# Source code for qutip.entropy

```__all__ = ['entropy_vn', 'entropy_linear', 'entropy_mutual', 'negativity',
'concurrence', 'entropy_conditional', 'entangling_power',
'entropy_relative']

from numpy import conj, e, inf, imag, inner, real, sort, sqrt
from numpy.lib.scimath import log, log2
from qutip.qobj import ptrace
from qutip.states import ket2dm
from qutip.tensor import tensor
from qutip.operators import sigmay
from qutip.sparse import sp_eigs
from qutip.partial_transpose import partial_transpose

[docs]def entropy_vn(rho, base=e, sparse=False):
"""
Von-Neumann entropy of density matrix

Parameters
----------
rho : qobj
Density matrix.
base : {e,2}
Base of logarithm.
sparse : {False,True}
Use sparse eigensolver.

Returns
-------
entropy : float
Von-Neumann entropy of `rho`.

Examples
--------
>>> rho=0.5*fock_dm(2,0)+0.5*fock_dm(2,1)
>>> entropy_vn(rho,2)
1.0

"""
if rho.type == 'ket' or rho.type == 'bra':
rho = ket2dm(rho)
vals = sp_eigs(rho.data, rho.isherm, vecs=False, sparse=sparse)
nzvals = vals[vals != 0]
if base == 2:
logvals = log2(nzvals)
elif base == e:
logvals = log(nzvals)
else:
raise ValueError("Base must be 2 or e.")
return float(real(-sum(nzvals * logvals)))

[docs]def entropy_linear(rho):
"""
Linear entropy of a density matrix.

Parameters
----------
rho : qobj
sensity matrix or ket/bra vector.

Returns
-------
entropy : float
Linear entropy of rho.

Examples
--------
>>> rho=0.5*fock_dm(2,0)+0.5*fock_dm(2,1)
>>> entropy_linear(rho)
0.5

"""
if rho.type == 'ket' or rho.type == 'bra':
rho = ket2dm(rho)
return float(real(1.0 - (rho ** 2).tr()))

[docs]def concurrence(rho):
"""
Calculate the concurrence entanglement measure for a two-qubit state.

Parameters
----------
state : qobj
Ket, bra, or density matrix for a two-qubit state.

Returns
-------
concur : float
Concurrence

References
----------

.. [1] https://en.wikipedia.org/wiki/Concurrence_(quantum_computing)

"""
if rho.isket and rho.dims != [[2, 2], [1, 1]]:
raise Exception("Ket must be tensor product of two qubits.")

elif rho.isbra and rho.dims != [[1, 1], [2, 2]]:
raise Exception("Bra must be tensor product of two qubits.")

elif rho.isoper and rho.dims != [[2, 2], [2, 2]]:
raise Exception("Density matrix must be tensor product of two qubits.")

if rho.isket or rho.isbra:
rho = ket2dm(rho)

sysy = tensor(sigmay(), sigmay())

rho_tilde = (rho * sysy) * (rho.conj() * sysy)

evals = rho_tilde.eigenenergies()

# abs to avoid problems with sqrt for very small negative numbers
evals = abs(sort(real(evals)))

lsum = sqrt(evals[3]) - sqrt(evals[2]) - sqrt(evals[1]) - sqrt(evals[0])

return max(0, lsum)

def negativity(rho, subsys, method='tracenorm', logarithmic=False):
"""
Compute the negativity for a multipartite quantum system described
by the density matrix rho. The subsys argument is an index that
indicates which system to compute the negativity for.

.. note::

Experimental.
"""
mask = [idx == subsys for idx, n in enumerate(rho.dims[0])]

if method == 'tracenorm':
N = ((rho_pt.dag() * rho_pt).sqrtm().tr().real - 1)/2.0
elif method == 'eigenvalues':
l = rho_pt.eigenenergies()
N = ((abs(l)-l)/2).sum()
else:
raise ValueError("Unknown method %s" % method)

if logarithmic:
return log2(2 * N + 1)
else:
return N

[docs]def entropy_mutual(rho, selA, selB, base=e, sparse=False):
"""
Calculates the mutual information S(A:B) between selection
components of a system density matrix.

Parameters
----------
rho : qobj
Density matrix for composite quantum systems
selA : int/list
`int` or `list` of first selected density matrix components.
selB : int/list
`int` or `list` of second selected density matrix components.
base : {e,2}
Base of logarithm.
sparse : {False,True}
Use sparse eigensolver.

Returns
-------
ent_mut : float
Mutual information between selected components.

"""
if isinstance(selA, int):
selA = [selA]
if isinstance(selB, int):
selB = [selB]
if rho.type != 'oper':
raise TypeError("Input must be a density matrix.")
if (len(selA) + len(selB)) != len(rho.dims[0]):
raise TypeError("Number of selected components must match " +
"total number.")

rhoA = ptrace(rho, selA)
rhoB = ptrace(rho, selB)
out = (entropy_vn(rhoA, base, sparse=sparse) +
entropy_vn(rhoB, base, sparse=sparse) -
entropy_vn(rho, base, sparse=sparse))
return out

[docs]def entropy_relative(rho, sigma, base=e, sparse=False, tol=1e-12):
"""
Calculates the relative entropy S(rho||sigma) between two density
matrices.

Parameters
----------
rho : :class:`qutip.Qobj`
First density matrix (or ket which will be converted to a density
matrix).
sigma : :class:`qutip.Qobj`
Second density matrix (or ket which will be converted to a density
matrix).
base : {e,2}
Base of logarithm. Defaults to e.
sparse : bool
Flag to use sparse solver when determining the eigenvectors
of the density matrices. Defaults to False.
tol : float
Tolerance to use to detect 0 eigenvalues or dot producted between
eigenvectors. Defaults to 1e-12.

Returns
-------
rel_ent : float
Value of relative entropy. Guaranteed to be greater than zero
and should equal zero only when rho and sigma are identical.

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

References
----------

See Nielsen & Chuang, "Quantum Computation and Quantum Information",
Section 11.3.1, pg. 511 for a detailed explanation of quantum relative
entropy.
"""
if rho.isket:
rho = ket2dm(rho)
if sigma.isket:
sigma = ket2dm(sigma)
if not rho.isoper or not sigma.isoper:
raise TypeError("Inputs must be density matrices.")
if rho.dims != sigma.dims:
raise ValueError("Inputs must have the same shape and dims.")
if base == 2:
log_base = log2
elif base == e:
log_base = log
else:
raise ValueError("Base must be 2 or e.")
# S(rho || sigma) = sum_i(p_i log p_i) - sum_ij(p_i P_ij log q_i)
#
# S is +inf if the kernel of sigma (i.e. svecs[svals == 0]) has non-trivial
# intersection with the support of rho (i.e. rvecs[rvals != 0]).
rvals, rvecs = sp_eigs(rho.data, rho.isherm, vecs=True, sparse=sparse)
if any(abs(imag(rvals)) >= tol):
raise ValueError("Input rho has non-real eigenvalues.")
rvals = real(rvals)
svals, svecs = sp_eigs(sigma.data, sigma.isherm, vecs=True, sparse=sparse)
if any(abs(imag(svals)) >= tol):
raise ValueError("Input sigma has non-real eigenvalues.")
svals = real(svals)
# Calculate inner products of eigenvectors and return +inf if kernel
# of sigma overlaps with support of rho.
P = abs(inner(rvecs, conj(svecs))) ** 2
if (rvals >= tol) @ (P >= tol) @ (svals < tol):
return inf
# Avoid -inf from log(0) -- these terms will be multiplied by zero later
# anyway
svals[abs(svals) < tol] = 1
nzrvals = rvals[abs(rvals) >= tol]
# Calculate S
S = nzrvals @ log_base(nzrvals) - rvals @ P @ log_base(svals)
# the relative entropy is guaranteed to be >= 0, so we clamp the
# calculated value to 0 to avoid small violations of the lower bound.
return max(0, S)

[docs]def entropy_conditional(rho, selB, base=e, sparse=False):
"""
Calculates the conditional entropy :math:`S(A|B)=S(A,B)-S(B)`
of a selected density matrix component.

Parameters
----------
rho : qobj
Density matrix of composite object
selB : int/list
Selected components for density matrix B
base : {e,2}
Base of logarithm.
sparse : {False,True}
Use sparse eigensolver.

Returns
-------
ent_cond : float
Value of conditional entropy

"""
if rho.type != 'oper':
raise TypeError("Input must be density matrix.")
if isinstance(selB, int):
selB = [selB]
B = ptrace(rho, selB)
out = (entropy_vn(rho, base, sparse=sparse) -
entropy_vn(B, base, sparse=sparse))
return out

def participation_ratio(rho):
"""
Returns the effective number of states for a density matrix.

The participation is unity for pure states, and maximally N,
where N is the Hilbert space dimensionality, for completely
mixed states.

Parameters
----------
rho : qobj
Density matrix

Returns
-------
pr : float
Effective number of states in the density matrix

"""
if rho.type == 'ket' or rho.type == 'bra':
return 1.0
else:
return 1.0 / (rho ** 2).tr()

def entangling_power(U):
"""
Calculate the entangling power of a two-qubit gate U, which
is zero of nonentangling gates and 1 and 2/9 for maximally
entangling gates.

Parameters
----------
U : qobj
Qobj instance representing a two-qubit gate.

Returns
-------
ep : float
The entanglement power of U (real number between 0 and 1)

References:

Explorations in Quantum Computing, Colin P. Williams (Springer, 2011)
"""

if not U.isoper:
raise Exception("U must be an operator.")

if U.dims != [[2, 2], [2, 2]]:
raise Exception("U must be a two-qubit gate.")

from qutip.qip.operations.gates import swap
a = (tensor(U, U).dag() * swap(N=4, targets=[1, 3]) *
tensor(U, U) * swap(N=4, targets=[1, 3]))
b = (tensor(swap() * U, swap() * U).dag() * swap(N=4, targets=[1, 3]) *
tensor(swap() * U, swap() * U) * swap(N=4, targets=[1, 3]))

return 5.0/9 - 1.0/36 * (a.tr() + b.tr()).real
```