toqito.channels.depolarizing
============================

.. py:module:: toqito.channels.depolarizing

.. autoapi-nested-parse::

   Generates the depolarizing channel.





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

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

   Produce the partially depolarizing channel.

   (Section: Replacement Channels and the Completely Depolarizing Channel from
   [@Watrous_2018_TQI]).

   The Choi matrix of the completely depolarizing channel [@WikiDepo] that acts on
   `dim`-by-`dim` matrices.

   The *completely depolarizing channel* is defined as

   \[
       \Omega(X) = \text{Tr}(X) \omega
   \]

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

   \[
       \omega = \frac{\mathbb{I}_{\mathcal{X}}}{\text{dim}(\mathcal{X})}
   \]

   denotes the completely mixed stated defined with respect to the space \(\mathcal{X}\).

   .. rubric:: Examples

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

   \[
       \rho = \frac{1}{2} \begin{pmatrix}
                            1 & 0 & 0 & 1 \\
                            0 & 0 & 0 & 0 \\
                            0 & 0 & 0 & 0 \\
                            1 & 0 & 0 & 1
                          \end{pmatrix}
   \]

   corresponding to one of the Bell states. Applying the depolarizing channel to \(\rho\) we
   have that

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

   This can be observed in `|toqito⟩` as follows.

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

   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]])

   print(apply_channel(test_input_mat, depolarizing(4)))
   ```

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

   test_input_mat = np.arange(1, 17).reshape(4, 4)

   print(apply_channel(test_input_mat, depolarizing(4, 0.5)))
   ```

   :raises ValueError: If `param_p` is outside the interval [0,1].

   :param dim: The dimensionality on which the channel acts.
   :param param_p: Depolarizing probability \(p \) \in [0,1] that mixes the input state with the maximally mixed state.
   :param Default 0.:

   :returns: The Choi matrix of the completely depolarizing channel.