channel_props.is_unitary¶

Determines if a channel is unitary.

Functions¶

is_unitary(phi)

Given a quantum channel, determine if it is unitary.

Module Contents¶

channel_props.is_unitary.is_unitary(phi)¶

Given a quantum channel, determine if it is unitary.

(Section 2.2.1: Definitions and Basic Notions Concerning Channels from [1]).

Let \(\mathcal{X}\) be a complex Euclidean space an let \(U \in U(\mathcal{X})\) be a unitary operator. Then a unitary channel is defined as:

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

Examples

The identity channel is one example of a unitary channel:

\[\begin{split}U = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}.\end{split}\]

We can verify this as follows:

import numpy as np
from toqito.channel_props import is_unitary

kraus_ops = [[np.identity(2), np.identity(2)]]

is_unitary(kraus_ops)

True

We can also specify the input as a Choi matrix. For instance, consider the Choi matrix corresponding to the \(2\)-dimensional completely depolarizing channel.

\[\begin{split}\Omega = \frac{1}{2} \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}.\end{split}\]

We may verify that this channel is not a unitary channel.

from toqito.channels import depolarizing
from toqito.channel_props import is_unitary

is_unitary(depolarizing(2))

False

References

Parameters:: phi (numpy.ndarray | list[list[numpy.ndarray]]) – The channel provided as either a Choi matrix or a list of Kraus operators.
Returns:: True if the channel is a unitary channel, and False otherwise.
Return type:: bool