channels.depolarizing

The depolarizing channel.

Module Contents

Functions

depolarizing(dim[, param_p])

Produce the partially depolarizing channel.

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.

>>> from toqito.channel_ops import apply_channel
>>> from toqito.channels import depolarizing
>>> import numpy as np
>>> 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]])

>>> from toqito.channel_ops import apply_channel
>>> from toqito.channels import depolarizing
>>> import numpy as np
>>> test_input_mat = np.array(
...     [[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]]
... )
>>> 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

[1]

John Watrous. The Theory of Quantum Information. Cambridge University Press, 2018. doi:10.1017/9781316848142.

[2]

Wikipedia. Quantum depolarizing channel. https://en.wikipedia.org/wiki/Quantum_depolarizing_channel.

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