channel_props.is_extremal

Determines whether a quantum channel is extremal.

Functions

is_extremal(phi[, tol])

Determine whether a quantum channel is extremal.

Module Contents

channel_props.is_extremal.is_extremal(phi, tol=1e-09)

Determine whether a quantum channel is extremal.

(Section 2.2.4: Extremal Channels from [1]).

Theorem 2.31 in [1] provides the characterization of extremal quantum channels as a channel \(\Phi\) is an extreme point of the convex set of quantum channels if and only if the collection:

\[\{ A_i^\dagger A_j \}_{i,j=1}^{r}\]

is linearly independent.

The channel can be provided in one of the following representations:

  • A Choi matrix, representing the quantum channel in the Choi representation. It will be converted internally to a set of Kraus operators.

  • A list of Kraus operators, representing the channel in Kraus form.

  • A nested list of Kraus operators, which will be flattened automatically.

Examples

The following demonstrates an example of an extremal quantum channel from Example 2.33 in [1].

import numpy as np
from toqito.channel_props import is_extremal
kraus_ops = [
    (1 / np.sqrt(6)) * np.array([[2, 0], [0, 1], [0, 1], [0, 0]]),
    (1 / np.sqrt(6)) * np.array([[0, 0], [1, 0], [1, 0], [0, 2]])
]

is_extremal(kraus_ops)

True

References

[1] (1,2,3)

John Watrous. The Theory of Quantum Information. Cambridge University Press, 2018. URL: https://johnwatrous.com/wp-content/uploads/TQI.pdf, doi:10.1017/9781316848142.

Parameters:

  • phi (list[numpy.ndarray] | list[list[numpy.ndarray]] | numpy.ndarray) – The quantum channel, which may be given as a Choi matrix or a list of Kraus operators.

  • tol (float) – Tolerance value for numerical precision in rank computation.

Raises:

ValueError – If the input is neither a valid list of Kraus operators nor a Choi matrix.

Returns:

True if the channel is extremal; False otherwise.

Return type:

bool