toqito.channel_props.choi_rank

toqito.channel_props.choi_rank(phi)

Calculate the rank of the Choi representation of a quantum channel [WatChoiRank18].

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:

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

and can be generated in toqito with the following list:

>>> import numpy as np
>>> 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],
>>> ]

To calculate its Choi rank, we proceed in the following way:

>>> from toqito.channel_props import choi_rank
>>> choi_rank(kraus_ops)
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)
4

References

[WatChoiRank18]

Watrous, John. “The Theory of Quantum Information.” Section: “2.2 Quantum Channels”. Cambridge University Press, 2018.

Raises:

ValueError – If matrix is not Choi.

Parameters:

phi – Either a Choi matrix or a list of Kraus operators

Returns:

The Choi rank of the provided channel representation.