toqito.channels.depolarizing

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

Produce the partially depolarizing channel [WikDepo], [WatDepo18].

The Choi matrix of the completely depolarizing channel 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

\[\begin{split}\Phi(\rho) = \frac{1}{4} \begin{pmatrix} \frac{1}{2} & 0 & 0 & \frac{1}{2} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \frac{1}{2} & 0 & 0 & \frac{1}{2} \end{pmatrix}.\end{split}\]

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))
[[0.125 0.    0.    0.125]
 [0.    0.    0.    0.   ]
 [0.    0.    0.    0.   ]
 [0.125 0.    0.    0.125]]

>>> 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))
[[17.125  0.25   0.375  0.5  ]
 [ 0.625 17.75   0.875  1.   ]
 [ 1.125  1.25  18.375  1.5  ]
 [ 1.625  1.75   1.875 19.   ]]

References

[WikDepo]

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

[WatDepo18]

Watrous, John. “The theory of quantum information.” Section: “Replacement channels and the completely depolarizing channel”. Cambridge University Press, 2018.

Parameters:

• dim – The dimensionality on which the channel acts.

• param_p – Default 0.

Returns:

The Choi matrix of the completely depolarizing channel.