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kraus_to_choi

Computes the Choi matrix of a list of Kraus operators.

kraus_to_choi 

kraus_to_choi(
    kraus_ops: list[ndarray] | list[list[ndarray]],
    sys: int = 2,
) -> ndarray

Compute the Choi matrix of a list of Kraus operators.

(Section: Kraus Representations of 1).

The Choi matrix of the list of Kraus operators, kraus_ops. The default convention is that the Choi matrix is the result of applying the map to the second subsystem of the standard maximally entangled (unnormalized) state. The Kraus operators are expected to be input as a list of numpy arrays (i.e. [[A_1, B_1],...,[A_n, B_n]]). In case the map is CP (completely positive), it suffices to input a flat list of operators omitting their conjugate transpose (i.e. [\(K_1\),..., \(K_n\)]).

This function was adapted from the QETLAB package.

Parameters:

  • kraus_ops (list[ndarray] | list[list[ndarray]]) –

    A list of Kraus operators.

  • sys (int, default: 2 ) –

    The subsystem on which the channel acts (default is 2).

Returns:

  • ndarray –

    The corresponding Choi matrix of the provided Kraus operators.

Examples:

The transpose map:

The Choi matrix of the transpose map is the swap operator. Notice that the transpose map is not completely positive.

import numpy as np
from toqito.channel_ops import kraus_to_choi
kraus_1 = np.array([[1, 0], [0, 0]])
kraus_2 = np.array([[1, 0], [0, 0]]).conj().T
kraus_3 = np.array([[0, 1], [0, 0]])
kraus_4 = np.array([[0, 1], [0, 0]]).conj().T
kraus_5 = np.array([[0, 0], [1, 0]])
kraus_6 = np.array([[0, 0], [1, 0]]).conj().T
kraus_7 = np.array([[0, 0], [0, 1]])
kraus_8 = np.array([[0, 0], [0, 1]]).conj().T

kraus_ops = [[kraus_1, kraus_2], [kraus_3, kraus_4], [kraus_5, kraus_6], [kraus_7, kraus_8]]
choi_op = kraus_to_choi(kraus_ops)
print(choi_op)

[[1. 0. 0. 0.]
 [0. 0. 1. 0.]
 [0. 1. 0. 0.]
 [0. 0. 0. 1.]]

See

choi_to_kraus

References

1 Watrous, John. The Theory of Quantum Information. (2018). doi:10.1017/9781316848142.

Source code in toqito/channel_ops/kraus_to_choi.py 
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def kraus_to_choi(kraus_ops: list[np.ndarray] | list[list[np.ndarray]], sys: int = 2) -> np.ndarray:
    r"""Compute the Choi matrix of a list of Kraus operators.

    (Section: Kraus Representations of [@watrous2018theory]).

    The Choi matrix of the list of Kraus operators, `kraus_ops`. The default convention is
    that the Choi matrix is the result of applying the map to the second subsystem of the
    standard maximally entangled (unnormalized) state. The Kraus operators are expected to be
    input as a list of numpy arrays (i.e. [[`A_1`, `B_1`],...,[`A_n`, `B_n`]]).
    In case the map is CP (completely positive), it suffices to input a flat list of operators omitting
    their conjugate transpose (i.e. [\(K_1\),..., \(K_n\)]).

    This function was adapted from the QETLAB package.

    Args:
        kraus_ops: A list of Kraus operators.
        sys: The subsystem on which the channel acts (default is 2).

    Returns:
        The corresponding Choi matrix of the provided Kraus operators.

    Examples:
        The transpose map:

        The Choi matrix of the transpose map is the swap operator. Notice that the transpose map
        is *not* completely positive.

        ```python exec="1" source="above" result="text"
        import numpy as np
        from toqito.channel_ops import kraus_to_choi
        kraus_1 = np.array([[1, 0], [0, 0]])
        kraus_2 = np.array([[1, 0], [0, 0]]).conj().T
        kraus_3 = np.array([[0, 1], [0, 0]])
        kraus_4 = np.array([[0, 1], [0, 0]]).conj().T
        kraus_5 = np.array([[0, 0], [1, 0]])
        kraus_6 = np.array([[0, 0], [1, 0]]).conj().T
        kraus_7 = np.array([[0, 0], [0, 1]])
        kraus_8 = np.array([[0, 0], [0, 1]]).conj().T

        kraus_ops = [[kraus_1, kraus_2], [kraus_3, kraus_4], [kraus_5, kraus_6], [kraus_7, kraus_8]]
        choi_op = kraus_to_choi(kraus_ops)
        print(choi_op)
        ```

        !!! See Also
            [choi_to_kraus][toqito.channel_ops.choi_to_kraus.choi_to_kraus]

    """
    if sys < 0:
        raise ValueError("The `sys` parameter must be non-negative.")

    dim_in, _, _ = channel_dim(kraus_ops)
    dim_op_1, dim_op_2 = dim_in

    choi_mat = partial_channel(
        max_entangled(dim_op_1, False, False) @ max_entangled(dim_op_2, False, False).conj().T,
        kraus_ops,
        sys,
        np.array([[dim_op_1, dim_op_1], [dim_op_2, dim_op_2]]),
    )

    return choi_mat