toqito.channels.choi

toqito.channels.choi(a_var=1, b_var=1, c_var=0)

Produce the Choi channel or one of its generalizations [Choi92].

The Choi channel is a positive map on 3-by-3 matrices that is capable of detecting some entanglement that the transpose map is not.

The standard Choi channel defined with a=1, b=1, and c=0 is the Choi matrix of the positive map defined in [Choi92]. Many of these maps are capable of detecting PPT entanglement.

Examples

The standard Choi channel is given as

\[\begin{split}\Phi_{1, 1, 0} = \begin{pmatrix} 1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 1 \end{pmatrix}\end{split}\]

We can generate the Choi channel in toqito as follows.

>>> from toqito.channels import choi
>>> import numpy as np
>>> choi()
[[ 1.,  0.,  0.,  0., -1.,  0.,  0.,  0., -1.],
 [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
 [ 0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.],
 [ 0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.],
 [-1.,  0.,  0.,  0.,  1.,  0.,  0.,  0., -1.],
 [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
 [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
 [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.],
 [-1.,  0.,  0.,  0., -1.,  0.,  0.,  0.,  1.]])

The reduction channel is the map \(R\) defined by:

\[R(X) = \text{Tr}(X) \mathbb{I} - X.\]

The matrix correspond to this is given as

\[\begin{split}\Phi_{0, 1, 1} = \begin{pmatrix} 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \end{pmatrix}\end{split}\]

The reduction channel is the Choi channel that arises when a = 0 and when b = c = 1. We can obtain this matrix using toqito as follows.

>>> from toqito.channels import choi
>>> import numpy as np
>>> choi(0, 1, 1)
[[ 0.,  0.,  0.,  0., -1.,  0.,  0.,  0., -1.],
 [ 0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
 [ 0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.],
 [ 0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.],
 [-1.,  0.,  0.,  0.,  0.,  0.,  0.,  0., -1.],
 [ 0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.],
 [ 0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.],
 [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.],
 [-1.,  0.,  0.,  0., -1.,  0.,  0.,  0.,  0.]])

See also

reduction

References

[Choi92] (1,2)

Cho, Sung Je, Seung-Hyeok Kye, and Sa Ge Lee. “Generalized Choi maps in three-dimensional matrix algebra.” Linear algebra and its applications 171 (1992): 213-224. https://www.sciencedirect.com/science/article/pii/002437959290260H

Parameters:

• a_var – Default integer for standard Choi map.

• b_var – Default integer for standard Choi map.

• c_var – Default integer for standard Choi map.

Returns:

The Choi channel (or one of its generalizations).