toqito.states.gisin
===================

.. py:module:: toqito.states.gisin

.. autoapi-nested-parse::

   Gisin states are the mixtures of separable mixed states and some purely entangled states.





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

.. py:function:: gisin(lambda_var, theta)

   Produce a Gisin state [@Gisin_1996_Hidden].

   Returns the Gisin state described in [@Gisin_1996_Hidden]. Specifically, the Gisin state can be defined as:

   \[
       \begin{equation}
           \rho_{\lambda, \theta} = \lambda
                                   \begin{pmatrix}
                                       0 & 0 & 0 & 0 \\
                                       0 & \sin^2(\theta) &
                                       -\sin(\theta)\cos(\theta) & 0 \\
                                       0 & -\sin(\theta)\cos(\theta) &
                                       \cos^2(\theta) & 0 \\
                                       0 & 0 & 0 & 0
                                   \end{pmatrix} +
                                   \frac{1 - \lambda}{2}
                                   \begin{pmatrix}
                                       1 & 0 & 0 & 0 \\
                                       0 & 0 & 0 & 0 \\
                                       0 & 0 & 0 & 0 \\
                                       0 & 0 & 0 & 1
                                   \end{pmatrix}.
       \end{equation}
   \]

   .. rubric:: Examples

   The following code generates the Gisin state \(\rho_{0.5, 1}\).

   ```python exec="1" source="above"
   from toqito.states import gisin
   print(gisin(0.5, 1))
   ```

   :raises ValueError: If `lambda_var` is not a real number.

   :param lambda_var: A real parameter in [0, 1].
   :param theta: A real parameter.

   :returns: Gisin state.