# 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.
#
# Significant parts of this code was contributed by Jonas Neergaard-Nielsen
###############################################################################
__all__ = ['partial_transpose']
import numpy as np
import scipy.sparse as sp
from qutip.qobj import Qobj
from qutip.states import (state_index_number, state_number_index,
state_number_enumerate)
[docs]def partial_transpose(rho, mask, method='dense'):
"""
Return the partial transpose of a Qobj instance `rho`,
where `mask` is an array/list with length that equals
the number of components of `rho` (that is, the length of
`rho.dims[0]`), and the values in `mask` indicates whether
or not the corresponding subsystem is to be transposed.
The elements in `mask` can be boolean or integers `0` or `1`,
where `True`/`1` indicates that the corresponding subsystem
should be tranposed.
Parameters
----------
rho : :class:`qutip.qobj`
A density matrix.
mask : *list* / *array*
A mask that selects which subsystems should be transposed.
method : str
choice of method, `dense` or `sparse`. The default method
is `dense`. The `sparse` implementation can be faster for
large and sparse systems (hundreds of quantum states).
Returns
-------
rho_pr: :class:`qutip.qobj`
A density matrix with the selected subsystems transposed.
"""
mask = [int(i) for i in mask]
if method == 'sparse':
return _partial_transpose_sparse(rho, mask)
else:
return _partial_transpose_dense(rho, mask)
def _partial_transpose_dense(rho, mask):
"""
Based on Jonas' implementation using numpy.
Very fast for dense problems.
"""
nsys = len(mask)
pt_dims = np.arange(2 * nsys).reshape(2, nsys).T
pt_idx = np.concatenate([[pt_dims[n, mask[n]] for n in range(nsys)],
[pt_dims[n, 1 - mask[n]] for n in range(nsys)]])
data = rho.data.toarray().reshape(
np.array(rho.dims).flatten()).transpose(pt_idx).reshape(rho.shape)
return Qobj(data, dims=rho.dims)
def _partial_transpose_sparse(rho, mask):
"""
Implement the partial transpose using the CSR sparse matrix.
"""
data = sp.lil_matrix((rho.shape[0], rho.shape[1]), dtype=complex)
for m in range(len(rho.data.indptr) - 1):
n1 = rho.data.indptr[m]
n2 = rho.data.indptr[m + 1]
psi_A = state_index_number(rho.dims[0], m)
for idx, n in enumerate(rho.data.indices[n1:n2]):
psi_B = state_index_number(rho.dims[1], n)
m_pt = state_number_index(
rho.dims[1], np.choose(mask, [psi_A, psi_B]))
n_pt = state_number_index(
rho.dims[0], np.choose(mask, [psi_B, psi_A]))
data[m_pt, n_pt] = rho.data.data[n1 + idx]
return Qobj(data.tocsr(), dims=rho.dims)
def _partial_transpose_reference(rho, mask):
"""
This is a reference implementation that explicitly loops over
all states and performs the transpose. It's slow but easy to
understand and useful for testing.
"""
A_pt = np.zeros(rho.shape, dtype=complex)
for psi_A in state_number_enumerate(rho.dims[0]):
m = state_number_index(rho.dims[0], psi_A)
for psi_B in state_number_enumerate(rho.dims[1]):
n = state_number_index(rho.dims[1], psi_B)
m_pt = state_number_index(
rho.dims[1], np.choose(mask, [psi_A, psi_B]))
n_pt = state_number_index(
rho.dims[0], np.choose(mask, [psi_B, psi_A]))
A_pt[m_pt, n_pt] = rho.data[m, n]
return Qobj(A_pt, dims=rho.dims)