# This file is part of QuTiP: Quantum Toolbox in Python.
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###############################################################################
__all__ = ['simdiag']
import numpy as np
import scipy.linalg as la
from qutip.qobj import Qobj
def _degen(tol, vecs, ops, i=0):
"""
Private function that finds eigen vals and vecs for degenerate matrices..
"""
if len(ops) == i:
return vecs
# New eigenvectors are sometime not orthogonal.
for j in range(1, vecs.shape[1]):
for k in range(j):
dot = vecs[:, j].dot(vecs[:, k].conj())
if np.abs(dot) > tol:
vecs[:, j] = ((vecs[:, j] - dot * vecs[:, k])
/ (1 - np.abs(dot)**2)**0.5)
subspace = vecs.conj().T @ ops[i].data @ vecs
eigvals, eigvecs = la.eig(subspace)
perm = np.argsort(eigvals)
eigvals = eigvals[perm]
vecs_new = vecs @ eigvecs[:, perm]
for k in range(len(eigvals)):
vecs_new[:, k] = vecs_new[:, k] / la.norm(vecs_new[:, k])
k = 0
while k < len(eigvals):
ttol = max(tol, tol * abs(eigvals[k]))
inds, = np.where(abs(eigvals - eigvals[k]) < ttol)
if len(inds) > 1: # if at least 2 eigvals are degenerate
vecs_new[:, inds] = _degen(tol, vecs_new[:, inds], ops, i+1)
k = inds[-1] + 1
return vecs_new
[docs]def simdiag(ops, evals: bool = True, *,
tol: float = 1e-14, safe_mode: bool = True):
"""Simultaneous diagonalization of commuting Hermitian matrices.
Parameters
----------
ops : list/array
``list`` or ``array`` of qobjs representing commuting Hermitian
operators.
evals : bool [True]
Whether to return the eigenvalues for each ops and eigenvectors or just
the eigenvectors.
tol : float [1e-14]
Tolerance for detecting degenerate eigenstates.
safe_mode : bool [True]
Whether to check that all ops are Hermitian and commuting. If set to
``False`` and operators are not commuting, the eigenvectors returned
will often be eigenvectors of only the first operator.
Returns
--------
eigs : tuple
Tuple of arrays representing eigvecs and eigvals of quantum objects
corresponding to simultaneous eigenvectors and eigenvalues for each
operator.
"""
if not ops:
raise ValueError("No input matrices.")
N = ops[0].shape[0]
num_ops = len(ops) if safe_mode else 0
for jj in range(num_ops):
A = ops[jj]
shape = A.shape
if shape[0] != shape[1]:
raise TypeError('Matricies must be square.')
if shape[0] != N:
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.')
eigvals, eigvecs = la.eigh(ops[0].full())
perm = np.argsort(eigvals)
eigvecs = eigvecs[:, perm]
eigvals = eigvals[perm]
k = 0
while k < N:
# find degenerate eigenvalues, get indicies of degenerate eigvals
ttol = max(tol, tol * abs(eigvals[k]))
inds, = np.where(abs(eigvals - eigvals[k]) < ttol)
if len(inds) > 1: # if at least 2 eigvals are degenerate
eigvecs[:, inds] = _degen(tol, eigvecs[:, inds], ops, 1)
k = inds[-1] + 1
for k in range(N):
eigvecs[:, k] = eigvecs[:, k] / la.norm(eigvecs[:, k])
kets_out = [
Qobj(eigvecs[:, j],
dims=[ops[0].dims[0], [1]], shape=[ops[0].shape[0], 1])
for j in range(N)
]
eigvals_out = np.zeros((len(ops), N), dtype=np.float64)
if not evals:
return kets_out
else:
for kk in range(len(ops)):
for j in range(N):
eigvals_out[kk, j] = ops[kk].matrix_element(kets_out[j],
kets_out[j]).real
return eigvals_out, kets_out