Source code for toqito.channels.choi

"""Generates the Choi channel."""

import numpy as np

from toqito.states import max_entangled




def choi(a_var: int = 1, b_var: int = 1, c_var: int = 0) -> np.ndarray:
    r"""Produce the Choi channel or one of its generalizations [@Choi_1992_Generalized].

    The *Choi channel* is a positive map on 3-by-3 matrices that is capable of detecting some
    entanglement that the transpose map is not.

    The standard Choi channel defined with `a=1`, `b=1`, and `c=0` is the Choi
    matrix of the positive map defined in [@Choi_1992_Generalized]. Many of these maps are capable of detecting
    PPT entanglement.

    Examples:
        The standard Choi channel is given as

        \[
            \Phi_{1, 1, 0} =
            \begin{pmatrix}
                1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 \\
                0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
                0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
                0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
                -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 \\
                0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
                0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
                0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
                -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 1
            \end{pmatrix}
        \]

        We can generate the Choi channel in `|toqito⟩` as follows.

        ```python exec="1" source="above"
        import numpy as np
        from toqito.channels import choi

        print(choi())
        ```

        The reduction channel is the map \(R\) defined by:

        \[
            R(X) = \text{Tr}(X) \mathbb{I} - X.
        \]

        The matrix correspond to this is given as

        \[
            \Phi_{0, 1, 1} =
            \begin{pmatrix}
                0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 \\
                0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
                0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
                0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
                -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\
                0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
                0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
                0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
                -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0
            \end{pmatrix}
        \]

        The reduction channel is the Choi channel that arises when `a = 0` and when `b =
        c = 1`. We can obtain this matrix using `|toqito⟩` as follows.

        ```python exec="1" source="above"
        import numpy as np
        from toqito.channels import choi

        print(choi(0, 1, 1))
        ```

        !!! See Also
            [reduction][toqito.channels.reduction.reduction]

    Args:
        a_var: Default integer for standard Choi map.
        b_var: Default integer for standard Choi map.
        c_var: Default integer for standard Choi map.

    Returns:
        The Choi channel (or one of its  generalizations).

    """
    psi = max_entangled(3, False, False)
    return np.diag([a_var + 1, c_var, b_var, b_var, a_var + 1, c_var, c_var, b_var, a_var + 1]) - psi @ psi.conj().T