toqito.channel_ops.dual_channel
===============================

.. py:module:: toqito.channel_ops.dual_channel

.. autoapi-nested-parse::

   Computes the dual of a map.





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

.. py:function:: dual_channel(phi_op, dims = None)

   Compute the dual of a map (quantum channel).

   (Section: Representations and Characterizations of Channels of [@Watrous_2018_TQI]).

   The map can be represented as a Choi matrix, with optional specification of input
   and output dimensions. If the input channel maps \(M_{r,c}\) to \(M_{x,y}\)
   then `dim` should be the list `[[r,x], [c,y]]`. If it maps \(M_m\)
   to \(M_n\), then `dim` can simply be the vector `[m,n]`. In this
   case the Choi matrix of the dual channel is returned, obtained by swapping input and
   output (see [`swap`][toqito.perms.swap.swap]), and complex conjugating all elements.

   The map can also be represented as a list of Kraus operators.
   A list of lists, each containing two elements, corresponds to the families
   of operators \(\{(A_a, B_a)\}\) representing the map

   \[
       \Phi(X) = \sum_a A_a X B^*_a.
   \]

   The dual map is obtained by taking the Hermitian adjoint of each operator.
   If `phi_op` is given as a one-dimensional list, \(\{A_a\}\),
   it is interpreted as the completely positive map

   \[
       \Phi(X) = \sum_a A_a X A^*_a.
   \]

   .. rubric:: Examples

   When a channel is represented by a 1-D list of of Kraus operators, the CPTP dual channel can be determined
   as shown below.

   ```python exec="1" source="above"
   import numpy as np
   from toqito.channel_ops import dual_channel
   kraus_1 = np.array([[1, 0, 1j, 0]])
   kraus_2 = np.array([[0, 1, 0, 1j]])
   kraus_list = [kraus_1, kraus_2]
   dual_kraus = dual_channel(kraus_list)
   for i, op in enumerate(dual_kraus, start=1):
      print(f"\nDual Kraus operator {i}:")
      print(op)
   ```


   If the input channel's dimensions are different from the output dual channel's dimensions,

   ```python exec="1" source="above"
   import numpy as np
   from toqito.channel_ops import dual_channel
   from toqito.perms import swap_operator
   input_op = swap_operator([2, 3])
   output = dual_channel(input_op, [[3, 2], [2, 3]])
   print(output)
   ```

   :raises ValueError: If matrices are not Choi matrix.

   :param phi_op: A superoperator. It should be provided either as a Choi matrix, or as a (1d or 2d) list of numpy arrays
   :param whose entries are its Kraus operators.:
   :param dims: Dimension of the input and output systems, for Choi matrix representation. If `None`, try to infer them
   :param from `phi_op.shape`.:

   :returns: The map dual to `phi_op`, in the same representation.