:py:mod:`state_props.is_separable`
==================================

.. py:module:: state_props.is_separable

.. autoapi-nested-parse::

   Check if state is separable.



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


Functions
~~~~~~~~~

.. autoapisummary::

   state_props.is_separable.is_separable



.. py:function:: is_separable(state, dim = None, level = 2, tol = 1e-08)

   Determine if a given state (given as a density matrix) is a separable state :cite:`WikiSepSt`.

   .. rubric:: Examples

   Consider the following separable (by construction) state:

   .. math::
       \rho = \rho_1 \otimes \rho_2.
       \rho_1 = \frac{1}{2} \left(
           |0 \rangle \langle 0| + |0 \rangle \langle 1| + |1 \rangle \langle 0| + |1 \rangle \langle 1| \right)
       \rho_2 = \frac{1}{2} \left( |0 \rangle \langle 0| + |1 \rangle \langle 1| \right)

   The resulting density matrix will be:

   .. math::
       \rho =  \frac{1}{4} \begin{pmatrix}
               1 & 0 & 1 & 0 \\
               0 & 1 & 0 & 1 \\
               1 & 0 & 1 & 0 \\
               0 & 1 & 0 & 1
               \end{pmatrix} \in \text{D}(\mathcal{X}).

   We provide the input as a denisty matrix :math:`\rho`.

   On the other hand, a random density matrix will be an entangled state (a separable state).
   >>> from toqito.rand import random_density_matrix
   >>> from toqito.state_props import is_separable
   >>> rho_separable = np.array([[1, 0, 1, 0],
   ...                           [0, 1, 0, 1],
   ...                           [1, 0, 1, 0],
   ...                           [0, 1, 0, 1]])
   >>> rho_random = random_density_matrix(4)
   >>> is_separable(rho_separable)
   True
   >>> is_separable(rho_random)
   False

   .. rubric:: References

   .. bibliography::
       :filter: docname in docnames

   :raises ValueError: If dimension is not specified.
   :param state: The matrix to check.
   :param dim: The dimension of the input.
   :param level: The level up to which to search for the symmetric extensions.
   :param tol: Numerical tolerance used.
   :return: :code:`True` if :code:`rho` is separabale and :code:`False` otherwise.