channels.depolarizing

Generates the depolarizing channel.

Functions

depolarizing(dim[, param_p])

Produce the partially depolarizing channel.

Module Contents

channels.depolarizing.depolarizing(dim, param_p=0)

Produce the partially depolarizing channel.

(Section: Replacement Channels and the Completely Depolarizing Channel from [1]).

The Choi matrix of the completely depolarizing channel [2] that acts on dim-by-dim matrices.

The completely depolarizing channel is defined as

\[\Omega(X) = \text{Tr}(X) \omega\]

for all \(X \in \text{L}(\mathcal{X})\), where

\[\omega = \frac{\mathbb{I}_{\mathcal{X}}}{\text{dim}(\mathcal{X})}\]

denotes the completely mixed stated defined with respect to the space \(\mathcal{X}\).

Examples

The completely depolarizing channel maps every density matrix to the maximally-mixed state. For example, consider the density operator

\[\begin{split}\rho = \frac{1}{2} \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \end{pmatrix}\end{split}\]

corresponding to one of the Bell states. Applying the depolarizing channel to \(\rho\) we have that

\[\begin{split}\Phi(\rho) = \frac{1}{4} \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}.\end{split}\]

This can be observed in |toqito⟩ as follows.

import numpy as np
from toqito.channels import depolarizing
from toqito.channel_ops import apply_channel

test_input_mat = np.array([[1 / 2, 0, 0, 1 / 2], [0, 0, 0, 0], [0, 0, 0, 0], [1 / 2, 0, 0, 1 / 2]])

apply_channel(test_input_mat, depolarizing(4))

array([[0.25, 0.  , 0.  , 0.  ],
       [0.  , 0.25, 0.  , 0.  ],
       [0.  , 0.  , 0.25, 0.  ],
       [0.  , 0.  , 0.  , 0.25]])

import numpy as np
from toqito.channels import depolarizing
from toqito.channel_ops import apply_channel

test_input_mat = np.arange(1, 17).reshape(4, 4)

apply_channel(test_input_mat, depolarizing(4, 0.5))

array([[ 4.75,  1.  ,  1.5 ,  2.  ],
       [ 2.5 ,  7.25,  3.5 ,  4.  ],
       [ 4.5 ,  5.  ,  9.75,  6.  ],
       [ 6.5 ,  7.  ,  7.5 , 12.25]])

References

Parameters:

  • dim (int) – The dimensionality on which the channel acts.

  • param_p (float) – Default 0.

Returns:

The Choi matrix of the completely depolarizing channel.

Return type:

numpy.ndarray