toqito.channels.choi
====================

.. py:module:: toqito.channels.choi

.. autoapi-nested-parse::

   Generates the Choi channel.





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

.. py:function:: choi(a_var = 1, b_var = 1, c_var = 0)

   Produce the Choi channel or one of its generalizations [@Choi_1992_Generalized].

   The *Choi channel* is a positive map on 3-by-3 matrices that is capable of detecting some
   entanglement that the transpose map is not.

   The standard Choi channel defined with `a=1`, `b=1`, and `c=0` is the Choi
   matrix of the positive map defined in [@Choi_1992_Generalized]. Many of these maps are capable of detecting
   PPT entanglement.

   .. rubric:: Examples

   The standard Choi channel is given as

   \[
       \Phi_{1, 1, 0} =
       \begin{pmatrix}
           1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
           -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
           -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 1
       \end{pmatrix}
   \]

   We can generate the Choi channel in `|toqito⟩` as follows.

   ```python exec="1" source="above"
   import numpy as np
   from toqito.channels import choi

   print(choi())
   ```

   The reduction channel is the map \(R\) defined by:

   \[
       R(X) = \text{Tr}(X) \mathbb{I} - X.
   \]

   The matrix correspond to this is given as

   \[
       \Phi_{0, 1, 1} =
       \begin{pmatrix}
           0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 \\
           0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
           -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\
           0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
           -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0
       \end{pmatrix}
   \]

   The reduction channel is the Choi channel that arises when `a = 0` and when `b =
   c = 1`. We can obtain this matrix using `|toqito⟩` as follows.

   ```python exec="1" source="above"
   import numpy as np
   from toqito.channels import choi

   print(choi(0, 1, 1))
   ```

   !!! See Also
       [reduction][toqito.channels.reduction.reduction]

   :param a_var: Default integer for standard Choi map.
   :param b_var: Default integer for standard Choi map.
   :param c_var: Default integer for standard Choi map.

   :returns: The Choi channel (or one of its  generalizations).