toqito.channel_metrics.channel_distinguishability
=================================================

.. py:module:: toqito.channel_metrics.channel_distinguishability

.. autoapi-nested-parse::

   Computes the maximum probability of distinguishing two quantum channels.





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

.. py:function:: channel_distinguishability(phi, psi, p, dim = None, strategy = 'bayesian', solver = 'cvxopt', primal_dual = 'dual', **kwargs)

   Compute the optimal probability of distinguishing two quantum channels.

   Bayesian and minimax discrimination of two quantum channels are implemented.

   For Bayesian discrimination, channels to be distinguished should have a given a priori probability distribution.
   The task of discriminating channels can be connected to the completely bounded trace norm
   (Section 3.3.3 of [@Watrous_2018_TQI]).
   The problem is finding POVMs for which error probability of discrimination of
   output states is minimized after input state is acted on by the two quantum channels.
   In the language of statistical decision theory, the problem is equivalent to minimizing quantum Bayes' risk.

   In the minimax problem, there are no a priori probabilities.
   Minimax discrimination of two channels consists of finding the
   optimal input state so that the two possible output states are discriminated
   with minimum risk. ([@d2005minimax]).

   QETLAB's functionality inspired the Bayesian option [@QETLAB_link]
   and the minimax option is adapted from QuTIpy [@QuTIpy_link].

   .. rubric:: Examples

   Optimal probability of distinguishing two amplitude damping channels in the Bayesian setting:

   ```python exec="1" source="above"
   from toqito.channels import amplitude_damping
   from toqito.channel_ops import kraus_to_choi
   from toqito.channel_metrics import channel_distinguishability
   # Define two amplitude damping channels with gamma=0.25 and gamma=0.5
   choi_ch_1 = kraus_to_choi(amplitude_damping(gamma=0.25))
   choi_ch_2 = kraus_to_choi(amplitude_damping(gamma=0.5))

   p = [0.5, 0.5]

   print(channel_distinguishability(choi_ch_1, choi_ch_2, p))
   ```

   Optimal probability of distinguishing two amplitude damping channels in the minimax setting:

   ```python exec="1" source="above"
   from toqito.channels import amplitude_damping
   from toqito.channel_ops import kraus_to_choi
   from toqito.channel_metrics import channel_distinguishability
   # Define two amplitude damping channels with gamma=0.25 and gamma=0.5
   choi_ch_1 = kraus_to_choi(amplitude_damping(gamma=0.25))
   choi_ch_2 = kraus_to_choi(amplitude_damping(gamma=0.5))

   print(channel_distinguishability(choi_ch_1, choi_ch_2, None, [2, 2], strategy="minimax",primal_dual="primal"))
   ```

   :raises ValueError: If prior probabilities not provided at all for Bayesian strategy.
   :raises ValueError: If strategy is neither Bayesian nor minimax.
   :raises ValueError: If channels have different input or output dimensions.
   :raises ValueError: If prior probabilities do not add up to 1.
   :raises ValueError: If number of prior probabilities not equal to 2.

   :param phi: A superoperator. It should be provided either as a Choi matrix,
               or as a (1d or 2d) list of numpy arrays whose entries are its Kraus operators.
   :param psi: A superoperator. It should be provided either as a Choi matrix,
               or as a (1d or 2d) list of numpy arrays whose entries are its Kraus operators.
   :param p: Prior probabilities of the two channels.
   :param dim: Input and output dimensions of the channels.
   :param strategy: Whether to perform Bayesian or minimax discrimination task. Possible
                    values are "Bayesian" and "minimax". Default option is `strategy="Bayesian"`.
   :param solver: Optimization option for `picos` solver. Default option is `solver="cvxopt"`.
   :param primal_dual: Option for the optimization problem. Default option is `solver="cvxopt"`.
   :param kwargs: Additional arguments to pass to picos' solve method.

   :returns: The optimal probability of discriminating two quantum channels.