"""
This module contains a collection functions for calculating continuous variable
quantities from fock-basis representation of the state of multi-mode fields.
"""
__all__ = ['correlation_matrix', 'covariance_matrix',
'correlation_matrix_field', 'correlation_matrix_quadrature',
'wigner_covariance_matrix', 'logarithmic_negativity']
from . import expect
import numpy as np
[docs]def correlation_matrix(basis, rho=None):
r"""
Given a basis set of operators :math:`\{a\}_n`, calculate the correlation
matrix:
.. math::
C_{mn} = \langle a_m a_n \rangle
Parameters
----------
basis : list
List of operators that defines the basis for the correlation matrix.
rho : Qobj, 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_mat : ndarray
A 2-dimensional *array* of correlation values or operators.
"""
if rho is None:
# return array of operators
out = np.empty((len(basis), len(basis)), dtype=object)
for i, op2 in enumerate(basis):
out[i, :] = [op1 * op2 for op1 in basis]
return out
else:
# return array of expectation values
return np.array([[expect(op1 * op2, rho)
for op1 in basis] for op2 in basis])
[docs]def covariance_matrix(basis, rho, symmetrized=True):
r"""
Given a basis set of operators :math:`\{a\}_n`, calculate the covariance
matrix:
.. math::
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`,
.. math::
V_{mn} = \langle a_m a_n\rangle -
\langle a_m \rangle \langle a_n\rangle
Parameters
----------
basis : list
List of operators that defines the basis for the covariance matrix.
rho : Qobj
Density matrix for which to calculate the covariance matrix.
symmetrized : bool, default: True
Flag indicating whether the symmetrized (default) or non-symmetrized
correlation matrix is to be calculated.
Returns
-------
corr_mat : ndarray
A 2-dimensional array of covariance values.
"""
if symmetrized:
return np.array([[0.5 * expect(op1 * op2 + op2 * op1, rho) -
expect(op1, rho) * expect(op2, rho)
for op1 in basis] for op2 in basis])
else:
return np.array([[expect(op1 * op2, rho) -
expect(op1, rho) * expect(op2, rho)
for op1 in basis] for op2 in basis])
[docs]def correlation_matrix_field(a1, a2, rho=None):
"""
Calculates the correlation matrix for given field operators :math:`a_1` and
:math:`a_2`. If a density matrix is given the expectation values are
calculated, otherwise a matrix with operators is returned.
Parameters
----------
a1 : Qobj
Field operator for mode 1.
a2 : Qobj
Field operator for mode 2.
rho : Qobj, optional
Density matrix for which to calculate the covariance matrix.
Returns
-------
cov_mat : ndarray
Array of complex numbers or Qobj's
A 2-dimensional *array* of covariance values, or, if rho=0, a matrix
of operators.
"""
basis = [a1, a1.dag(), a2, a2.dag()]
return correlation_matrix(basis, rho)
[docs]def correlation_matrix_quadrature(a1, a2, rho=None, g=np.sqrt(2)):
"""
Calculate the quadrature correlation matrix with given field operators
:math:`a_1` and :math:`a_2`. If a density matrix is given the expectation
values are calculated, otherwise a matrix with operators is returned.
Parameters
----------
a1 : Qobj
Field operator for mode 1.
a2 : Qobj
Field operator for mode 2.
rho : Qobj, optional
Density matrix for which to calculate the covariance matrix.
g : float, 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``.
Returns
-------
corr_mat : ndarray
Array of complex numbers or Qobj's
A 2-dimensional *array* of covariance values for the field quadratures,
or, if rho=0, a matrix of operators.
"""
x1 = (a1 + a1.dag()) / g
p1 = -1j * (a1 - a1.dag()) / g
x2 = (a2 + a2.dag()) / g
p2 = -1j * (a2 - a2.dag()) / g
basis = [x1, p1, x2, p2]
return correlation_matrix(basis, rho)
[docs]def wigner_covariance_matrix(a1=None, a2=None, R=None, rho=None, g=np.sqrt(2)):
r"""
Calculates the Wigner covariance matrix
:math:`V_{ij} = \frac{1}{2}(R_{ij} + R_{ji})`, given
the quadrature correlation matrix
:math:`R_{ij} = \langle R_{i} R_{j}\rangle -
\langle R_{i}\rangle \langle R_{j}\rangle`, where
:math:`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 operators ``a1`` and
``a2`` 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
----------
a1 : Qobj, optional
Field operator for mode 1.
a2 : Qobj, optional
Field operator for mode 2.
R : ndarray, optional
The quadrature correlation matrix.
rho : Qobj, optional
Density matrix for which to calculate the covariance matrix.
g : float, 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``.
Returns
-------
cov_mat : ndarray
A 2-dimensional array of covariance values.
"""
if R is not None:
if rho is None:
return np.array([[0.5 * np.real(R[i, j] + R[j, i])
for i in range(4)]
for j in range(4)], dtype=np.float64)
else:
return np.array([[0.5 * np.real(expect(R[i, j] + R[j, i], rho))
for i in range(4)]
for j in range(4)], dtype=np.float64)
elif a1 is not None and a2 is not None:
if rho is not None:
x1 = (a1 + a1.dag()) / g
p1 = -1j * (a1 - a1.dag()) / g
x2 = (a2 + a2.dag()) / g
p2 = -1j * (a2 - a2.dag()) / g
return covariance_matrix([x1, p1, x2, p2], rho)
else:
raise ValueError("Must give rho if using field operators " +
"(a1 and a2)")
else:
raise ValueError("Must give either field operators (a1 and a2) " +
"or a precomputed correlation matrix (R)")
[docs]def logarithmic_negativity(V, g=np.sqrt(2)):
"""
Calculates the logarithmic negativity given a symmetrized covariance
matrix, see :func:`qutip.continuous_variables.covariance_matrix`. Note that
the two-mode field state that is described by `V` must be Gaussian for this
function to applicable.
Parameters
----------
V : ndarray
The covariance matrix.
g : float, 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``.
Returns
-------
N : float
The logarithmic negativity for the two-mode Gaussian state
that is described by the the Wigner covariance matrix V.
"""
A = 0.5 * V[0:2, 0:2] * g ** 2
B = 0.5 * V[2:4, 2:4] * g ** 2
C = 0.5 * V[0:2, 2:4] * g ** 2
sigma = np.linalg.det(A) + np.linalg.det(B) - 2 * np.linalg.det(C)
nu_ = sigma / 2 - np.sqrt(sigma ** 2 - 4 * np.linalg.det(V)) / 2
if nu_ < 0.0:
return 0.0
nu = np.sqrt(nu_)
lognu = -np.log(2 * nu)
logneg = max(0, lognu)
return logneg