toqito.channel_ops.dual_channel

Computes the dual of a map.

Module Contents

toqito.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 [@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.

]

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.

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

• operators. (whose entries are its Kraus)

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

• phi_op.shape. (from)

Returns:

The map dual to phi_op, in the same representation.

Return type:

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