# Copyright (c) 2024, Kerry He, James Saunderson, and Hamza Fawzi
# This Python package QICS is licensed under the MIT license; see LICENSE.md
# file in the root directory or at https://github.com/kerry-he/qics
import numpy as np
[docs]
def entropy(x):
r"""Computes either the classical (Shannon) entropy
.. math::
H(x) = -\sum_{i=1}^n x_i \log(x_i),
for nonnegative vector :math:`x`, or the quantum (von Neumann) entropy
.. math::
S(X) = -\text{tr}[X \log(X)],
for positive semidefinite matrix :math:`X`.
Parameters
----------
x : :class:`~numpy.ndarray`
If this is a nonnegative array of size ``(n,)`` or ``(n, 1)``, we
compute the classical entropy of :math:`x`. If this is a symmetric
or Hermitian positive semidefinite array of size ``(n, n)``, then
we compute the quantum entropy of :math:`X`
Returns
-------
:obj:`float`
Classical entropy of :math:`x` or quantum entropy of :math:`X`.
"""
if x.size > 1 and len(x.shape) == 2 and x.shape[0] == x.shape[1]:
eig = np.linalg.eigvalsh(x)
return entropy(eig)
x = x[x > 0]
return -sum(x * np.log(x))
[docs]
def purify(X):
r"""Returns a purification of a quantum state :math:`X`. If :math:`X`
has spectral decomposition
.. math::
X = \sum_{i=1}^n \lambda_i | v_i \rangle\langle v_i |,
then the purification is :math:`| \psi \rangle\langle \psi |` where
.. math::
| \psi \rangle = \sum_{i=1}^n \sqrt{\lambda_i} (| v_i \rangle
\otimes | v_i \rangle).
Parameters
----------
X : :class:`~numpy.ndarray`
Symmetric or Hermitian array of size ``(n, n)`` representing the
matrix :math:`X` we want to find the purification of.
Returns
-------
:class:`~numpy.ndarray`
Symmetric or Hermitian array of size ``(n*n, n*n)`` representing the
purification of :math:`X`.
"""
n = X.shape[0]
D, U = np.linalg.eigh(X)
vec = np.zeros((n * n, 1), dtype=X.dtype)
for i in range(n):
vec += np.sqrt(max(0.0, D[i])) * np.kron(U[:, [i]], U[:, [i]])
return vec @ vec.conj().T
QICS