channel_props.choi_rank
=======================

.. py:module:: channel_props.choi_rank

.. autoapi-nested-parse::

   Calculates the Choi rank of a channel.



Functions
---------

.. autoapisummary::

   channel_props.choi_rank.choi_rank


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

.. 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}
           \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}, \quad
           \begin{pmatrix}
               1 & 0 \\
               0 & 0
           \end{pmatrix}, \quad
           \begin{pmatrix}
               0 & 0 \\
               0 & 1
           \end{pmatrix}.
       \end{equation}

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

   .. jupyter-execute::

    import numpy as np
    from toqito.channel_props import choi_rank

    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]]

    choi_rank(kraus_ops)

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

   .. jupyter-execute::

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

   .. 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.