state_props.in_separable_ball¶

Checks whether operator is in the ball of separability centered at the maximally-mixed state.

Functions¶

in_separable_ball(mat)

Check whether an operator is contained in ball of separability [1].

Module Contents¶

state_props.in_separable_ball.in_separable_ball(mat)¶

Check whether an operator is contained in ball of separability [1].

Determines whether 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 [1].

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 \(\mathbb{C}^m \otimes \mathbb{C}^n\) in the separable ball that do not have full rank are those with exactly 1 zero eigenvalue, and the \(mn - 1\) non-zero eigenvalues equal to each other.

The following is an example of generating a random density matrix with eigenvalues [1, 1, 1, 0]/3. This example yields a matrix that is contained within the separable ball.

from toqito.rand 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)

np.True_

The following is an example of generating a random density matrix with eigenvalues [1.01, 1, 0.99, 0]/3. This example yields a matrix that is not contained within the separable ball.

from toqito.rand 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)

np.False_

References

Parameters:: mat (numpy.ndarray) – A positive semidefinite matrix or a vector of the eigenvalues of a positive semidefinite matrix.
Returns:: True if the matrix mat is contained within the separable ball, and False otherwise.
Return type:: bool