# 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__ = ['simdiag']
import numpy as np
import scipy.linalg as la
from qutip.qobj import Qobj
[docs]def simdiag(ops, evals=True):
"""Simultaneous diagonalization of commuting Hermitian matrices.
Parameters
----------
ops : list/array
``list`` or ``array`` of qobjs representing commuting Hermitian
operators.
Returns
--------
eigs : tuple
Tuple of arrays representing eigvecs and eigvals of quantum objects
corresponding to simultaneous eigenvectors and eigenvalues for each
operator.
"""
tol = 1e-14
start_flag = 0
if not any(ops):
raise ValueError('Need at least one input operator.')
if not isinstance(ops, (list, np.ndarray)):
ops = np.array([ops])
num_ops = len(ops)
for jj in range(num_ops):
A = ops[jj]
shape = A.shape
if shape[0] != shape[1]:
raise TypeError('Matricies must be square.')
if start_flag == 0:
s = shape[0]
if s != shape[0]:
raise TypeError('All matrices. must be the same shape')
if not A.isherm:
raise TypeError('Matricies must be Hermitian')
for kk in range(jj):
B = ops[kk]
if (A * B - B * A).norm() / (A * B).norm() > tol:
raise TypeError('Matricies must commute.')
A = ops[0]
eigvals, eigvecs = la.eig(A.full())
zipped = list(zip(-eigvals, range(len(eigvals))))
zipped.sort()
ds, perm = zip(*zipped)
ds = -np.real(np.array(ds))
perm = np.array(perm)
eigvecs_array = np.array(
[np.zeros((A.shape[0], 1), dtype=complex) for k in range(A.shape[0])])
for kk in range(len(perm)): # matrix with sorted eigvecs in columns
eigvecs_array[kk][:, 0] = eigvecs[:, perm[kk]]
k = 0
rng = np.arange(len(eigvals))
while k < len(ds):
# find degenerate eigenvalues, get indicies of degenerate eigvals
inds = np.array(abs(ds - ds[k]) < max(tol, tol * abs(ds[k])))
inds = rng[inds]
if len(inds) > 1: # if at least 2 eigvals are degenerate
eigvecs_array[inds] = degen(
tol, eigvecs_array[inds],
np.array([ops[kk] for kk in range(1, num_ops)]))
k = max(inds) + 1
eigvals_out = np.zeros((num_ops, len(ds)), dtype=float)
kets_out = np.array([Qobj(eigvecs_array[j] / la.norm(eigvecs_array[j]),
dims=[ops[0].dims[0], [1]],
shape=[ops[0].shape[0], 1])
for j in range(len(ds))])
if not evals:
return kets_out
else:
for kk in range(num_ops):
for j in range(len(ds)):
eigvals_out[kk, j] = np.real(np.dot(
eigvecs_array[j].conj().T,
ops[kk].data * eigvecs_array[j]))
return eigvals_out, kets_out
def degen(tol, in_vecs, ops):
"""
Private function that finds eigen vals and vecs for degenerate matrices..
"""
n = len(ops)
if n == 0:
return in_vecs
A = ops[0]
vecs = np.column_stack(in_vecs)
eigvals, eigvecs = la.eig(np.dot(vecs.conj().T, A.data.dot(vecs)))
zipped = list(zip(-eigvals, range(len(eigvals))))
zipped.sort()
ds, perm = zip(*zipped)
ds = -np.real(np.array(ds))
perm = np.array(perm)
vecsperm = np.zeros(eigvecs.shape, dtype=complex)
for kk in range(len(perm)): # matrix with sorted eigvecs in columns
vecsperm[:, kk] = eigvecs[:, perm[kk]]
vecs_new = np.dot(vecs, vecsperm)
vecs_out = np.array(
[np.zeros((A.shape[0], 1), dtype=complex) for k in range(len(ds))])
for kk in range(len(perm)): # matrix with sorted eigvecs in columns
vecs_out[kk][:, 0] = vecs_new[:, kk]
k = 0
rng = np.arange(len(ds))
while k < len(ds):
inds = np.array(abs(ds - ds[k]) < max(
tol, tol * abs(ds[k]))) # get indicies of degenerate eigvals
inds = rng[inds]
if len(inds) > 1: # if at least 2 eigvals are degenerate
vecs_out[inds] = degen(tol, vecs_out[inds],
np.array([ops[jj] for jj in range(1, n)]))
k = max(inds) + 1
return vecs_out