toqito.channel_metrics.channel_fidelity
=======================================

.. py:module:: toqito.channel_metrics.channel_fidelity

.. autoapi-nested-parse::

   Computes the channel fidelity between two quantum channels.





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

.. py:function:: channel_fidelity(choi_1, choi_2, eps = 1e-07)

   Compute the channel fidelity between two quantum channels [@Katariya_2021_Geometric].

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

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

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

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

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

   .. rubric:: Examples

   For two identical channels, we should expect that the channel fidelity should yield a value of
   \(1\).

   ```python exec="1" source="above"
   import numpy as np
   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)
   print(channel_fidelity(choi_1, choi_2))
   ```

   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.

   ```python exec="1" source="above"
   import numpy as np
   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)
   print(channel_fidelity(choi_1, choi_2))
   ```

   :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.
   :param eps: The solver tolerance for convergence to feasability.

   :returns: The channel fidelity between the channels specified by the quantum channels corresponding to the Choi matrices
             `choi_1` and `choi_2`.