channel_ops.choi_to_kraus¶

Computes a list of Kraus operators from the Choi matrix.

Functions¶

choi_to_kraus(choi_mat[, tol, dim])

Compute a list of Kraus operators from the Choi matrix from [1].

Module Contents¶

channel_ops.choi_to_kraus.choi_to_kraus(choi_mat, tol=1e-09, dim=None)¶

Compute a list of Kraus operators from the Choi matrix from [1].

Note that unlike the Choi or natural representation of operators, the Kraus representation is not unique.

If the input channel maps \(M_{r,c}\) to \(M_{x,y}\) then dim should be the list [[r,x], [c,y]]. If it maps \(M_m\) to \(M_n\), then dim can simply be the vector [m,n].

For completely positive maps the output is a single flat list of numpy arrays since the left and right Kraus maps are the same.

This function has been adapted from [1] and QETLAB [1].

Examples

Consider taking the Kraus operators of the Choi matrix that characterizes the “swap operator” defined as

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

The corresponding Kraus operators of the swap operator are given as follows,

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

This can be verified in toqito as follows.

>>> import numpy as np
>>> from toqito.channel_ops import choi_to_kraus
>>> choi_mat = np.array([[1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0], [0, 0, 0, 1]])
>>> kraus_ops = choi_to_kraus(choi_mat)
>>> kraus_ops
[[array([[ 0.        ,  0.70710678],
       [-0.70710678,  0.        ]]), array([[-0.        , -0.70710678],
       [ 0.70710678, -0.        ]])], [array([[0.        , 0.70710678],
       [0.70710678, 0.        ]]), array([[0.        , 0.70710678],
       [0.70710678, 0.        ]])], [array([[1., 0.],
       [0., 0.]]), array([[1., 0.],
       [0., 0.]])], [array([[0., 0.],
       [0., 1.]]), array([[0., 0.],
       [0., 1.]])]]