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))