toqito.channel_ops.apply_channel
================================

.. py:module:: toqito.channel_ops.apply_channel

.. autoapi-nested-parse::

   Applies a quantum channel to an operator.





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

.. py:function:: apply_channel(mat, phi_op)

   Apply a quantum channel to an operator.

   (Section: Representations and Characterizations of Channels of [@Watrous_2018_TQI]).

   Specifically, an application of the channel is defined as

   \[
       \Phi(X) = \text{Tr}_{\mathcal{X}} \left(J(\Phi)
       \left(\mathbb{I}_{\mathcal{Y}} \otimes X^{T}\right)\right),
   \]

   where

   \[
       J(\Phi): \text{T}(\mathcal{X}, \mathcal{Y}) \rightarrow
       \text{L}(\mathcal{Y} \otimes \mathcal{X})
   \]

   is the Choi representation of \(\Phi\).

   We assume the quantum channel given as `phi_op` is provided as either the Choi matrix
   of the channel or a set of Kraus operators that define the quantum channel.

   This function is adapted from the QETLAB package.

   .. rubric:: Examples

   The swap operator is the Choi matrix of the transpose map. The following is a (non-ideal,
   but illustrative) way of computing the transpose of a matrix.

   Consider the following matrix

   \[
       X = \begin{pmatrix}
               1 & 4 & 7 \\
               2 & 5 & 8 \\
               3 & 6 & 9
           \end{pmatrix}
   \]

   Applying the swap operator given as

   \[
       \Phi =
       \begin{pmatrix}
           1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
           0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
           0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
           0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1
        \end{pmatrix}
   \]

   to the matrix \(X\), we have the resulting matrix of

   \[
       \Phi(X) = \begin{pmatrix}
                       1 & 2 & 3 \\
                       4 & 5 & 6 \\
                       7 & 8 & 9
                  \end{pmatrix}
   \]

   Using `|toqito⟩`, we can obtain the above matrices as follows.

   ```python exec="1" source="above"
   from toqito.channel_ops import apply_channel
   from toqito.perms import swap_operator
   import numpy as np
   test_input_mat = np.array([[1, 4, 7], [2, 5, 8], [3, 6, 9]])
   print(apply_channel(test_input_mat, swap_operator(3)))
   ```

   :raises ValueError: If matrix is not Choi matrix.

   :param mat: A matrix.
   :param phi_op: A superoperator. `phi_op` should be provided either as a Choi matrix, or as a list of numpy arrays with
   :param either 1 or 2 columns whose entries are its Kraus operators.:

   :returns: The result of applying the superoperator `phi_op` to the operator `mat`.