channel_ops.dual_channel

Computes the dual of a map.

Functions

dual_channel(phi_op[, dims])

Compute the dual of a map (quantum channel).

Module Contents

channel_ops.dual_channel.dual_channel(phi_op, dims=None)

Compute the dual of a map (quantum channel).

(Section: Representations and Characterizations of Channels of [1]).

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 toqito.perms.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

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

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.

>>> 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_channel(kraus_list)
[array([[1.-0.j],
       [0.-0.j],
       [0.-1.j],
       [0.-0.j]]), array([[0.-0.j],
       [1.-0.j],
       [0.-0.j],
       [0.-1.j]])]

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

>>> import numpy as np
>>> from toqito.channel_ops import dual_channel
>>> from toqito.perms import swap_operator
>>> input_op = swap_operator([2, 3])
>>> dual_channel(input_op, [[3, 2], [2, 3]])
array([[1., 0., 0., 0., 0., 0.],
       [0., 0., 1., 0., 0., 0.],
       [0., 0., 0., 0., 1., 0.],
       [0., 1., 0., 0., 0., 0.],
       [0., 0., 0., 1., 0., 0.],
       [0., 0., 0., 0., 0., 1.]])

References

[1]

John Watrous. The Theory of Quantum Information. Cambridge University Press, 2018. URL: https://johnwatrous.com/wp-content/uploads/TQI.pdf, doi:10.1017/9781316848142.

Raises:

ValueError – If matrices are not Choi matrix.

Parameters:

• phi_op (numpy.ndarray | list[numpy.ndarray] | list[list[numpy.ndarray]]) – 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.

• dims (list[int]) – Dimension of the input and output systems, for Choi matrix representation. If None, try to infer them from phi_op.shape.

Returns:

The map dual to phi_op, in the same representation.

Return type:

numpy.ndarray | list[list[numpy.ndarray]]