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 [2].

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)
for i, pair in enumerate(kraus_ops):
   print(f"\nKraus Pair {i+1}:")
   for j, op in enumerate(pair):
       print(f"  Operator {j+1}:\n{np.array_str(op, precision=4, suppress_small=True)}")

Kraus Pair 1:
  Operator 1:
[[ 0.      0.7071]
 [-0.7071  0.    ]]
  Operator 2:
[[-0.     -0.7071]
 [ 0.7071 -0.    ]]

Kraus Pair 2:
  Operator 1:
[[0.     0.7071]
 [0.7071 0.    ]]
  Operator 2:
[[0.     0.7071]
 [0.7071 0.    ]]

Kraus Pair 3:
  Operator 1:
[[1. 0.]
 [0. 0.]]
  Operator 2:
[[1. 0.]
 [0. 0.]]

Kraus Pair 4:
  Operator 1:
[[0. 0.]
 [0. 1.]]
  Operator 2:
[[0. 0.]
 [0. 1.]]