toqito.channel_props.is_completely_positive
===========================================

.. py:module:: toqito.channel_props.is_completely_positive

.. autoapi-nested-parse::

   Determines if a channel is completely positive.





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

.. py:function:: is_completely_positive(phi, rtol = 1e-05, atol = 1e-08)

   Determine whether the given channel is completely positive.

   (Section: Linear Maps Of Square Operators from [@Watrous_2018_TQI]).

   A map \(\Phi \in \text{T} \left(\mathcal{X}, \mathcal{Y} \right)\) is *completely
   positive* if it holds that

   \[
       \Phi \otimes \mathbb{I}_{\text{L}(\mathcal{Z})}
   \]

   is a positive map for every complex Euclidean space \(\mathcal{Z}\).

   Alternatively, a channel is completely positive if the corresponding Choi matrix of the
   channel is both Hermitian-preserving and positive semidefinite.

   .. rubric:: Examples

   We can specify the input as a list of Kraus operators. Consider the map \(\Phi\) defined as

   \[
       \Phi(X) = X - U X U^*
   \]

   where

   \[
       U = \frac{1}{\sqrt{2}}
       \begin{pmatrix}
           1 & 1 \\
           -1 & 1
       \end{pmatrix}.
   \]

   This map is not completely positive, as we can verify as follows.

   ```python exec="1" source="above"
   import numpy as np
   from toqito.channel_props import is_completely_positive

   unitary_mat = np.array([[1, 1], [-1, 1]]) / np.sqrt(2)
   kraus_ops = [[np.identity(2), np.identity(2)], [unitary_mat, -unitary_mat]]

   print(is_completely_positive(kraus_ops))
   ```

   We can also specify the input as a Choi matrix. For instance, consider the Choi matrix
   corresponding to the \(2\)-dimensional completely depolarizing channel

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

   We may verify that this channel is completely positive

   ```python exec="1" source="above"
   from toqito.channels import depolarizing
   from toqito.channel_props import is_completely_positive

   print(is_completely_positive(depolarizing(2)))
   ```

   :param phi: The channel provided as either a Choi matrix or a list of Kraus operators.
   :param rtol: The relative tolerance parameter (default 1e-05).
   :param atol: The absolute tolerance parameter (default 1e-08).

   :returns: True if the channel is completely positive, and False otherwise.