# Source code for qutip.metrics

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"""
This module contains a collection of functions for calculating metrics
(distance measures) between states and operators.
"""

__all__ = ['fidelity', 'tracedist', 'bures_dist', 'bures_angle',
'hilbert_dist', 'average_gate_fidelity', 'process_fidelity']

import numpy as np
from qutip.sparse import sp_eigs
from qutip.states import ket2dm
from qutip.superop_reps import to_kraus

[docs]def fidelity(A, B):
"""
Calculates the fidelity (pseudo-metric) between two density matrices.
See: Nielsen & Chuang, "Quantum Computation and Quantum Information"

Parameters
----------
A : qobj
Density matrix or state vector.
B : qobj
Density matrix or state vector with same dimensions as A.

Returns
-------
fid : float
Fidelity pseudo-metric between A and B.

Examples
--------
>>> x = fock_dm(5,3)
>>> y = coherent_dm(5,1)
>>> fidelity(x,y)
0.24104350624628332

"""
if A.isket or A.isbra:
A = ket2dm(A)
if B.isket or B.isbra:
B = ket2dm(B)

if A.dims != B.dims:
raise TypeError('Density matrices do not have same dimensions.')

A = A.sqrtm()
return float(np.real((A * (B * A)).sqrtm().tr()))

[docs]def process_fidelity(U1, U2, normalize=True):
"""
Calculate the process fidelity given two process operators.
"""
if normalize:
return (U1 * U2).tr() / (U1.tr() * U2.tr())
else:
return (U1 * U2).tr()

[docs]def average_gate_fidelity(oper):
"""
Given a Qobj representing the supermatrix form of a map, returns the
average gate fidelity (pseudo-metric) of that map.

Parameters
----------
A : Qobj
Quantum object representing a superoperator.

Returns
-------
fid : float
Fidelity pseudo-metric between A and the identity superoperator.
"""
kraus_form = to_kraus(oper)
d = kraus_form[0].shape[0]

if kraus_form[0].shape[1] != d:
return TypeError("Average gate fielity only implemented for square "
"superoperators.")

return (d + np.sum([np.abs(A_k.tr())**2
for A_k in kraus_form])) / (d**2 + d)

[docs]def tracedist(A, B, sparse=False, tol=0):
"""
Calculates the trace distance between two density matrices..
See: Nielsen & Chuang, "Quantum Computation and Quantum Information"

Parameters
----------!=
A : qobj
Density matrix or state vector.
B : qobj
Density matrix or state vector with same dimensions as A.
tol : float
Tolerance used by sparse eigensolver, if used. (0=Machine precision)
sparse : {False, True}
Use sparse eigensolver.

Returns
-------
tracedist : float
Trace distance between A and B.

Examples
--------
>>> x=fock_dm(5,3)
>>> y=coherent_dm(5,1)
>>> tracedist(x,y)
0.9705143161472971

"""
if A.isket or A.isbra:
A = ket2dm(A)
if B.isket or B.isbra:
B = ket2dm(B)

if A.dims != B.dims:
raise TypeError("A and B do not have same dimensions.")

diff = A - B
diff = diff.dag() * diff
vals = sp_eigs(diff.data, diff.isherm, vecs=False, sparse=sparse, tol=tol)
return float(np.real(0.5 * np.sum(np.sqrt(np.abs(vals)))))

[docs]def hilbert_dist(A, B):
"""
Returns the Hilbert-Schmidt distance between two density matrices A & B.

Parameters
----------
A : qobj
Density matrix or state vector.
B : qobj
Density matrix or state vector with same dimensions as A.

Returns
-------
dist : float
Hilbert-Schmidt distance between density matrices.

Notes
-----
See V. Vedral and M. B. Plenio, Phys. Rev. A 57, 1619 (1998).

"""
if A.isket or A.isbra:
A = ket2dm(A)
if B.isket or B.isbra:
B = ket2dm(B)

if A.dims != B.dims:
raise TypeError('A and B do not have same dimensions.')

return ((A - B)**2).tr()

[docs]def bures_dist(A, B):
"""
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
----------
A : qobj
Density matrix or state vector.
B : qobj
Density matrix or state vector with same dimensions as A.

Returns
-------
dist : float
Bures distance between density matrices.
"""
if A.isket or A.isbra:
A = ket2dm(A)
if B.isket or B.isbra:
B = ket2dm(B)

if A.dims != B.dims:
raise TypeError('A and B do not have same dimensions.')

dist = np.sqrt(2.0 * (1.0 - fidelity(A, B)))
return dist

[docs]def bures_angle(A, B):
"""
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
----------
A : qobj
Density matrix or state vector.
B : qobj
Density matrix or state vector with same dimensions as A.

Returns
-------
angle : float
Bures angle between density matrices.
"""
if A.isket or A.isbra:
A = ket2dm(A)
if B.isket or B.isbra:
B = ket2dm(B)

if A.dims != B.dims:
raise TypeError('A and B do not have same dimensions.')

return np.arccos(fidelity(A, B))