Source code for toqito.channel_props.is_herm_preserving

"""Determines if a channel is Hermiticity-preserving."""

import numpy as np

from toqito.channel_ops import kraus_to_choi
from toqito.matrix_props import is_hermitian




def is_herm_preserving(
    phi: np.ndarray | list[list[np.ndarray]],
    rtol: float = 1e-05,
    atol: float = 1e-08,
) -> bool:
    r"""Determine whether the given channel is Hermitian-preserving.

    (Section: Linear Maps Of Square Operators from [@Watrous_2018_TQI]).

    A map \(\Phi \in \text{T} \left(\mathcal{X}, \mathcal{Y} \right)\) is
    *Hermitian-preserving* if it holds that

    \[
        \Phi(H) \in \text{Herm}(\mathcal{Y})
    \]

    for every Hermitian operator \(H \in \text{Herm}(\mathcal{X})\).

    Examples:
        The map \(\Phi\) defined as

        \[
            \Phi(X) = X - U X U^*
        \]

        is Hermitian-preserving, where

        \[
            U = \frac{1}{\sqrt{2}}
            \begin{pmatrix}
                1 & 1 \\
                -1 & 1
            \end{pmatrix}.
        \]

        ```python exec="1" source="above"
        import numpy as np
        from toqito.channel_props import is_herm_preserving

        unitary_mat = np.array([[1, 1], [-1, 1]]) / np.sqrt(2)
        kraus_ops = [[np.identity(2), np.identity(2)], [unitary_mat, -unitary_mat]]

        print(is_herm_preserving(kraus_ops))
        ```


        We may also verify whether the corresponding Choi matrix of a given map is
        Hermitian-preserving. The swap operator is the Choi matrix of the transpose map, which is
        Hermitian-preserving as can be seen as follows:

        ```python exec="1" source="above"
        import numpy as np
        from toqito.perms import swap_operator
        from toqito.channel_props import is_herm_preserving

        unitary_mat = np.array([[1, 1], [-1, 1]]) / np.sqrt(2)
        choi_mat = swap_operator(3)

        print(is_herm_preserving(choi_mat))
        ```

    Args:
        phi: The channel provided as either a Choi matrix or a list of Kraus operators.
        rtol: The relative tolerance parameter (default 1e-05).
        atol: The absolute tolerance parameter (default 1e-08).

    Returns:
        True if the channel is Hermitian-preserving, and False otherwise.

    """
    # If the variable `phi` is provided as a list, we assume this is a list
    # of Kraus operators.
    if isinstance(phi, list):
        phi = kraus_to_choi(phi)

    # Phi is Hermiticity-preserving if and only if its Choi matrix is Hermitian.
    if phi.shape[0] != phi.shape[1]:
        return False
    return is_hermitian(phi, rtol=rtol, atol=atol)