def fidelity(state1, state2):
""" Computes the fidelity between two quantum states
(http://en.wikipedia.org/wiki/Fidelity_of_quantum_states)
The arguments provided to this function should be a square matrix or a
Density object. If it is a square matrix, it is assumed to be diagonalizable.
Parameters:
==========
state1, state2 : a density matrix or Matrix
Examples:
=========
>>> from sympy import S, sqrt
>>> from sympsi.dagger import Dagger
>>> from sympsi.spin import JzKet
>>> from sympsi.density import Density, fidelity
>>> from sympsi.represent import represent
>>>
>>> up = JzKet(S(1)/2,S(1)/2)
>>> down = JzKet(S(1)/2,-S(1)/2)
>>> amp = 1/sqrt(2)
>>> updown = (amp * up) + (amp * down)
>>>
>>> # represent turns Kets into matrices
>>> up_dm = represent(up * Dagger(up))
>>> down_dm = represent(down * Dagger(down))
>>> updown_dm = represent(updown * Dagger(updown))
>>>
>>> fidelity(up_dm, up_dm)
1
>>> fidelity(up_dm, down_dm) #orthogonal states
0
>>> fidelity(up_dm, updown_dm).evalf().round(3)
0.707
"""
state1 = represent(state1) if isinstance(state1, Density) else state1
state2 = represent(state2) if isinstance(state2, Density) else state2
if (not isinstance(state1, Matrix) or
not isinstance(state2, Matrix)):
raise ValueError("state1 and state2 must be of type Density or Matrix "
"received type=%s for state1 and type=%s for state2" %
(type(state1), type(state2)))
if ( state1.shape != state2.shape and state1.is_square):
raise ValueError("The dimensions of both args should be equal and the "
"matrix obtained should be a square matrix")
sqrt_state1 = state1**Rational(1, 2)
return Tr((sqrt_state1 * state2 * sqrt_state1)**Rational(1, 2)).doit()
def fidelity(state1, state2):
""" Computes the fidelity between two quantum states
(http://en.wikipedia.org/wiki/Fidelity_of_quantum_states)
The arguments provided to this function should be a square matrix or a
Density object. If it is a square matrix, it is assumed to be diagonalizable.
Parameters:
==========
state1, state2 : a density matrix or Matrix
Examples:
=========
>>> from sympy import S, sqrt
>>> from sympsi.dagger import Dagger
>>> from sympsi.spin import JzKet
>>> from sympsi.density import Density, fidelity
>>> from sympsi.represent import represent
>>>
>>> up = JzKet(S(1)/2,S(1)/2)
>>> down = JzKet(S(1)/2,-S(1)/2)
>>> amp = 1/sqrt(2)
>>> updown = (amp * up) + (amp * down)
>>>
>>> # represent turns Kets into matrices
>>> up_dm = represent(up * Dagger(up))
>>> down_dm = represent(down * Dagger(down))
>>> updown_dm = represent(updown * Dagger(updown))
>>>
>>> fidelity(up_dm, up_dm)
1
>>> fidelity(up_dm, down_dm) #orthogonal states
0
>>> fidelity(up_dm, updown_dm).evalf().round(3)
0.707
"""
state1 = represent(state1) if isinstance(state1, Density) else state1
state2 = represent(state2) if isinstance(state2, Density) else state2
if (not isinstance(state1, Matrix) or not isinstance(state2, Matrix)):
raise ValueError("state1 and state2 must be of type Density or Matrix "
"received type=%s for state1 and type=%s for state2" %
(type(state1), type(state2)))
if (state1.shape != state2.shape and state1.is_square):
raise ValueError("The dimensions of both args should be equal and the "
"matrix obtained should be a square matrix")
sqrt_state1 = state1**Rational(1, 2)
return Tr((sqrt_state1 * state2 * sqrt_state1)**Rational(1, 2)).doit()
def entropy(density):
"""Compute the entropy of a matrix/density object.
This computes -Tr(density*ln(density)) using the eigenvalue decomposition
of density, which is given as either a Density instance or a matrix
(numpy.ndarray, sympy.Matrix or scipy.sparse).
Parameters
==========
density : density matrix of type Density, sympy matrix,
scipy.sparse or numpy.ndarray
Examples:
========
>>> from sympsi.density import Density, entropy
>>> from sympsi.represent import represent
>>> from sympsi.matrixutils import scipy_sparse_matrix
>>> from sympsi.spin import JzKet, Jz
>>> from sympy import S, log
>>> up = JzKet(S(1)/2,S(1)/2)
>>> down = JzKet(S(1)/2,-S(1)/2)
>>> d = Density((up,0.5),(down,0.5))
>>> entropy(d)
log(2)/2
"""
if isinstance(density, Density):
density = represent(density) # represent in Matrix
if isinstance(density, scipy_sparse_matrix):
density = to_numpy(density)
if isinstance(density, Matrix):
eigvals = density.eigenvals().keys()
return expand(-sum(e*log(e) for e in eigvals))
elif isinstance(density, numpy_ndarray):
import numpy as np
eigvals = np.linalg.eigvals(density)
return -np.sum(eigvals*np.log(eigvals))
else:
raise ValueError(
"numpy.ndarray, scipy.sparse or sympy matrix expected")
def entropy(density):
"""Compute the entropy of a matrix/density object.
This computes -Tr(density*ln(density)) using the eigenvalue decomposition
of density, which is given as either a Density instance or a matrix
(numpy.ndarray, sympy.Matrix or scipy.sparse).
Parameters
==========
density : density matrix of type Density, sympy matrix,
scipy.sparse or numpy.ndarray
Examples:
========
>>> from sympsi.density import Density, entropy
>>> from sympsi.represent import represent
>>> from sympsi.matrixutils import scipy_sparse_matrix
>>> from sympsi.spin import JzKet, Jz
>>> from sympy import S, log
>>> up = JzKet(S(1)/2,S(1)/2)
>>> down = JzKet(S(1)/2,-S(1)/2)
>>> d = Density((up,0.5),(down,0.5))
>>> entropy(d)
log(2)/2
"""
if isinstance(density, Density):
density = represent(density) # represent in Matrix
if isinstance(density, scipy_sparse_matrix):
density = to_numpy(density)
if isinstance(density, Matrix):
eigvals = density.eigenvals().keys()
return expand(-sum(e * log(e) for e in eigvals))
elif isinstance(density, numpy_ndarray):
import numpy as np
eigvals = np.linalg.eigvals(density)
return -np.sum(eigvals * np.log(eigvals))
else:
raise ValueError(
"numpy.ndarray, scipy.sparse or sympy matrix expected")
def _represent(self, **options):
return represent(self.doit(), **options)
def _represent(self, **options):
return represent(self.doit(), **options)
