# This file is part of QuTiP: Quantum Toolbox in Python.
#
# Copyright (c) 2011 and later, Paul D. Nation and Robert J. Johansson.
# All rights reserved.
#
# Redistribution and use in source and binary forms, with or without
# modification, are permitted provided that the following conditions are
# met:
#
# 1. Redistributions of source code must retain the above copyright notice,
# this list of conditions and the following disclaimer.
#
# 2. Redistributions in binary form must reproduce the above copyright
# notice, this list of conditions and the following disclaimer in the
# documentation and/or other materials provided with the distribution.
#
# 3. Neither the name of the QuTiP: Quantum Toolbox in Python nor the names
# of its contributors may be used to endorse or promote products derived
# from this software without specific prior written permission.
#
# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
# "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
# LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
# PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
# HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
# SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
# LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
# DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
# THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
# (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
# OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
###############################################################################
__all__ = ['entropy_vn', 'entropy_linear', 'entropy_mutual', 'negativity',
'concurrence', 'entropy_conditional', 'entangling_power']
from numpy import e, 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] http://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])]
rho_pt = partial_transpose(rho, mask)
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
def _entropy_relative(rho, sigma, base=e, sparse=False):
"""
****NEEDS TO BE WORKED ON****
Calculates the relative entropy S(rho||sigma) between two density
matrices.
Parameters
----------
rho : qobj
First density matrix.
sigma : qobj
Second density matrix.
base : {e,2}
Base of logarithm.
Returns
-------
rel_ent : float
Value of relative entropy.
"""
if rho.type != 'oper' or sigma.type != 'oper':
raise TypeError("Inputs must be density matrices..")
# sigma terms
svals = sp_eigs(sigma.data, sigma.isherm, vecs=False, sparse=sparse)
snzvals = svals[svals != 0]
if base == 2:
slogvals = log2(snzvals)
elif base == e:
slogvals = log(snzvals)
else:
raise ValueError("Base must be 2 or e.")
# rho terms
rvals = sp_eigs(rho.data, rho.isherm, vecs=False, sparse=sparse)
rnzvals = rvals[rvals != 0]
# calculate tr(rho*log sigma)
rel_trace = float(real(sum(rnzvals * slogvals)))
return -entropy_vn(rho, base, sparse) - rel_trace
[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