:py:mod:`channel_props.is_positive`
===================================

.. py:module:: channel_props.is_positive

.. autoapi-nested-parse::

   Is channel positive.



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


Functions
~~~~~~~~~

.. autoapisummary::

   channel_props.is_positive.is_positive



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

   Determine whether the given channel is positive.

   (Section: Linear Maps Of Square Operators from :cite:`Watrous_2018_TQI`).

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

   .. math::
       \Phi(P) \in \text{Pos}(\mathcal{Y})

   for every positive semidefinite operator :math:`P \in \text{Pos}(\mathcal{X})`.

   Alternatively, a channel is 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 :math:`\Phi` defined as

   .. math::
       \Phi(X) = X - U X U^*

   where

   .. math::
       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.

   >>> from toqito.channel_props import is_positive
   >>> import numpy as np
   >>> unitary_mat = np.array([[1, 1], [-1, -1]]) / np.sqrt(2)
   >>> kraus_ops = [[np.identity(2), np.identity(2)], [unitary_mat, -unitary_mat]]
   >>> is_positive(kraus_ops)
   False

   We can also specify the input as a Choi matrix. For instance, consider the Choi matrix
   corresponding to the :math:`4`-dimensional completely depolarizing channel and may verify
   that this channel is positive.

   >>> from toqito.channels import depolarizing
   >>> from toqito.channel_props import is_positive
   >>> is_positive(depolarizing(4))
   True

   .. rubric:: References

   .. bibliography::
       :filter: docname in docnames

   :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).
   :return: True if the channel is positive, and False otherwise.