channel_props.choi_rank¶

Calculates the Choi rank of a channel.

Functions¶

choi_rank(phi)

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

Module Contents¶

channel_props.choi_rank.choi_rank(phi)¶

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

(Section 2.2: Quantum Channels from [1]).

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:

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

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]]

choi_rank(kraus_ops)

np.int64(4)

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

choi_matrix = np.array([[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]])
choi_rank(choi_matrix)

np.int64(4)

References

Raises:: ValueError – If matrix is not Choi.
Parameters:: phi (numpy.ndarray | list[list[numpy.ndarray]]) – Either a Choi matrix or a list of Kraus operators
Returns:: The Choi rank of the provided channel representation.
Return type:: int