:py:mod:`channel_metrics.channel_fidelity`
==========================================

.. py:module:: channel_metrics.channel_fidelity

.. autoapi-nested-parse::

   Compute the channel fidelity between two quantum channels.



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


Functions
~~~~~~~~~

.. autoapisummary::

   channel_metrics.channel_fidelity.channel_fidelity



.. py:function:: channel_fidelity(choi_1, choi_2)

   Compute the channel fidelity between two quantum channels :cite:`Katariya_2021_Geometric`.

   Let :math:`\Phi : \text{L}(\mathcal{Y}) \rightarrow \text{L}(\mathcal{X})` and
   :math:`\Psi: \text{L}(\mathcal{Y}) \rightarrow \text{L}(\mathcal{X})` be quantum channels. Then
   the root channel fidelity defined as

   .. math::
       \sqrt{F}(\Phi, \Psi) := \text{inf}_{\rho} \sqrt{F}(\Phi(\rho), \Psi(\rho))

   where :math:`\rho \in \text{D}(\mathcal{Z} \otimes \mathcal{X})` can be calculated by means of
   the following semidefinite program (Proposition 50) in :cite:`Katariya_2021_Geometric`,

   .. math::
       \begin{align*}
           \text{maximize:} \quad & \lambda \\
           \text{subject to:} \quad & \lambda \mathbb{I}_{\mathcal{Z}} \leq
               \text{Re}\left( \text{Tr}_{\mathcal{Y}} \left( Q \right) \right),\\
               & \begin{pmatrix}
                   J(\Phi) & Q^* \\
                   Q & J(\Psi)
               \end{pmatrix} \geq 0
       \end{align*}

   where :math:`Q \in \text{L}(\mathcal{Z} \otimes \mathcal{X})`.

   .. rubric:: Examples

   For two identical channels, we should expect that the channel fidelity should yield a value of
   :math:`1`.

   >>> from toqito.channels import dephasing
   >>> from toqito.channel_metrics import channel_fidelity
   >>>
   >>> # The Choi matrices of dimension-4 for the dephasing channel
   >>> choi_1 = dephasing(4)
   >>> choi_2 = dephasing(4)
   >>> '%.2f' % channel_fidelity(choi_1, choi_2)
   '1.00'

   .. note::
       You do not need to use `'%.2f' %` when you use this function.

       We use this to format our output such that `doctest` compares the calculated output to the
       expected output upto two decimal points only. The accuracy of the solvers can calculate the
       `float` output to a certain amount of precision such that the value deviates after a few digits
       of accuracy.

   We can also compute the channel fidelity between two different channels. For example, we can
   compute the channel fidelity between the dephasing and depolarizing channels.

   >>> from toqito.channels import dephasing, depolarizing
   >>> from toqito.channel_metrics import channel_fidelity
   >>>
   >>> # The Choi matrices of dimension-4 for the dephasing and depolarizing channels
   >>> choi_1 = dephasing(4)
   >>> choi_2 = depolarizing(4)
   >>> '%.2f' % channel_fidelity(choi_1, choi_2)
   '0.50'

   .. note::
       You do not need to use `'%.2f' %` when you use this function.

       We use this to format our output such that `doctest` compares the calculated output to the
       expected output upto two decimal points only. The accuracy of the solvers can calculate the
       `float` output to a certain amount of precision such that the value deviates after a few digits
       of accuracy.


   .. rubric:: References

   .. bibliography::
       :filter: docname in docnames

   :raises ValueError: If matrices are not of equal dimension.
   :raises ValueError: If matrices are not square.
   :param choi_1: The Choi matrix of the first quantum channel.
   :param choi_2: The Choi matrix of the second quantum channel.
   :return: The channel fidelity between the channels specified by the quantum channels
            corresponding to the Choi matrices :code:`choi_1` and :code:`choi_2`.