"""Checks whether operator is in the ball of separability centered at the maximally-mixed state."""
import numpy as np
[docs]
def in_separable_ball(mat: np.ndarray) -> bool:
r"""
Check whether an operator is contained in ball of separability [GB02]_.
Determines whether :code:`mat` is contained within the ball of separable operators centered
at the identity matrix (i.e. the maximally-mixed state). The size of this ball was derived in
[GB02]_.
This function can be used as a method for separability testing of states in certain scenarios.
This function is adapted from QETLAB.
Examples
==========
The only states acting on :math:`\mathbb{C}^m \otimes \mathbb{C}^n` in the
separable ball that do not have full rank are those with exactly 1 zero
eigenvalue, and the :math:`mn - 1` non-zero eigenvalues equal to each
other.
The following is an example of generating a random density matrix with eigenvalues
:code:`[1, 1, 1, 0]/3`. This example yields a matrix that is contained within the separable
ball.
>>> from toqito.random import random_unitary
>>> from toqito.state_props import in_separable_ball
>>> import numpy as np
>>>
>>> U = random_unitary(4)
>>> lam = np.array([1, 1, 1, 0]) / 3
>>> rho = U @ np.diag(lam) @ U.conj().T
>>> in_separable_ball(rho)
True
The following is an example of generating a random density matrix with eigenvalues
:code:`[1.01, 1, 0.99, 0]/3`. This example yields a matrix that is not contained within the
separable ball.
>>> from toqito.random import random_unitary
>>> from toqito.state_props import in_separable_ball
>>> import numpy as np
>>>
>>> U = random_unitary(4)
>>> lam = np.array([1.01, 1, 0,.99, 0]) / 3
>>> rho = U @ np.diag(lam) @ U.conj().T
>>> in_separable_ball(rho)
False
References
==========
.. [GB02] Gurvits, Leonid, and Barnum, Howard.
"Largest separable balls around the maximally mixed bipartite quantum state."
Physical Review A 66.6 (2002): 062311.
https://arxiv.org/pdf/quant-ph/0204159.pdf
:param mat: A positive semidefinite matrix or a vector of the eigenvalues of a positive
semidefinite matrix.
:return: :code:`True` if the matrix :code:`mat` is contained within the separable ball, and
:code:`False` otherwise.
"""
mat_dims = mat.shape
max_dim = max(mat_dims)
# If the matrix is a vector, turn it into a matrix. We could instead turn every matrix into a
# vector of eigenvalues, but that would make the computation take O(n^3) time instead of the
# current method which is O(n^2).
# Case: Vector of eigenvalues.
if len(mat_dims) == 1 or min(mat_dims) == 1:
mat = np.diag(mat)
# If the matrix has trace equal to 0 or less, it cannot be in the separable ball.
if np.trace(mat) < max_dim * np.finfo(float).eps:
return False
mat = mat / np.trace(mat)
# The following check relies on the fact that we scaled the matrix so that trace(mat) = 1.
# The following condition is then exactly the condition mentioned in [GB02]_.
return np.linalg.norm(mat / np.linalg.norm(mat, "fro") ** 2 - np.eye(max_dim), "fro") <= 1