toqito.channels.depolarizing¶

Generates the depolarizing channel.

Module Contents¶

toqito.channels.depolarizing.depolarizing(dim, param_p=0)[source]¶

Produce the partially depolarizing channel.

(Section: Replacement Channels and the Completely Depolarizing Channel from [@Watrous_2018_TQI]).

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

The completely depolarizing channel is defined as

]

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

]

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

[

rho = frac{1}{2} begin{pmatrix}: 1 & 0 & 0 & 1 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 1

end{pmatrix}

]

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

[

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}.

]

This can be observed in |toqito⟩ as follows.

```python exec=”1” source=”above” 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]])

print(apply_channel(test_input_mat, depolarizing(4))) ```

```python exec=”1” source=”above” 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)

print(apply_channel(test_input_mat, depolarizing(4, 0.5))) ```

Raises:

ValueError – If param_p is outside the interval [0,1].

Parameters:

dim (int) – The dimensionality on which the channel acts.
param_p (float) – Depolarizing probability (p ) in [0,1] that mixes the input state with the maximally mixed state.
0. (Default)

Returns:

The Choi matrix of the completely depolarizing channel.

Return type:

numpy.ndarray