:py:mod:`states.gen_bell`
=========================

.. py:module:: states.gen_bell

.. autoapi-nested-parse::

   Generalized Bell state.



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


Functions
~~~~~~~~~

.. autoapisummary::

   states.gen_bell.gen_bell



.. 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.