toqito.state_props.in_separable_ball

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

Module Contents

toqito.state_props.in_separable_ball.in_separable_ball(mat)

Check whether an operator is contained in ball of separability [@Gurvits_2002_Largest].

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 [@Gurvits_2002_Largest].

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.

`python exec="1" source="above" 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 print(in_separable_ball(rho)) `

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.

`python exec="1" source="above" 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 print(in_separable_ball(rho)) `

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 | numpy.bool_