Skip to content

is_quantum_channel

Determines if an input is a quantum channel.

is_quantum_channel 

is_quantum_channel(
    phi: ndarray | list[list[ndarray]],
    rtol: float = 1e-05,
    atol: float = 1e-08,
) -> bool

Determine whether the given input is a quantum channel.

For more info, see Section 2.2.1: Definitions and Basic Notions Concerning Channels from 1.

A map \(\Phi \in \text{T} \left(\mathcal{X}, \mathcal{Y} \right)\) is a quantum channel for some choice of complex Euclidean spaces \(\mathcal{X}\) and \(\mathcal{Y}\), if it holds that:

  1. \(\Phi\) is completely positive.
  2. \(\Phi\) is trace preserving.

Parameters:

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

    The channel provided as either a Choi matrix or a list of Kraus operators.

  • rtol (float, default: 1e-05 ) –

    The relative tolerance parameter (default 1e-05).

  • atol (float, default: 1e-08 ) –

    The absolute tolerance parameter (default 1e-08).

Returns:

  • bool –

    True if the channel is a quantum channel, and False otherwise.

Examples:

We can specify the input as a list of Kraus operators. Consider the map \(\Phi\) defined as

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

where

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

To check if this is a valid quantum channel or not,

import numpy as np
from toqito.matrices import pauli
from toqito.channel_props import is_quantum_channel

U = (1/np.sqrt(2))*np.array([[1, 1],[-1, 1]])
X = pauli("X")
phi = X - np.matmul(U, np.matmul(X, np.conjugate(U)))

print(is_quantum_channel(phi))

False

If we instead check for the validity of depolarizing channel being a valid quantum channel,

from toqito.channels import depolarizing
from toqito.channel_props import is_quantum_channel

choi_depolarizing = depolarizing(dim=2, param_p=0.2)

print(is_quantum_channel(choi_depolarizing))

True

References

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

Source code in toqito/channel_props/is_quantum_channel.py 
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
def is_quantum_channel(
    phi: np.ndarray | list[list[np.ndarray]],
    rtol: float = 1e-05,
    atol: float = 1e-08,
) -> bool:
    r"""Determine whether the given input is a quantum channel.

    For more info, see Section 2.2.1: Definitions and Basic Notions Concerning Channels from
    [@watrous2018theory].

    A map \(\Phi \in \text{T} \left(\mathcal{X}, \mathcal{Y} \right)\) is a *quantum
    channel* for some choice of complex Euclidean spaces \(\mathcal{X}\)
    and \(\mathcal{Y}\), if it holds that:

    1. \(\Phi\) is completely positive.
    2. \(\Phi\) is trace preserving.

    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 a quantum channel, and `False` otherwise.

    Examples:
        We can specify the input as a list of Kraus operators. Consider the map \(\Phi\) defined as

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

        where

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

        To check if this is a valid quantum channel or not,

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

        U = (1/np.sqrt(2))*np.array([[1, 1],[-1, 1]])
        X = pauli("X")
        phi = X - np.matmul(U, np.matmul(X, np.conjugate(U)))

        print(is_quantum_channel(phi))
        ```

        If we instead check for the validity of depolarizing channel being a valid quantum channel,

        ```python exec="1" source="above" result="text"
        from toqito.channels import depolarizing
        from toqito.channel_props import is_quantum_channel

        choi_depolarizing = depolarizing(dim=2, param_p=0.2)

        print(is_quantum_channel(choi_depolarizing))
        ```

    """
    # If the variable `phi` is provided as a list, we assume this is a list
    # of Kraus operators.
    if not (
        isinstance(phi, np.ndarray)
        or (
            isinstance(phi, list)
            and all(isinstance(row, list) and all(isinstance(op, np.ndarray) for op in row) for row in phi)
        )
    ):
        raise TypeError(
            "phi must be either a numpy array (Choi matrix) or a list of lists of numpy arrays (Kraus operators)."
        )
    if isinstance(phi, list):
        phi = kraus_to_choi(phi)

    # A valid quantum channel is a superoperator that is both completely
    # positive and trace-preserving.
    try:
        return is_completely_positive(phi, rtol, atol) and is_trace_preserving(phi, rtol, atol)
    except Exception:
        return False