toqito.state_props.in_separable_ball
====================================

.. py:module:: toqito.state_props.in_separable_ball

.. autoapi-nested-parse::

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





Module Contents
---------------

.. py:function:: 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.

   .. rubric:: 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))
   ```

   :param mat: 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.