dagger \sigma_n\) Default: True.
 H
 Returns:
 scattering_probfloat
The probability of scattering n photons from the system over the time range specified.
 temporal_basis_vector(waveguide_emission_indices, n_time_bins)[source]
Generate a temporal basis vector for emissions at specified time bins into specified waveguides.
 Parameters:
 waveguide_emission_indiceslist or tuple
List of indices where photon emission occurs for each waveguide, e.g. [[t1_wg1], [t1_wg2, t2_wg2], [], [t1_wg4, t2_wg4, t3_wg4]].
 n_time_binsint
Number of time bins; the range over which each index can vary.
 Returns:
 temporal_basis_vector
Qobj
A basis vector representing photon scattering at the specified indices. If there are W waveguides, T times, and N photon emissions, then the basis vector has dimensionality (W*T)^N.
 temporal_basis_vector
 temporal_scattered_state(H, psi0, n_emissions, c_ops, tlist, system_zero_state=None, construct_effective_hamiltonian=True)[source]
Compute the scattered nphoton state projected onto the temporal basis.
 Parameters:
 H
Qobj
or list Systemwaveguide(s) Hamiltonian or effective Hamiltonian in Qobj or listcallback format. If construct_effective_hamiltonian is not specified, an effective Hamiltonian is constructed from H and c_ops.
 psi0
Qobj
Initial state density matrix \(\rho(t_0)\) or state vector \(\psi(t_0)\).
 n_emissionsint
Number of photon emissions to calculate.
 c_opslist
List of collapse operators for each waveguide; these are assumed to include spontaneous decay rates, e.g. \(\sigma = \sqrt \gamma \cdot a\)
 tlistarray_like
List of times for \(\tau_i\). tlist should contain 0 and exceed the pulse duration / temporal region of interest.
 system_zero_state
Qobj
, optional State representing zero excitations in the system. Defaults to \(\psi(t_0)\)
 construct_effective_hamiltonianbool, default: True
Whether an effective Hamiltonian should be constructed from H and c_ops: \(H_{eff} = H  \frac{i}{2} \sum_n \sigma_n^\dagger \sigma_n\) Default: True.
 H
 Returns:
 phi_n
Qobj
The scattered bath state projected onto the temporal basis given by tlist. If there are W waveguides, T times, and N photon emissions, then the state is a tensor product state with dimensionality T^(W*N).
 phi_n
Permutational Invariance
Permutational Invariant Quantum Solver (PIQS)
This module calculates the Liouvillian for the dynamics of ensembles of identical twolevel systems (TLS) in the presence of local and collective processes by exploiting permutational symmetry and using the Dicke basis. It also allows to characterize nonlinear functions of the density matrix.
 am(j, m)[source]
Calculate the operator
am
used later.The action of
ap
is given by: \(J_{}\lvert j,m\rangle = A_{}(jm)\lvert j,m1\rangle\) Parameters:
 j: float
The value for j.
 m: float
The value for m.
 Returns:
 a_minus: float
The value of \(a_{}\).
 ap(j, m)[source]
Calculate the coefficient
ap
by applying \(J_+\lvert j,m\rangle\).The action of ap is given by: \(J_{+}\lvert j, m\rangle = A_{+}(j, m) \lvert j, m+1\rangle\)
 Parameters:
 j, m: float
The value for j and m in the dicke basis \(\lvert j, m\rangle\).
 Returns:
 a_plus: float
The value of \(a_{+}\).
 block_matrix(N, elements='ones')[source]
Construct the blockdiagonal matrix for the Dicke basis.
 Parameters:
 Nint
Number of twolevel systems.
 elementsstr {‘ones’, ‘degeneracy’}, default: ‘ones’
 Returns:
 block_matrndarray
A 2D blockdiagonal matrix with dimension (nds,nds), where nds is the number of Dicke states for N twolevel systems. Filled with ones or the value of degeneracy at each matrix element.
 collapse_uncoupled(N, emission=0.0, dephasing=0.0, pumping=0.0, collective_emission=0.0, collective_dephasing=0.0, collective_pumping=0.0)[source]
Create the collapse operators (c_ops) of the Lindbladian in the uncoupled basis
These operators are in the uncoupled basis of the twolevel system (TLS) SU(2) Pauli matrices.
 Parameters:
 N: int
The number of twolevel systems.
 emission: float, default: 0.0
Incoherent emission coefficient (also nonradiative emission).
 dephasing: float, default: 0.0
Local dephasing coefficient.
 pumping: float, default: 0.0
Incoherent pumping coefficient.
 collective_emission: float, default: 0.0
Collective (superradiant) emmission coefficient.
 collective_pumping: float, default: 0.0
Collective pumping coefficient.
 collective_dephasing: float, default: 0.0
Collective dephasing coefficient.
 Returns:
 c_ops: list
The list of collapse operators as
Qobj
for the system.
Notes
The collapse operator list can be given to qutip.mesolve. Notice that the operators are placed in a Hilbert space of dimension \(2^N\). Thus the method is suitable only for small N (of the order of 10).
 css(N, x=0.7071067811865475, y=0.7071067811865475, basis='dicke', coordinates='cartesian')[source]
Generate the density matrix of the Coherent Spin State (CSS).
It can be defined as, \(\lvert CSS\rangle = \prod_i^N(a\lvert1\rangle_i+b\lvert0\rangle_i)\) with \(a = sin(\frac{\theta}{2})\), \(b = e^{i \phi}\cos(\frac{\theta}{2})\). The default basis is that of Dicke space \(\lvert j, m\rangle \langle j, m'\rvert\). The default state is the symmetric CSS, \(\lvert CSS\rangle = \lvert+\rangle\).
 Parameters:
 N: int
The number of twolevel systems.
 x, y: float, default: sqrt(1/2)
The coefficients of the CSS state.
 basis: str {“dicke”, “uncoupled”}, default: “dicke”
The basis to use.
 coordinates: str {“cartesian”, “polar”}, default: “cartesian”
If polar then the coefficients are constructed as \(sin(x/2), cos(x/2)e^(iy)`\).
 Returns:
 rho:
Qobj
The CSS state density matrix.
 rho:
 dicke(N, j, m)[source]
Generate a Dicke state as a pure density matrix in the Dicke basis.
For instance, the superradiant state given by \(\lvert j, m\rangle = \lvert 1, 0\rangle\) for N = 2, and the state is represented as a density matrix of size (nds, nds) or (4, 4), with the (1, 1) element set to 1.
 Parameters:
 N: int
The number of twolevel systems.
 j: float
The eigenvalue j of the Dicke state (j, m).
 m: float
The eigenvalue m of the Dicke state (j, m).
 Returns:
 rho:
Qobj
The density matrix.
 rho:
 dicke_basis(N, jmm1)[source]
Initialize the density matrix of a Dicke state for several (j, m, m1).
This function can be used to build arbitrary states in the Dicke basis \(\lvert j, m\rangle\langle j, m'\rvert\). We create coefficients for each (j, m, m1) value in the dictionary jmm1. The mapping for the (i, k) index of the density matrix to the \(\lvert j, m\rangle\) values is given by the cythonized function jmm1_dictionary. A density matrix is created from the given dictionary of coefficients for each (j, m, m1).
 Parameters:
 N: int
The number of twolevel systems.
 jmm1: dict
A dictionary of {(j, m, m1): p} that gives a density p for the (j, m, m1) matrix element.
 Returns:
 rho:
Qobj
The density matrix in the Dicke basis.
 rho:
 dicke_blocks(rho)[source]
Create the list of blocks for blockdiagonal density matrix in the Dicke basis.
 Parameters:
 rho
Qobj
A 2D blockdiagonal matrix of ones with dimension (nds,nds), where nds is the number of Dicke states for N twolevel systems.
 rho
 Returns:
 square_blocks: list of np.ndarray
Give back the blocks list.
 dicke_blocks_full(rho)[source]
Give the full (2^Ndimensional) list of blocks for a Dickebasis matrix.
 Parameters:
 rho
Qobj
A 2D blockdiagonal matrix of ones with dimension (nds,nds), where nds is the number of Dicke states for N twolevel systems.
 rho
 Returns:
 full_blockslist
The list of blocks expanded in the 2^N space for N qubits.
 dicke_function_trace(f, rho)[source]
Calculate the trace of a function on a Dicke density matrix. :param f: A Taylorexpandable function of rho. :type f: function :param rho: A density matrix in the Dicke basis. :type rho:
Qobj
 Returns:
 resfloat
Trace of a nonlinear function on rho.
 energy_degeneracy(N, m)[source]
Calculate the number of Dicke states with same energy.
The use of the
Decimals
class allows to explore N > 1000, unlike the builtin functionscipy.special.binom
. Parameters:
 N: int
The number of twolevel systems.
 m: float
Total spin zaxis projection eigenvalue. This is proportional to the total energy.
 Returns:
 degeneracy: int
The energy degeneracy
 entropy_vn_dicke(rho)[source]
Von Neumann Entropy of a Dickebasis density matrix.
 Parameters:
 rho
Qobj
A 2D blockdiagonal matrix of ones with dimension (nds, nds), where nds is the number of Dicke states for N twolevel systems.
 rho
 Returns:
 entropy_dm: float
Entropy. Use degeneracy to multiply each block.
 excited(N, basis='dicke')[source]
Generate the density matrix for the excited state.
This state is given by (N/2, N/2) in the default Dicke basis. If the argument
basis
is “uncoupled” then it generates the state in a 2**N dim Hilbert space. Parameters:
 N: int
The number of twolevel systems.
 basis: str, {“dicke”, “uncoupled”}, default: “dicke”
The basis to use.
 Returns:
 state:
Qobj
The excited state density matrix in the requested basis.
 state:
 ghz(N, basis='dicke')[source]
Generate the density matrix of the GHZ state.
If the argument
basis
is “uncoupled” then it generates the state in a \(2^N\)dimensional Hilbert space. Parameters:
 N: int
The number of twolevel systems.
 basis: str, {“dicke”, “uncoupled”}, default: “dicke”
The basis to use.
 Returns:
 state:
Qobj
The GHZ state density matrix in the requested basis.
 state:
 ground(N, basis='dicke')[source]
Generate the density matrix of the ground state.
This state is given by (N/2, N/2) in the Dicke basis. If the argument
basis
is “uncoupled” then it generates the state in a \(2^N\)dimensional Hilbert space. Parameters:
 N: int
The number of twolevel systems.
 basis: str, {“dicke”, “uncoupled”}, default: “dicke”
The basis to use.
 Returns:
 state:
Qobj
The ground state density matrix in the requested basis.
 state:
 identity_uncoupled(N)[source]
Generate the identity in a \(2^N\)dimensional Hilbert space.
The identity matrix is formed from the tensor product of N TLSs.
 Parameters:
 N: int
The number of twolevel systems.
 Returns:
 identity:
Qobj
The identity matrix.
 identity:
 isdiagonal(mat)[source]
Check if the input matrix is diagonal.
 Parameters:
 mat: ndarray/Qobj
A 2D numpy array
 Returns:
 diag: bool
True/False depending on whether the input matrix is diagonal.
 jspin(N, op=None, basis='dicke')[source]
Calculate the list of collective operators of the total algebra.
The Dicke basis \(\lvert j,m\rangle\langle j,m'\rvert\) is used by default. Otherwise with “uncoupled” the operators are in a \(2^N\) space.
 Parameters:
 N: int
Number of twolevel systems.
 op: str {‘x’, ‘y’, ‘z’, ‘+’, ‘‘}, optional
The operator to return ‘x’,’y’,’z’,’+’,’‘. If no operator given, then output is the list of operators for [‘x’,’y’,’z’].
 basis: str {“dicke”, “uncoupled”}, default: “dicke”
The basis of the operators.
 Returns:
 j_alg: list or
Qobj
A list of qutip.Qobj representing all the operators in the “dicke” or “uncoupled” basis or a single operator requested.
 j_alg: list or
 m_degeneracy(N, m)[source]
Calculate the number of Dicke states \(\lvert j, m\rangle\) with same energy.
 Parameters:
 N: int
The number of twolevel systems.
 m: float
Total spin zaxis projection eigenvalue (proportional to the total energy).
 Returns:
 degeneracy: int
The mdegeneracy.
 num_dicke_ladders(N)[source]
Calculate the total number of ladders in the Dicke space.
For a collection of N twolevel systems it counts how many different “j” exist or the number of blocks in the blockdiagonal matrix.
 Parameters:
 N: int
The number of twolevel systems.
 Returns:
 Nj: int
The number of Dicke ladders.
 num_dicke_states(N)[source]
Calculate the number of Dicke states.
 Parameters:
 N: int
The number of twolevel systems.
 Returns:
 nds: int
The number of Dicke states.
 num_tls(nds)[source]
Calculate the number of twolevel systems.
 Parameters:
 nds: int
The number of Dicke states.
 Returns:
 N: int
The number of twolevel systems.
 purity_dicke(rho)[source]
Calculate purity of a density matrix in the Dicke basis. It accounts for the degenerate blocks in the density matrix.
 Parameters:
 rho
Qobj
Density matrix in the Dicke basis of qutip.piqs.jspin(N), for N spins.
 rho
 Returns:
 purityfloat
The purity of the quantum state. It’s 1 for pure states, 0<=purity<1 for mixed states.
 spin_algebra(N, op=None)[source]
Create the list [sx, sy, sz] with the spin operators.
The operators are constructed for a collection of N twolevel systems (TLSs). Each element of the list, i.e., sx, is a vector of qutip.Qobj objects (spin matrices), as it cointains the list of the SU(2) Pauli matrices for the N TLSs. Each TLS operator sx[i], with i = 0, …, (N1), is placed in a \(2^N\)dimensional Hilbert space.
 Parameters:
 N: int
The number of twolevel systems.
 Returns:
 spin_operators: list or
Qobj
A list of qutip.Qobj operators  [sx, sy, sz] or the requested operator.
 spin_operators: list or
Notes
sx[i] is \(\frac{\sigma_x}{2}\) in the composite Hilbert space.
 state_degeneracy(N, j)[source]
Calculate the degeneracy of the Dicke state.
Each state \(\lvert j, m\rangle\) includes D(N,j) irreducible representations \(\lvert j, m, \alpha\rangle\).
Uses Decimals to calculate higher numerator and denominators numbers.
 Parameters:
 N: int
The number of twolevel systems.
 j: float
Total spin eigenvalue (cooperativity).
 Returns:
 degeneracy: int
The state degeneracy.
 superradiant(N, basis='dicke')[source]
Generate the density matrix of the superradiant state.
This state is given by (N/2, 0) or (N/2, 0.5) in the Dicke basis. If the argument basis is “uncoupled” then it generates the state in a 2**N dim Hilbert space.
 Parameters:
 N: int
The number of twolevel systems.
 basis: str, {“dicke”, “uncoupled”}, default: “dicke”
The basis to use.
 Returns:
 state:
Qobj
The superradiant state density matrix in the requested basis.
 state:
 tau_column(tau, k, j)[source]
Determine the column index for the nonzero elements of the matrix for a particular row k and the value of j from the Dicke space.
 Parameters:
 tau: str
The tau function to check for this k and j.
 k: int
The row of the matrix M for which the non zero elements have to be calculated.
 j: float
The value of j for this row.
Visualization
Pseudoprobability Functions
 qfunc(state: Qobj, xvec, yvec, g: float = 1.4142135623730951, precompute_memory: float = 1024)[source]
HusimiQ function of a given state vector or density matrix at phasespace points
0.5 * g * (xvec + i*yvec)
. Parameters:
 state
Qobj
A state vector or density matrix. This cannot have tensorproduct structure.
 xvec, yvecarray_like
x and ycoordinates at which to calculate the HusimiQ function.
 gfloat, default: sqrt(2)
Scaling factor for
a = 0.5 * g * (x + iy)
. The value of g is related to the value of \(\hbar\) in the commutation relation \([x,\,y] = i\hbar\) via \(\hbar=2/g^2\), so the default corresponds to \(\hbar=1\). precompute_memoryreal, default: 1024
Size in MB that may be used during calculations as working space when dealing with densitymatrix inputs. This is ignored for statevector inputs. The bound is not quite exact due to other, orderofmagnitude smaller, intermediaries being necessary, but is a good approximation. If you want to use the same iterative algorithm for density matrices that is used for single kets, set
precompute_memory=None
.
 state
 Returns:
 ndarray
Values representing the HusimiQ function calculated over the specified range
[xvec, yvec]
.
See also
QFunc
a classbased version, more efficient if you want to calculate the HusimiQ function for several states over the same coordinates.
 spin_q_function(rho, theta, phi)[source]
The Husimi Q function for spins is defined as
Q(theta, phi) = SCS.dag() * rho * SCS
for the spin coherent stateSCS = spin_coherent( j, theta, phi)
where j is the spin length. The implementation here is more efficient as it doesn’t generate all of the SCS at theta and phi (see references).The spin Q function is normal when integrated over the surface of the sphere
\[\frac{4 \pi}{2j + 1}\int_\phi \int_\theta Q(\theta, \phi) \sin(\theta) d\theta d\phi = 1\] Parameters:
 stateqobj
A state vector or density matrix for a spinj quantum system.
 thetaarray_like
Polar (colatitude) angle at which to calculate the HusimiQ function.
 phiarray_like
Azimuthal angle at which to calculate the HusimiQ function.
 Returns:
 Q, THETA, PHI2darray
Values representing the spin Husimi Q function at the values specified by THETA and PHI.
References
[1] Lee Loh, Y., & Kim, M. (2015). American J. of Phys., 83(1), 30–35. https://doi.org/10.1119/1.4898595
 spin_wigner(rho, theta, phi)[source]
Wigner function for a spinj system.
The spin W function is normal when integrated over the surface of the sphere
\[\sqrt{\frac{4 \pi}{2j + 1}}\int_\phi \int_\theta W(\theta,\phi) \sin(\theta) d\theta d\phi = 1\] Parameters:
 stateqobj
A state vector or density matrix for a spinj quantum system.
 thetaarray_like
Polar (colatitude) angle at which to calculate the W function.
 phiarray_like
Azimuthal angle at which to calculate the W function.
 Returns:
 W, THETA, PHI2darray
Values representing the spin Wigner function at the values specified by THETA and PHI.
References
[1] Agarwal, G. S. (1981). Phys. Rev. A, 24(6), 2889–2896. https://doi.org/10.1103/PhysRevA.24.2889
[2] Dowling, J. P., Agarwal, G. S., & Schleich, W. P. (1994). Phys. Rev. A, 49(5), 4101–4109. https://doi.org/10.1103/PhysRevA.49.4101
[3] Conversion between Wigner 3j symbol and ClebschGordan coefficients taken from Wikipedia (https://en.wikipedia.org/wiki/3j_symbol)
 wigner(psi, xvec, yvec=None, method='clenshaw', g=1.4142135623730951, sparse=False, parfor=False)[source]
Wigner function for a state vector or density matrix at points xvec + i * yvec.
 Parameters:
 stateqobj
A state vector or density matrix.
 xvecarray_like
xcoordinates at which to calculate the Wigner function.
 yvecarray_like
ycoordinates at which to calculate the Wigner function. Does not apply to the ‘fft’ method.
 gfloat, default: sqrt(2)
Scaling factor for a = 0.5 * g * (x + iy), default g = sqrt(2). The value of g is related to the value of hbar in the commutation relation [x, y] = i * hbar via hbar=2/g^2 giving the default value hbar=1.
 methodstring {‘clenshaw’, ‘iterative’, ‘laguerre’, ‘fft’}, default: ‘clenshaw’
Select method ‘clenshaw’ ‘iterative’, ‘laguerre’, or ‘fft’, where ‘clenshaw’ and ‘iterative’ use an iterative method to evaluate the Wigner functions for density matrices \(m><n\), while ‘laguerre’ uses the Laguerre polynomials in scipy for the same task. The ‘fft’ method evaluates the Fourier transform of the density matrix. The ‘iterative’ method is default, and in general recommended, but the ‘laguerre’ method is more efficient for very sparse density matrices (e.g., superpositions of Fock states in a large Hilbert space). The ‘clenshaw’ method is the preferred method for dealing with density matrices that have a large number of excitations (>~50). ‘clenshaw’ is a fast and numerically stable method.
 sparsebool, optional
Tells the default solver whether or not to keep the input density matrix in sparse format. As the dimensions of the density matrix grow, setthing this flag can result in increased performance.
 parforbool, optional
Flag for calculating the Laguerre polynomial based Wigner function method=’laguerre’ in parallel using the parfor function.
 Returns:
 Warray
Values representing the Wigner function calculated over the specified range [xvec,yvec].
 yvexarray
FFT ONLY. Returns the ycoordinate values calculated via the Fourier transform.
Notes
The ‘fft’ method accepts only an xvec input for the xcoordinate. The ycoordinates are calculated internally.
References
Ulf Leonhardt, Measuring the Quantum State of Light, (Cambridge University Press, 1997)
Graphs and Visualization
Functions for visualizing results of quantum dynamics simulations, visualizations of quantum states and processes.
 hinton(rho, x_basis=None, y_basis=None, color_style='scaled', label_top=True, *, cmap=None, colorbar=True, fig=None, ax=None)[source]
Draws a Hinton diagram to visualize a density matrix or superoperator.
 Parameters:
 rhoqobj
Input density matrix or superoperator.
Note
Hinton plots of superoperators are currently only supported for qubits.
 x_basislist of strings, optional
list of x ticklabels to represent x basis of the input.
 y_basislist of strings, optional
list of y ticklabels to represent y basis of the input.
 color_stylestr, {“scaled”, “threshold”, “phase”}, default: “scaled”
Determines how colors are assigned to each square:
If set to
"scaled"
(default), each color is chosen by passing the absolute value of the corresponding matrix element into cmap with the sign of the real part.If set to
"threshold"
, each square is plotted as the maximum of cmap for the positive real part and as the minimum for the negative part of the matrix element; note that this generalizes “threshold” to complex numbers.If set to
"phase"
, each color is chosen according to the angle of the corresponding matrix element.
 label_topbool, default: True
If True, x ticklabels will be placed on top, otherwise they will appear below the plot.
 cmapa matplotlib colormap instance, optional
Color map to use when plotting.
 colorbarbool, default: True
Whether (True) or not (False) a colorbar should be attached.
 figa matplotlib Figure instance, optional
The Figure canvas in which the plot will be drawn.
 axa matplotlib axes instance, optional
The ax context in which the plot will be drawn.
 Returns:
 fig, outputtuple
A tuple of the matplotlib figure and the axes instance or animation instance used to produce the figure.
 Raises:
 ValueError
Input argument is not a quantum object.
Examples
>>> import qutip
>>> dm = qutip.rand_dm(4) >>> fig, ax = qutip.hinton(dm) >>> fig.show()
>>> qutip.settings.colorblind_safe = True >>> fig, ax = qutip.hinton(dm, color_style="threshold") >>> fig.show() >>> qutip.settings.colorblind_safe = False
>>> fig, ax = qutip.hinton(dm, color_style="phase") >>> fig.show()
 matrix_histogram(M, x_basis=None, y_basis=None, limits=None, bar_style='real', color_limits=None, color_style='real', options=None, *, cmap=None, colorbar=True, fig=None, ax=None)[source]
Draw a histogram for the matrix M, with the given x and y labels and title.
 Parameters:
 MMatrix of Qobj
The matrix to visualize
 x_basislist of strings, optional
list of x ticklabels
 y_basislist of strings, optional
list of y ticklabels
 limitslist/array with two float numbers, optional
The zaxis limits [min, max]
 bar_stylestr, {“real”, “img”, “abs”, “phase”}, default: “real”
If set to
"real"
(default), each bar is plotted as the real part of the corresponding matrix elementIf set to
"img"
, each bar is plotted as the imaginary part of the corresponding matrix elementIf set to
"abs"
, each bar is plotted as the absolute value of the corresponding matrix elementIf set to
"phase"
(default), each bar is plotted as the angle of the corresponding matrix element
 color_limitslist/array with two float numbers, optional
The limits of colorbar [min, max]
 color_stylestr, {“real”, “img”, “abs”, “phase”}, default: “real”
Determines how colors are assigned to each square:
If set to
"real"
(default), each color is chosen according to the real part of the corresponding matrix element.If set to
"img"
, each color is chosen according to the imaginary part of the corresponding matrix element.If set to
"abs"
, each color is chosen according to the absolute value of the corresponding matrix element.If set to
"phase"
, each color is chosen according to the angle of the corresponding matrix element.
 cmapa matplotlib colormap instance, optional
Color map to use when plotting.
 colorbarbool, default: True
show colorbar
 figa matplotlib Figure instance, optional
The Figure canvas in which the plot will be drawn.
 axa matplotlib axes instance, optional
The axes context in which the plot will be drawn.
 optionsdict, optional
A dictionary containing extra options for the plot. The names (keys) and values of the options are described below:
 ‘zticks’list of numbers, optional
A list of zaxis tick locations.
 ‘bars_spacing’float, default: 0.1
spacing between bars.
 ‘bars_alpha’float, default: 1.
transparency of bars, should be in range 0  1
 ‘bars_lw’float, default: 0.5
linewidth of bars’ edges.
 ‘bars_edgecolor’color, default: ‘k’
The colors of the bars’ edges. Examples: ‘k’, (0.1, 0.2, 0.5) or ‘#0f0f0f80’.
 ‘shade’bool, default: True
Whether to shade the dark sides of the bars (True) or not (False). The shading is relative to plot’s source of light.
 ‘azim’float, default: 35
The azimuthal viewing angle.
 ‘elev’float, default: 35
The elevation viewing angle.
 ‘stick’bool, default: False
Changes xlim and ylim in such a way that bars next to XZ and YZ planes will stick to those planes. This option has no effect if
ax
is passed as a parameter. ‘cbar_pad’float, default: 0.04
The fraction of the original axes between the colorbar and the new image axes. (i.e. the padding between the 3D figure and the colorbar).
 ‘cbar_to_z’bool, default: False
Whether to set the color of maximum and minimum zvalues to the maximum and minimum colors in the colorbar (True) or not (False).
 ‘threshold’: float, optional
Threshold for when bars of smaller height should be transparent. If not set, all bars are colored according to the color map.
 Returns:
 fig, outputtuple
A tuple of the matplotlib figure and the axes instance or animation instance used to produce the figure.
 Raises:
 ValueError
Input argument is not valid.
 plot_energy_levels(H_list, h_labels=None, energy_levels=None, N=0, *, fig=None, ax=None)[source]
Plot the energy level diagrams for a list of Hamiltonians. Include up to N energy levels. For each element in H_list, the energy levels diagram for the cummulative Hamiltonian sum(H_list[0:n]) is plotted, where n is the index of an element in H_list.
 Parameters:
 H_listList of Qobj
A list of Hamiltonians.
 h_lablesList of string, optional
A list of xticklabels for each Hamiltonian
 energy_levelsList of string, optional
A list of yticklabels to the left of energy levels of the initial Hamiltonian.
 Nint, default: 0
The number of energy levels to plot
 figa matplotlib Figure instance, optional
The Figure canvas in which the plot will be drawn.
 axa matplotlib axes instance, optional
The axes context in which the plot will be drawn.
 Returns:
 fig, axtuple
A tuple of the matplotlib figure and axes instances used to produce the figure.
 Raises:
 ValueError
Input argument is not valid.
 plot_expectation_values(results, ylabels=None, *, fig=None, axes=None)[source]
Visualize the results (expectation values) for an evolution solver. results is assumed to be an instance of Result, or a list of Result instances.
 Parameters:
 results(list of)
Result
List of results objects returned by any of the QuTiP evolution solvers.
 ylabelslist of strings, optional
The yaxis labels. List should be of the same length as results.
 figa matplotlib Figure instance, optional
The Figure canvas in which the plot will be drawn.
 axes(list of) axes instances, optional
The axes context in which the plot will be drawn.
 results(list of)
 Returns:
 fig, axestuple
A tuple of the matplotlib figure and array of axes instances used to produce the figure.
 plot_fock_distribution(rho, fock_numbers=None, color='green', unit_y_range=True, *, fig=None, ax=None)[source]
Plot the Fock distribution for a density matrix (or ket) that describes an oscillator mode.
 Parameters:
 rho
Qobj
The density matrix (or ket) of the state to visualize.
 fock_numberslist of strings, optional
list of x ticklabels to represent fock numbers
 colorcolor or list of colors, default: “green”
The colors of the bar faces.
 unit_y_rangebool, default: True
Set yaxis limits [0, 1] or not
 figa matplotlib Figure instance, optional
The Figure canvas in which the plot will be drawn.
 axa matplotlib axes instance, optional
The axes context in which the plot will be drawn.
 rho
 Returns:
 fig, outputtuple
A tuple of the matplotlib figure and the axes instance or animation instance used to produce the figure.
 plot_qubism(ket, theme='light', how='pairs', grid_iteration=1, legend_iteration=0, *, fig=None, ax=None)[source]
Qubism plot for pure states of many qudits. Works best for spin chains, especially with even number of particles of the same dimension. Allows to see entanglement between first 2k particles and the rest.
Note
colorblind_safe does not apply because of its unique colormap
 Parameters:
 ketQobj
Pure state for plotting.
 themestr {‘light’, ‘dark’}, default: ‘light’
Set coloring theme for mapping complex values into colors. See: complex_array_to_rgb.
 howstr {‘pairs’, ‘pairs_skewed’ or ‘before_after’}, default: ‘pairs’
Type of Qubism plotting. Options:
‘pairs’  typical coordinates,
‘pairs_skewed’  for ferromagnetic/antriferromagnetic plots,
‘before_after’  related to Schmidt plot (see also: plot_schmidt).
 grid_iterationint, default: 1
Helper lines to be drawn on plot. Show tiles for 2*grid_iteration particles vs all others.
 legend_iterationint or ‘grid_iteration’ or ‘all’, default: 0
Show labels for first
2*legend_iteration
particles. Option ‘grid_iteration’ sets the same number of particles as for grid_iteration. Option ‘all’ makes label for all particles. Typically it should be 0, 1, 2 or perhaps 3. figa matplotlib figure instance, optional
The figure canvas on which the plot will be drawn.
 axa matplotlib axis instance, optional
The axis context in which the plot will be drawn.
 Returns:
 fig, outputtuple
A tuple of the matplotlib figure and the axes instance or animation instance used to produce the figure.
Notes
See also [1].
References
[1]J. RodriguezLaguna, P. Migdal, M. Ibanez Berganza, M. Lewenstein and G. Sierra, Qubism: selfsimilar visualization of manybody wavefunctions, New J. Phys. 14 053028, arXiv:1112.3560 (2012), open access.
 plot_schmidt(ket, theme='light', splitting=None, labels_iteration=(3, 2), *, fig=None, ax=None)[source]
Plotting scheme related to Schmidt decomposition. Converts a state into a matrix (A_ij > A_i^j), where rows are first particles and columns  last.
See also: plot_qubism with how=’before_after’ for a similar plot.
Note
colorblind_safe does not apply because of its unique colormap
 Parameters:
 ketQobj
Pure state for plotting.
 themestr {‘light’, ‘dark’}, default: ‘light’
Set coloring theme for mapping complex values into colors. See: complex_array_to_rgb.
 splittingint, optional
Plot for a number of first particles versus the rest. If not given, it is (number of particles + 1) // 2.
 labels_iterationint or pair of ints, default: (3, 2)
Number of particles to be shown as tick labels, for first (vertical) and last (horizontal) particles, respectively.
 figa matplotlib figure instance, optional
The figure canvas on which the plot will be drawn.
 axa matplotlib axis instance, optional
The axis context in which the plot will be drawn.
 Returns:
 fig, outputtuple
A tuple of the matplotlib figure and the axes instance or animation instance used to produce the figure.
 plot_spin_distribution(P, THETA, PHI, projection='2d', *, cmap=None, colorbar=False, fig=None, ax=None)[source]
Plots a spin distribution (given as meshgrid data).
 Parameters:
 Pmatrix
Distribution values as a meshgrid matrix.
 THETAmatrix
Meshgrid matrix for the theta coordinate. Its range is between 0 and pi
 PHImatrix
Meshgrid matrix for the phi coordinate. Its range is between 0 and 2*pi
 projection: str {‘2d’, ‘3d’}, default: ‘2d’
Specify whether the spin distribution function is to be plotted as a 2D projection where the surface of the unit sphere is mapped on the unit disk (‘2d’) or surface plot (‘3d’).
 cmapa matplotlib cmap instance, optional
The colormap.
 colorbarbool, default: False
Whether (True) or not (False) a colorbar should be attached to the Wigner function graph.
 figa matplotlib figure instance, optional
The figure canvas on which the plot will be drawn.
 axa matplotlib axis instance, optional
The axis context in which the plot will be drawn.
 Returns:
 fig, outputtuple
A tuple of the matplotlib figure and the axes instance or animation instance used to produce the figure.
 plot_wigner(rho, xvec=None, yvec=None, method='clenshaw', projection='2d', g=1.4142135623730951, sparse=False, parfor=False, *, cmap=None, colorbar=False, fig=None, ax=None)[source]
Plot the the Wigner function for a density matrix (or ket) that describes an oscillator mode.
 Parameters:
 rho
Qobj
The density matrix (or ket) of the state to visualize.
 xvecarray_like, optional
xcoordinates at which to calculate the Wigner function.
 yvecarray_like, optional
ycoordinates at which to calculate the Wigner function. Does not apply to the ‘fft’ method.
 methodstr {‘clenshaw’, ‘iterative’, ‘laguerre’, ‘fft’}, default: ‘clenshaw’
The method used for calculating the wigner function. See the documentation for qutip.wigner for details.
 projection: str {‘2d’, ‘3d’}, default: ‘2d’
Specify whether the Wigner function is to be plotted as a contour graph (‘2d’) or surface plot (‘3d’).
 gfloat
Scaling factor for a = 0.5 * g * (x + iy), default g = sqrt(2). See the documentation for qutip.wigner for details.
 sparsebool {False, True}
Flag for sparse format. See the documentation for qutip.wigner for details.
 parforbool {False, True}
Flag for parallel calculation. See the documentation for qutip.wigner for details.
 cmapa matplotlib cmap instance, optional
The colormap.
 colorbarbool, default: False
Whether (True) or not (False) a colorbar should be attached to the Wigner function graph.
 figa matplotlib Figure instance, optional
The Figure canvas in which the plot will be drawn.
 axa matplotlib axes instance, optional
The axes context in which the plot will be drawn.
 rho
 Returns:
 fig, outputtuple
A tuple of the matplotlib figure and the axes instance or animation instance used to produce the figure.
 plot_wigner_sphere(wigner, reflections=False, *, cmap=None, colorbar=True, fig=None, ax=None)[source]
Plots a coloured Bloch sphere.
 Parameters:
 wignera wigner transformation
The wigner transformation at steps different theta and phi.
 reflectionsbool, default: False
If the reflections of the sphere should be plotted as well.
 cmapa matplotlib colormap instance, optional
Color map to use when plotting.
 colorbarbool, default: True
Whether (True) or not (False) a colorbar should be attached.
 figa matplotlib Figure instance, optional
The Figure canvas in which the plot will be drawn.
 axa matplotlib axes instance, optional
The ax context in which the plot will be drawn.
 Returns:
 fig, outputtuple
A tuple of the matplotlib figure and the axes instance or animation instance used to produce the figure.
Notes
Special thanks to Russell P Rundle for writing this function.
 sphereplot(values, theta, phi, *, cmap=None, colorbar=True, fig=None, ax=None)[source]
Plots a matrix of values on a sphere
 Parameters:
 valuesarray
Data set to be plotted
 thetafloat
Angle with respect to zaxis. Its range is between 0 and pi
 phifloat
Angle in xy plane. Its range is between 0 and 2*pi
 cmapa matplotlib colormap instance, optional
Color map to use when plotting.
 colorbarbool, default: True
Whether (True) or not (False) a colorbar should be attached.
 figa matplotlib Figure instance, optional
The Figure canvas in which the plot will be drawn.
 axa matplotlib axes instance, optional
The axes context in which the plot will be drawn.
 Returns:
 fig, outputtuple
A tuple of the matplotlib figure and the axes instance or animation instance used to produce the figure.
Functions to animate results of quantum dynamics simulations,
 anim_fock_distribution(rhos, fock_numbers=None, color='green', unit_y_range=True, *, fig=None, ax=None)[source]
Animation of the Fock distribution for a density matrix (or ket) that describes an oscillator mode.
 Parameters:
 rhos
Result
or list ofQobj
The density matrix (or ket) of the state to visualize.
 fock_numberslist of strings, optional
list of x ticklabels to represent fock numbers
 colorcolor or list of colors, default: “green”
The colors of the bar faces.
 unit_y_rangebool, default: True
Set yaxis limits [0, 1] or not
 figa matplotlib Figure instance, optional
The Figure canvas in which the plot will be drawn.
 axa matplotlib axes instance, optional
The axes context in which the plot will be drawn.
 rhos
 Returns:
 fig, anituple
A tuple of the matplotlib figure and the animation instance used to produce the figure.
 anim_hinton(rhos, x_basis=None, y_basis=None, color_style='scaled', label_top=True, *, cmap=None, colorbar=True, fig=None, ax=None)[source]
Draws an animation of Hinton diagram.
 Parameters:
 rhos
Result
or list ofQobj
Input density matrix or superoperator.
Note
Hinton plots of superoperators are currently only supported for qubits.
 x_basislist of strings, optional
list of x ticklabels to represent x basis of the input.
 y_basislist of strings, optional
list of y ticklabels to represent y basis of the input.
 color_stylestr, {“scaled”, “threshold”, “phase”}, default: “scaled”
Determines how colors are assigned to each square:
If set to
"scaled"
(default), each color is chosen by passing the absolute value of the corresponding matrix element into cmap with the sign of the real part.If set to
"threshold"
, each square is plotted as the maximum of cmap for the positive real part and as the minimum for the negative part of the matrix element; note that this generalizes “threshold” to complex numbers.If set to
"phase"
, each color is chosen according to the angle of the corresponding matrix element.
 label_topbool, default: True
If True, x ticklabels will be placed on top, otherwise they will appear below the plot.
 cmapa matplotlib colormap instance, optional
Color map to use when plotting.
 colorbarbool, default: True
Whether (True) or not (False) a colorbar should be attached.
 figa matplotlib Figure instance, optional
The Figure canvas in which the plot will be drawn.
 axa matplotlib axes instance, optional
The ax context in which the plot will be drawn.
 rhos
 Returns:
 fig, anituple
A tuple of the matplotlib figure and the animation instance used to produce the figure.
 Raises:
 ValueError
Input argument is not a quantum object.
 anim_matrix_histogram(Ms, x_basis=None, y_basis=None, limits=None, bar_style='real', color_limits=None, color_style='real', options=None, *, cmap=None, colorbar=True, fig=None, ax=None)[source]
Draw an animation of a histogram for the matrix M, with the given x and y labels.
 Parameters:
 Mslist of matrices or
Result
The matrix to visualize
 x_basislist of strings, optional
list of x ticklabels
 y_basislist of strings, optional
list of y ticklabels
 limitslist/array with two float numbers, optional
The zaxis limits [min, max]
 bar_stylestr, {“real”, “img”, “abs”, “phase”}, default: “real”
If set to
"real"
(default), each bar is plotted as the real part of the corresponding matrix elementIf set to
"img"
, each bar is plotted as the imaginary part of the corresponding matrix elementIf set to
"abs"
, each bar is plotted as the absolute value of the corresponding matrix elementIf set to
"phase"
(default), each bar is plotted as the angle of the corresponding matrix element
 color_limitslist/array with two float numbers, optional
The limits of colorbar [min, max]
 color_stylestr, {“real”, “img”, “abs”, “phase”}, default: “real”
Determines how colors are assigned to each square:
If set to
"real"
(default), each color is chosen according to the real part of the corresponding matrix element.If set to
"img"
, each color is chosen according to the imaginary part of the corresponding matrix element.If set to
"abs"
, each color is chosen according to the absolute value of the corresponding matrix element.If set to
"phase"
, each color is chosen according to the angle of the corresponding matrix element.
 cmapa matplotlib colormap instance, optional
Color map to use when plotting.
 colorbarbool, default: True
show colorbar
 figa matplotlib Figure instance, optional
The Figure canvas in which the plot will be drawn.
 axa matplotlib axes instance, optional
The axes context in which the plot will be drawn.
 optionsdict, optional
A dictionary containing extra options for the plot. The names (keys) and values of the options are described below:
 ‘zticks’list of numbers, optional
A list of zaxis tick locations.
 ‘bars_spacing’float, default: 0.1
spacing between bars.
 ‘bars_alpha’float, default: 1.
transparency of bars, should be in range 0  1
 ‘bars_lw’float, default: 0.5
linewidth of bars’ edges.
 ‘bars_edgecolor’color, default: ‘k’
The colors of the bars’ edges. Examples: ‘k’, (0.1, 0.2, 0.5) or ‘#0f0f0f80’.
 ‘shade’bool, default: True
Whether to shade the dark sides of the bars (True) or not (False). The shading is relative to plot’s source of light.
 ‘azim’float, default: 35
The azimuthal viewing angle.
 ‘elev’float, default: 35
The elevation viewing angle.
 ‘stick’bool, default: False
Changes xlim and ylim in such a way that bars next to XZ and YZ planes will stick to those planes. This option has no effect if
ax
is passed as a parameter. ‘cbar_pad’float, default: 0.04
The fraction of the original axes between the colorbar and the new image axes. (i.e. the padding between the 3D figure and the colorbar).
 ‘cbar_to_z’bool, default: False
Whether to set the color of maximum and minimum zvalues to the maximum and minimum colors in the colorbar (True) or not (False).
 ‘threshold’: float, optional
Threshold for when bars of smaller height should be transparent. If not set, all bars are colored according to the color map.
 Mslist of matrices or
 Returns:
 fig, anituple
A tuple of the matplotlib figure and the animation instance used to produce the figure.
 Raises:
 ValueError
Input argument is not valid.
 anim_qubism(kets, theme='light', how='pairs', grid_iteration=1, legend_iteration=0, *, fig=None, ax=None)[source]
Animation of Qubism plot for pure states of many qudits. Works best for spin chains, especially with even number of particles of the same dimension. Allows to see entanglement between first 2k particles and the rest.
Note
colorblind_safe does not apply because of its unique colormap
 Parameters:
 kets
Result
or list ofQobj
Pure states for animation.
 themestr {‘light’, ‘dark’}, default: ‘light’
Set coloring theme for mapping complex values into colors. See: complex_array_to_rgb.
 howstr {‘pairs’, ‘pairs_skewed’, ‘before_after’}, default: ‘pairs’
Type of Qubism plotting. Options:
‘pairs’  typical coordinates,
‘pairs_skewed’  for ferromagnetic/antriferromagnetic plots,
‘before_after’  related to Schmidt plot (see also: plot_schmidt).
 grid_iterationint, default: 1
Helper lines to be drawn on plot. Show tiles for 2*grid_iteration particles vs all others.
 legend_iterationint or ‘grid_iteration’ or ‘all’, default: 0
Show labels for first
2*legend_iteration
particles. Option ‘grid_iteration’ sets the same number of particles as for grid_iteration. Option ‘all’ makes label for all particles. Typically it should be 0, 1, 2 or perhaps 3. figa matplotlib figure instance, optional
The figure canvas on which the plot will be drawn.
 axa matplotlib axis instance, optional
The axis context in which the plot will be drawn.
 kets
 Returns:
 fig, anituple
A tuple of the matplotlib figure and the animation instance used to produce the figure.
Notes
See also [1].
References
[1]J. RodriguezLaguna, P. Migdal, M. Ibanez Berganza, M. Lewenstein and G. Sierra, Qubism: selfsimilar visualization of manybody wavefunctions, New J. Phys. 14 053028, arXiv:1112.3560 (2012), open access.
 anim_schmidt(kets, theme='light', splitting=None, labels_iteration=(3, 2), *, fig=None, ax=None)[source]
Animation of Schmidt decomposition. Converts a state into a matrix (A_ij > A_i^j), where rows are first particles and columns  last.
See also: plot_qubism with how=’before_after’ for a similar plot.
Note
colorblind_safe does not apply because of its unique colormap
 Parameters:
 ket
Result
or list ofQobj
Pure states for animation.
 themestr {‘light’, ‘dark’}, default: ‘light’
Set coloring theme for mapping complex values into colors. See: complex_array_to_rgb.
 splittingint, optional
Plot for a number of first particles versus the rest. If not given, it is (number of particles + 1) // 2.
 labels_iterationint or pair of ints, default: (3, 2)
Number of particles to be shown as tick labels, for first (vertical) and last (horizontal) particles, respectively.
 figa matplotlib figure instance, optional
The figure canvas on which the plot will be drawn.
 axa matplotlib axis instance, optional
The axis context in which the plot will be drawn.
 ket
 Returns:
 fig, anituple
A tuple of the matplotlib figure and the animation instance used to produce the figure.
 anim_sphereplot(V, theta, phi, *, cmap=None, colorbar=True, fig=None, ax=None)[source]
animation of a matrices of values on a sphere
 Parameters:
 Vlist of array instances
Data set to be plotted
 thetafloat
Angle with respect to zaxis. Its range is between 0 and pi
 phifloat
Angle in xy plane. Its range is between 0 and 2*pi
 cmapa matplotlib colormap instance, optional
Color map to use when plotting.
 colorbarbool, default: True
Whether (True) or not (False) a colorbar should be attached.
 figa matplotlib Figure instance, optional
The Figure canvas in which the plot will be drawn.
 axa matplotlib axes instance, optional
The axes context in which the plot will be drawn.
 Returns:
 fig, anituple
A tuple of the matplotlib figure and the animation instance used to produce the figure.
 anim_spin_distribution(Ps, THETA, PHI, projection='2d', *, cmap=None, colorbar=False, fig=None, ax=None)[source]
Animation of a spin distribution (given as meshgrid data).
 Parameters:
 Pslist of matrices
Distribution values as a meshgrid matrix.
 THETAmatrix
Meshgrid matrix for the theta coordinate. Its range is between 0 and pi
 PHImatrix
Meshgrid matrix for the phi coordinate. Its range is between 0 and 2*pi
 projection: str {‘2d’, ‘3d’}, default: ‘2d’
Specify whether the spin distribution function is to be plotted as a 2D projection where the surface of the unit sphere is mapped on the unit disk (‘2d’) or surface plot (‘3d’).
 cmapa matplotlib cmap instance, optional
The colormap.
 colorbarbool, default: False
Whether (True) or not (False) a colorbar should be attached to the Wigner function graph.
 figa matplotlib figure instance, optional
The figure canvas on which the plot will be drawn.
 axa matplotlib axis instance, optional
The axis context in which the plot will be drawn.
 Returns:
 fig, anituple
A tuple of the matplotlib figure and the animation instance used to produce the figure.
 anim_wigner(rhos, xvec=None, yvec=None, method='clenshaw', projection='2d', g=1.4142135623730951, sparse=False, parfor=False, *, cmap=None, colorbar=False, fig=None, ax=None)[source]
Animation of the Wigner function for a density matrix (or ket) that describes an oscillator mode.
 Parameters:
 rhos
Result
or list ofQobj
The density matrix (or ket) of the state to visualize.
 xvecarray_like, optional
xcoordinates at which to calculate the Wigner function.
 yvecarray_like, optional
ycoordinates at which to calculate the Wigner function. Does not apply to the ‘fft’ method.
 methodstr {‘clenshaw’, ‘iterative’, ‘laguerre’, ‘fft’}, default: ‘clenshaw’
The method used for calculating the wigner function. See the documentation for qutip.wigner for details.
 projection: str {‘2d’, ‘3d’}, default: ‘2d’
Specify whether the Wigner function is to be plotted as a contour graph (‘2d’) or surface plot (‘3d’).
 gfloat
Scaling factor for a = 0.5 * g * (x + iy), default g = sqrt(2). See the documentation for qutip.wigner for details.
 sparsebool {False, True}
Flag for sparse format. See the documentation for qutip.wigner for details.
 parforbool {False, True}
Flag for parallel calculation. See the documentation for qutip.wigner for details.
 cmapa matplotlib cmap instance, optional
The colormap.
 colorbarbool, default: False
Whether (True) or not (False) a colorbar should be attached to the Wigner function graph.
 figa matplotlib Figure instance, optional
The Figure canvas in which the plot will be drawn.
 axa matplotlib axes instance, optional
The axes context in which the plot will be drawn.
 rhos
 Returns:
 fig, anituple
A tuple of the matplotlib figure and the animation instance used to produce the figure.
 anim_wigner_sphere(wigners, reflections=False, *, cmap=None, colorbar=True, fig=None, ax=None)[source]
Animate a coloured Bloch sphere.
 Parameters:
 wignerslist of transformations
The wigner transformation at steps different theta and phi.
 reflectionsbool, default: False
If the reflections of the sphere should be plotted as well.
 cmapa matplotlib colormap instance, optional
Color map to use when plotting.
 colorbarbool, default: True
Whether (True) or not (False) a colorbar should be attached.
 figa matplotlib Figure instance, optional
The Figure canvas in which the plot will be drawn.
 axa matplotlib axes instance, optional
The ax context in which the plot will be drawn.
 Returns:
 fig, anituple
A tuple of the matplotlib figure and the animation instance used to produce the figure.
Notes
Special thanks to Russell P Rundle for writing this function.
This module contains utility functions that enhance Matplotlib in one way or another.
 complex_phase_cmap()[source]
Create a cyclic colormap for representing the phase of complex variables
 Returns:
 cmap
A matplotlib linear segmented colormap.
 wigner_cmap(W, levels=1024, shift=0, max_color='#09224F', mid_color='#FFFFFF', min_color='#530017', neg_color='#FF97D4', invert=False)[source]
A custom colormap that emphasizes negative values by creating a nonlinear colormap.
 Parameters:
 Warray
Wigner function array, or any array.
 levelsint, default: 1024
Number of color levels to create.
 shiftfloat, default: 0
Shifts the value at which Wigner elements are emphasized. This parameter should typically be negative and small (i.e 1e5).
 max_colorstr, default: ‘#09224F’
String for color corresponding to maximum value of data. Accepts any string format compatible with the Matplotlib.colors.ColorConverter.
 mid_colorstr, default: ‘#FFFFFF’
Color corresponding to zero values. Accepts any string format compatible with the Matplotlib.colors.ColorConverter.
 min_colorstr, default: ‘#530017’
Color corresponding to minimum data values. Accepts any string format compatible with the Matplotlib.colors.ColorConverter.
 neg_colorstr, default: ‘#FF97D4’
Color that starts highlighting negative values. Accepts any string format compatible with the Matplotlib.colors.ColorConverter.
 invertbool, default: False
Invert the color scheme for negative values so that smaller negative values have darker color.
 Returns:
 Returns a Matplotlib colormap instance for use in plotting.
Notes
The ‘shift’ parameter allows you to vary where the colormap begins to highlight negative colors. This is beneficial in cases where there are small negative Wigner elements due to numerical roundoff and/or truncation.
Quantum Process Tomography
 qpt(U, op_basis_list)[source]
Calculate the quantum process tomography chi matrix for a given (possibly nonunitary) transformation matrix U, which transforms a density matrix in vector form according to:
vec(rho) = U * vec(rho0)
or
rho = unstack_columns(U * stack_columns(rho0))
U can be calculated for an open quantum system using the QuTiP propagator function.
 Parameters:
 UQobj
Transformation operator. Can be calculated using QuTiP propagator function.
 op_basis_listlist
A list of Qobj’s representing the basis states.
 Returns:
 chiarray
QPT chi matrix
 qpt_plot(chi, lbls_list, title=None, fig=None, axes=None)[source]
Visualize the quantum process tomography chi matrix. Plot the real and imaginary parts separately.
 Parameters:
 chiarray
Input QPT chi matrix.
 lbls_listlist
List of labels for QPT plot axes.
 titlestr, optional
Plot title.
 figfigure instance, optional
User defined figure instance used for generating QPT plot.
 axeslist of figure axis instance, optional
User defined figure axis instance (list of two axes) used for generating QPT plot.
 Returns:
 fig, axtuple
A tuple of the matplotlib figure and axes instances used to produce the figure.
 qpt_plot_combined(chi, lbls_list, title=None, fig=None, ax=None, figsize=(8, 6), threshold=None)[source]
Visualize the quantum process tomography chi matrix. Plot bars with height and color corresponding to the absolute value and phase, respectively.
 Parameters:
 chiarray
Input QPT chi matrix.
 lbls_listlist
List of labels for QPT plot axes.
 titlestr, optional
Plot title.
 figfigure instance, optional
User defined figure instance used for generating QPT plot.
 figsize(int, int), default: (8, 6)
Size of the figure when the
fig
is not provided. axfigure axis instance, optional
User defined figure axis instance used for generating QPT plot (alternative to the fig argument).
 threshold: float, optional
Threshold for when bars of smaller height should be transparent. If not set, all bars are colored according to the color map.
 Returns:
 fig, axtuple
A tuple of the matplotlib figure and axes instances used to produce the figure.
NonMarkovian Solvers
This module contains an implementation of the nonMarkovian transfer tensor method (TTM), introduced in [1].
[1] Javier Cerrillo and Jianshu Cao, Phys. Rev. Lett 112, 110401 (2014)
 ttmsolve(dynmaps, state0, times, e_ops=(), num_learning=0, options=None)[source]
Expand timeevolution using the Transfer Tensor Method [1], based on a set of precomputed dynamical maps.
 Parameters:
 dynmapslist of
Qobj
, callable List of precomputed dynamical maps (superoperators) for the first times of
times
or a callback function that returns the superoperator at a given time. state0
Qobj
Initial density matrix or state vector (ket).
 timesarray_like
List of times \(t_n\) at which to compute results. Must be uniformily spaced.
 e_ops
Qobj
, callable, or list, optional Single operator or list of operators for which to evaluate expectation values or callable or list of callable. Callable signature must be, f(t: float, state: Qobj). See
expect
for more detail of operator expectation. num_learningint, default: 0
Number of times used to construct the dynmaps operators when
dynmaps
is a callable. optionsdictionary, optional
Dictionary of options for the solver.
store_final_state : bool Whether or not to store the final state of the evolution in the result class.
store_states : bool, None Whether or not to store the state vectors or density matrices. On None the states will be saved if no expectation operators are given.
normalize_output : bool Normalize output state to hide ODE numerical errors.
threshold : float Threshold for halting. Halts if \(T_{n}T_{n1}\) is below treshold.
 dynmapslist of
 Returns:

[1]
Javier Cerrillo and Jianshu Cao, Phys. Rev. Lett 112, 110401 (2014) ..
Utility Functions
Utility Functions
This module contains utility functions that are commonly needed in other qutip modules.
 clebsch(j1, j2, j3, m1, m2, m3)[source]
Calculates the ClebschGordon coefficient for coupling (j1,m1) and (j2,m2) to give (j3,m3).
 Parameters:
 j1float
Total angular momentum 1.
 j2float
Total angular momentum 2.
 j3float
Total angular momentum 3.
 m1float
zcomponent of angular momentum 1.
 m2float
zcomponent of angular momentum 2.
 m3float
zcomponent of angular momentum 3.
 Returns:
 cg_coefffloat
Requested ClebschGordan coefficient.
 convert_unit(value, orig='meV', to='GHz')[source]
Convert an energy from unit orig to unit to.
 Parameters:
 valuefloat / array
The energy in the old unit.
 origstr, {“J”, “eV”, “meV”, “GHz”, “mK”}, default: “meV”
The name of the original unit.
 tostr, {“J”, “eV”, “meV”, “GHz”, “mK”}, default: “GHz”
The name of the new unit.
 Returns:
 value_new_unitfloat / array
The energy in the new unit.
 n_thermal(w, w_th)[source]
Return the number of photons in thermal equilibrium for an harmonic oscillator mode with frequency ‘w’, at the temperature described by ‘w_th’ where \(\omega_{\rm th} = k_BT/\hbar\).
 Parameters:
 wfloat or ndarray
Frequency of the oscillator.
 w_thfloat
The temperature in units of frequency (or the same units as w).
 Returns:
 n_avgfloat or array
Return the number of average photons in thermal equilibrium for a an oscillator with the given frequency and temperature.
File I/O Functions
 file_data_read(filename, sep=None)[source]
Retrieves an array of data from the requested file.
 Parameters:
 filenamestr or pathlib.Path
Name of file containing reqested data.
 sepstr, optional
Seperator used to store data.
 Returns:
 dataarray_like
Data from selected file.
 file_data_store(filename, data, numtype='complex', numformat='decimal', sep=',')[source]
Stores a matrix of data to a file to be read by an external program.
 Parameters:
 filenamestr or pathlib.Path
Name of data file to be stored, including extension.
 data: array_like
Data to be written to file.
 numtypestr {‘complex, ‘real’}, default: ‘complex’
Type of numerical data.
 numformatstr {‘decimal’,’exp’}, default: ‘decimal’
Format for written data.
 sepstr, default: ‘,’
Singlecharacter field seperator. Usually a tab, space, comma, or semicolon.
Parallelization
This module provides functions for parallel execution of loops and function mappings, using the builtin Python module multiprocessing or the loky parallel execution library.
 loky_pmap(task, values, task_args=None, task_kwargs=None, reduce_func=None, map_kw=None, progress_bar=None, progress_bar_kwargs={})[source]
Parallel execution of a mapping of
values
to the functiontask
. This is functionally equivalent to:result = [task(value, *task_args, **task_kwargs) for value in values]
Use the loky module instead of multiprocessing.
 Parameters:
 taska Python function
The function that is to be called for each value in
task_vec
. valuesarray / list
The list or array of values for which the
task
function is to be evaluated. task_argslist, optional
The optional additional arguments to the
task
function. task_kwargsdictionary, optional
The optional additional keyword arguments to the
task
function. reduce_funcfunc, optional
If provided, it will be called with the output of each task instead of storing them in a list. Note that the order in which results are passed to
reduce_func
is not defined. It should return None or a number. When returning a number, it represents the estimation of the number of tasks left. On a return <= 0, the map will end early. progress_barstr, optional
Progress bar options’s string for showing progress.
 progress_bar_kwargsdict, optional
Options for the progress bar.
 map_kw: dict, optional
Dictionary containing entry for:  timeout: float, Maximum time (sec) for the whole map.  num_cpus: int, Number of jobs to run at once.  fail_fast: bool, Abort at the first error.
 Returns:
 resultlist
The result list contains the value of
task(value, *task_args, **task_kwargs)
for each value invalues
. If areduce_func
is provided, and empty list will be returned.
 mpi_pmap(task, values, task_args=None, task_kwargs=None, reduce_func=None, map_kw=None, progress_bar=None, progress_bar_kwargs={})[source]
Parallel execution of a mapping of
values
to the functiontask
. This is functionally equivalent to:result = [task(value, *task_args, **task_kwargs) for value in values]
Uses the mpi4py module to execute the tasks asynchronously with MPI processes. For more information, consult the documentation of mpi4py and the mpi4py.MPIPoolExecutor class.
Note: in keeping consistent with the API of parallel_map, the parameter determining the number of requested worker processes is called num_cpus. The value of map_kw[‘num_cpus’] is passed to the MPIPoolExecutor as its max_workers argument. If this parameter is not provided, the environment variable QUTIP_NUM_PROCESSES is used instead. If this environment variable is not set either, QuTiP will use default values that might be unsuitable for MPI applications.
 Parameters:
 taska Python function
The function that is to be called for each value in
task_vec
. valuesarray / list
The list or array of values for which the
task
function is to be evaluated. task_argslist, optional
The optional additional arguments to the
task
function. task_kwargsdictionary, optional
The optional additional keyword arguments to the
task
function. reduce_funcfunc, optional
If provided, it will be called with the output of each task instead of storing them in a list. Note that the order in which results are passed to
reduce_func
is not defined. It should return None or a number. When returning a number, it represents the estimation of the number of tasks left. On a return <= 0, the map will end early. progress_barstr, optional
Progress bar options’s string for showing progress.
 progress_bar_kwargsdict, optional
Options for the progress bar.
 map_kw: dict, optional
Dictionary containing entry for:  timeout: float, Maximum time (sec) for the whole map.  num_cpus: int, Number of jobs to run at once.  fail_fast: bool, Abort at the first error. All remaining entries of map_kw will be passed to the mpi4py.MPIPoolExecutor constructor.
 Returns:
 resultlist
The result list contains the value of
task(value, *task_args, **task_kwargs)
for each value invalues
. If areduce_func
is provided, and empty list will be returned.
 parallel_map(task, values, task_args=None, task_kwargs=None, reduce_func=None, map_kw=None, progress_bar=None, progress_bar_kwargs={})[source]
Parallel execution of a mapping of
values
to the functiontask
. This is functionally equivalent to:result = [task(value, *task_args, **task_kwargs) for value in values]
 Parameters:
 taska Python function
The function that is to be called for each value in
task_vec
. valuesarray / list
The list or array of values for which the
task
function is to be evaluated. task_argslist, optional
The optional additional arguments to the
task
function. task_kwargsdictionary, optional
The optional additional keyword arguments to the
task
function. reduce_funcfunc, optional
If provided, it will be called with the output of each task instead of storing them in a list. Note that the order in which results are passed to
reduce_func
is not defined. It should return None or a number. When returning a number, it represents the estimation of the number of tasks left. On a return <= 0, the map will end early. progress_barstr, optional
Progress bar options’s string for showing progress.
 progress_bar_kwargsdict, optional
Options for the progress bar.
 map_kw: dict, optional
Dictionary containing entry for:  timeout: float, Maximum time (sec) for the whole map.  num_cpus: int, Number of jobs to run at once.  fail_fast: bool, Abort at the first error.
 Returns:
 resultlist
The result list contains the value of
task(value, *task_args, **task_kwargs)
for each value invalues
. If areduce_func
is provided, and empty list will be returned.
 serial_map(task, values, task_args=None, task_kwargs=None, reduce_func=None, map_kw=None, progress_bar=None, progress_bar_kwargs={})[source]
Serial mapping function with the same call signature as parallel_map, for easy switching between serial and parallel execution. This is functionally equivalent to:
result = [task(value, *task_args, **task_kwargs) for value in values]
This function work as a dropin replacement of
parallel_map
. Parameters:
 taska Python function
The function that is to be called for each value in
task_vec
. valuesarray / list
The list or array of values for which the
task
function is to be evaluated. task_argslist, optional
The optional additional argument to the
task
function. task_kwargsdictionary, optional
The optional additional keyword argument to the
task
function. reduce_funcfunc, optional
If provided, it will be called with the output of each tasks instead of storing a them in a list. It should return None or a number. When returning a number, it represent the estimation of the number of task left. On a return <= 0, the map will end early.
 progress_barstr, optional
Progress bar options’s string for showing progress.
 progress_bar_kwargsdict, optional
Options for the progress bar.
 map_kw: dict, optional
Dictionary containing:  timeout: float, Maximum time (sec) for the whole map.  fail_fast: bool, Raise an error at the first.
 Returns:
 resultlist
The result list contains the value of
task(value, *task_args, **task_kwargs)
for each value invalues
. If areduce_func
is provided, and empty list will be returned.
IPython Notebook Tools
This module contains utility functions for using QuTiP with IPython notebooks.
 version_table(verbose=False)[source]
Print an HTMLformatted table with version numbers for QuTiP and its dependencies. Use it in a IPython notebook to show which versions of different packages that were used to run the notebook. This should make it possible to reproduce the environment and the calculation later on.
 Parameters:
 verbosebool, default: False
Add extra information about install location.
 Returns:
 version_table: str
Return an HTMLformatted string containing version information for QuTiP dependencies.
Miscellaneous
 about()[source]
About box for QuTiP. Gives version numbers for QuTiP, NumPy, SciPy, Cython, and MatPlotLib.
 simdiag(ops, evals: bool = True, *, tol: float = 1e14, safe_mode: bool = True)[source]
Simultaneous diagonalization of commuting Hermitian matrices.
 Parameters:
 opslist, array
list
orarray
of qobjs representing commuting Hermitian operators. evalsbool, default: True
Whether to return the eigenvalues for each ops and eigenvectors or just the eigenvectors.
 tolfloat, default: 1e14
Tolerance for detecting degenerate eigenstates.
 safe_modebool, default: True
Whether to check that all ops are Hermitian and commuting. If set to
False
and operators are not commuting, the eigenvectors returned will often be eigenvectors of only the first operator.
 Returns:
 eigstuple
Tuple of arrays representing eigvecs and eigvals of quantum objects corresponding to simultaneous eigenvectors and eigenvalues for each operator.