:py:mod:`channel_props.choi_rank`
=================================

.. py:module:: channel_props.choi_rank

.. autoapi-nested-parse::

   Calculate the Choi rank of a channel.



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


Functions
~~~~~~~~~

.. autoapisummary::

   channel_props.choi_rank.choi_rank



.. py:function:: choi_rank(phi)

   Calculate the rank of the Choi representation of a quantum channel.

   (Section 2.2: Quantum Channels from :cite:`Watrous_2018_TQI`).

   .. rubric:: Examples

   The transpose map can be written either in Choi representation (as a
   SWAP operator) or in Kraus representation. If we choose the latter, it
   will be given by the following matrices:

   .. math::
       \begin{equation}
           \begin{aligned}
               \frac{1}{\sqrt{2}}
               \begin{pmatrix}
                   0 & i \\ -i & 0
               \end{pmatrix}, &\quad
               \frac{1}{\sqrt{2}}
               \begin{pmatrix}
                   0 & 1 \\
                   1 & 0
               \end{pmatrix}, \\
               \begin{pmatrix}
                   1 & 0 \\
                   0 & 0
               \end{pmatrix}, &\quad
               \begin{pmatrix}
                   0 & 0 \\
                   0 & 1
               \end{pmatrix}.
           \end{aligned}
       \end{equation}

   and can be generated in :code:`toqito` with the following list:

   >>> import numpy as np
   >>> kraus_1 = np.array([[1, 0], [0, 0]])
   >>> kraus_2 = np.array([[1, 0], [0, 0]]).conj().T
   >>> kraus_3 = np.array([[0, 1], [0, 0]])
   >>> kraus_4 = np.array([[0, 1], [0, 0]]).conj().T
   >>> kraus_5 = np.array([[0, 0], [1, 0]])
   >>> kraus_6 = np.array([[0, 0], [1, 0]]).conj().T
   >>> kraus_7 = np.array([[0, 0], [0, 1]])
   >>> kraus_8 = np.array([[0, 0], [0, 1]]).conj().T
   >>> kraus_ops = [[kraus_1, kraus_2], [kraus_3, kraus_4],[kraus_5, kraus_6],[kraus_7, kraus_8]]

   To calculate its Choi rank, we proceed in the following way:

   >>> from toqito.channel_props import choi_rank
   >>> choi_rank(kraus_ops)
   4

   We can the verify the associated Choi representation (the SWAP gate)
   gets the same Choi rank:

   >>> choi_matrix = np.array([[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]])
   >>> choi_rank(choi_matrix)
   4

   .. rubric:: References

   .. bibliography::
       :filter: docname in docnames

   :raises ValueError: If matrix is not Choi.
   :param phi: Either a Choi matrix or a list of Kraus operators
   :return: The Choi rank of the provided channel representation.