states.gen_bell
===============

.. py:module:: states.gen_bell

.. autoapi-nested-parse::

   Generalized Bell state represents a bigger set of Bell states.

   This set includes the standard bell states and other higher dimensional bell states as well.
   Generalized Bell states are the basis of multidimensional bipartite states having maximum entanglement.



Functions
---------

.. autoapisummary::

   states.gen_bell.gen_bell


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

.. py:function:: gen_bell(k_1, k_2, dim)

   Produce a generalized Bell state :cite:`Sych_2009_AComplete`.

   Produces a generalized Bell state. Note that the standard Bell states can be recovered as:

   .. code-block:: python
       :linenos:

       bell(0) : gen_bell(0, 0, 2)

       bell(1) : gen_bell(0, 1, 2)

       bell(2) : gen_bell(1, 0, 2)

       bell(3) : gen_bell(1, 1, 2)


   .. rubric:: Examples

   For :math:`d = 2` and :math:`k_1 = k_2 = 0`, this generates the following matrix

   .. math::
       G = \frac{1}{2} \begin{pmatrix}
                       1 & 0 & 0 & 1 \\
                       0 & 0 & 0 & 0 \\
                       0 & 0 & 0 & 0 \\
                       1 & 0 & 0 & 1
                   \end{pmatrix}

   which is equivalent to :math:`|\phi_0 \rangle \langle \phi_0 |` where

   .. math::
       |\phi_0\rangle = \frac{1}{\sqrt{2}} \left( |00 \rangle + |11 \rangle \right)

   is one of the four standard Bell states. This can be computed via :code:`toqito` as follows.

   >>> from toqito.states import gen_bell
   >>> dim = 2
   >>> k_1 = 0
   >>> k_2 = 0
   >>> gen_bell(k_1, k_2, dim)
   array([[0.5+0.j, 0. +0.j, 0. +0.j, 0.5+0.j],
          [0. +0.j, 0. +0.j, 0. +0.j, 0. +0.j],
          [0. +0.j, 0. +0.j, 0. +0.j, 0. +0.j],
          [0.5+0.j, 0. +0.j, 0. +0.j, 0.5+0.j]])

   It is possible for us to consider higher dimensional Bell states. For instance, we can consider
   the :math:`3`-dimensional Bell state for :math:`k_1 = k_2 = 0` as follows.

   >>> from toqito.states import gen_bell
   >>> dim = 3
   >>> k_1 = 0
   >>> k_2 = 0
   >>> gen_bell(k_1, k_2, dim)
   array([[0.33333333+0.j, 0.        +0.j, 0.        +0.j, 0.        +0.j,
           0.33333333+0.j, 0.        +0.j, 0.        +0.j, 0.        +0.j,
           0.33333333+0.j],
          [0.        +0.j, 0.        +0.j, 0.        +0.j, 0.        +0.j,
           0.        +0.j, 0.        +0.j, 0.        +0.j, 0.        +0.j,
           0.        +0.j],
          [0.        +0.j, 0.        +0.j, 0.        +0.j, 0.        +0.j,
           0.        +0.j, 0.        +0.j, 0.        +0.j, 0.        +0.j,
           0.        +0.j],
          [0.        +0.j, 0.        +0.j, 0.        +0.j, 0.        +0.j,
           0.        +0.j, 0.        +0.j, 0.        +0.j, 0.        +0.j,
           0.        +0.j],
          [0.33333333+0.j, 0.        +0.j, 0.        +0.j, 0.        +0.j,
           0.33333333+0.j, 0.        +0.j, 0.        +0.j, 0.        +0.j,
           0.33333333+0.j],
          [0.        +0.j, 0.        +0.j, 0.        +0.j, 0.        +0.j,
           0.        +0.j, 0.        +0.j, 0.        +0.j, 0.        +0.j,
           0.        +0.j],
          [0.        +0.j, 0.        +0.j, 0.        +0.j, 0.        +0.j,
           0.        +0.j, 0.        +0.j, 0.        +0.j, 0.        +0.j,
           0.        +0.j],
          [0.        +0.j, 0.        +0.j, 0.        +0.j, 0.        +0.j,
           0.        +0.j, 0.        +0.j, 0.        +0.j, 0.        +0.j,
           0.        +0.j],
          [0.33333333+0.j, 0.        +0.j, 0.        +0.j, 0.        +0.j,
           0.33333333+0.j, 0.        +0.j, 0.        +0.j, 0.        +0.j,
           0.33333333+0.j]])

   .. rubric:: References

   .. bibliography::
       :filter: docname in docnames

   :param k_1: An integer 0 <= k_1 <= n.
   :param k_2: An integer 0 <= k_2 <= n.
   :param dim: The dimension of the generalized Bell state.