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choi_rank

Calculates the Choi rank of a channel.

choi_rank 

choi_rank(phi: ndarray | list[list[ndarray]]) -> int

Calculate the rank of the Choi representation of a quantum channel.

(Section 2.2: Quantum Channels from 1).

Parameters:

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

    Either a Choi matrix or a list of Kraus operators

Returns:

  • int –

    The Choi rank of the provided channel representation.

Raises:

  • ValueError –

    If matrix is not Choi.

Examples:

The transpose map can be written either in Choi representation (as a SWAP operator) or in Kraus representation. If we choose the latter, it will be given by the following matrices:

\[ \begin{equation} \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}, \quad \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \quad \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}. \end{equation} \]

and can be generated in |toqito⟩ with the following list:

import numpy as np
from toqito.channel_props import choi_rank

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]]

print(choi_rank(kraus_ops))

4

We can the verify the associated Choi representation (the SWAP gate) gets the same Choi rank:

import numpy as np
from toqito.channel_props import choi_rank
choi_matrix = np.array([[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]])
print(choi_rank(choi_matrix))

4

References

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

Source code in toqito/channel_props/choi_rank.py 
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def choi_rank(phi: np.ndarray | list[list[np.ndarray]]) -> int:
    r"""Calculate the rank of the Choi representation of a quantum channel.

    (Section 2.2: Quantum Channels from [@watrous2018theory]).

    Args:
        phi: Either a Choi matrix or a list of Kraus operators

    Returns:
        The Choi rank of the provided channel representation.

    Raises:
        ValueError: If matrix is not Choi.

    Examples:
        The transpose map can be written either in Choi representation (as a
        SWAP operator) or in Kraus representation. If we choose the latter, it
        will be given by the following matrices:

        \[
            \begin{equation}
                \frac{1}{\sqrt{2}}
                \begin{pmatrix}
                    0 & i \\ -i & 0
                \end{pmatrix}, \quad
                \frac{1}{\sqrt{2}}
                \begin{pmatrix}
                    0 & 1 \\
                    1 & 0
                \end{pmatrix}, \quad
                \begin{pmatrix}
                    1 & 0 \\
                    0 & 0
                \end{pmatrix}, \quad
                \begin{pmatrix}
                    0 & 0 \\
                    0 & 1
                \end{pmatrix}.
            \end{equation}
        \]

        and can be generated in `|toqito⟩` with the following list:

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

        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]]

        print(choi_rank(kraus_ops))
        ```

        We can the verify the associated Choi representation (the SWAP gate)
        gets the same Choi rank:

        ```python exec="1" source="above" result="text"
        import numpy as np
        from toqito.channel_props import choi_rank
        choi_matrix = np.array([[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]])
        print(choi_rank(choi_matrix))
        ```

    """
    if isinstance(phi, list):
        phi = kraus_to_choi(phi)
    elif not isinstance(phi, np.ndarray):
        raise ValueError("Not a valid Choi matrix.")

    return np.linalg.matrix_rank(phi)