:py:mod:`channels.depolarizing`
===============================

.. py:module:: channels.depolarizing

.. autoapi-nested-parse::

   The depolarizing channel.



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


Functions
~~~~~~~~~

.. autoapisummary::

   channels.depolarizing.depolarizing



.. py:function:: depolarizing(dim, param_p = 0)

   Produce the partially depolarizing channel.

   (Section: Replacement Channels and the Completely Depolarizing Channel from
   :cite:`Watrous_2018_TQI`).

   The Choi matrix of the completely depolarizing channel :cite:`WikiDepo` that acts on
   :code:`dim`-by-:code:`dim` matrices.

   The *completely depolarizing channel* is defined as

   .. math::
       \Omega(X) = \text{Tr}(X) \omega

   for all :math:`X \in \text{L}(\mathcal{X})`, where

   .. math::
       \omega = \frac{\mathbb{I}_{\mathcal{X}}}{\text{dim}(\mathcal{X})}

   denotes the completely mixed stated defined with respect to the space :math:`\mathcal{X}`.

   .. rubric:: Examples

   The completely depolarizing channel maps every density matrix to the maximally-mixed state.
   For example, consider the density operator

   .. math::
       \rho = \frac{1}{2} \begin{pmatrix}
                            1 & 0 & 0 & 1 \\
                            0 & 0 & 0 & 0 \\
                            0 & 0 & 0 & 0 \\
                            1 & 0 & 0 & 1
                          \end{pmatrix}

   corresponding to one of the Bell states. Applying the depolarizing channel to :math:`\rho` we
   have that

   .. math::
       \Phi(\rho) = \frac{1}{4} \begin{pmatrix}
                                   1 & 0 & 0 & 0 \\
                                   0 & 1 & 0 & 0 \\
                                   0 & 0 & 1 & 0 \\
                                   0 & 0 & 0 & 1
                                \end{pmatrix}.

   This can be observed in :code:`toqito` as follows.

   >>> from toqito.channel_ops import apply_channel
   >>> from toqito.channels import depolarizing
   >>> import numpy as np
   >>> test_input_mat = np.array([[1 / 2, 0, 0, 1 / 2], [0, 0, 0, 0], [0, 0, 0, 0], [1 / 2, 0, 0, 1 / 2]])
   >>> apply_channel(test_input_mat, depolarizing(4))
   array([[0.25, 0.  , 0.  , 0.  ],
          [0.  , 0.25, 0.  , 0.  ],
          [0.  , 0.  , 0.25, 0.  ],
          [0.  , 0.  , 0.  , 0.25]])

   >>> from toqito.channel_ops import apply_channel
   >>> from toqito.channels import depolarizing
   >>> import numpy as np
   >>> test_input_mat = np.array(
   ...     [[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]]
   ... )
   >>> apply_channel(test_input_mat, depolarizing(4, 0.5))
   array([[ 4.75,  1.  ,  1.5 ,  2.  ],
          [ 2.5 ,  7.25,  3.5 ,  4.  ],
          [ 4.5 ,  5.  ,  9.75,  6.  ],
          [ 6.5 ,  7.  ,  7.5 , 12.25]])

   .. rubric:: References

   .. bibliography::
       :filter: docname in docnames

   :param dim: The dimensionality on which the channel acts.
   :param param_p: Default 0.
   :return: The Choi matrix of the completely depolarizing channel.