toqito.channels.dephasing

Generates the dephasing channel.

Module Contents

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

Produce the partially dephasing channel.

(Section: The Completely Dephasing Channel from [@Watrous_2018_TQI]).

The Choi matrix of the completely dephasing channel that acts on dim-by-dim matrices.

Let (Sigma) be an alphabet and let (mathcal{X} = mathbb{C}^{Sigma}). The map (Delta in text{T}(mathcal{X})) defined as

[

Delta(X) = sum_{a in Sigma} X(a, a) E_{a,a}

]

for every (X in text{L}(mathcal{X})) is defined as the completely dephasing channel.

Examples

The completely dephasing channel maps kills everything off the diagonal. Consider the following matrix

[

rho = begin{pmatrix}

1 & 2 & 3 & 4 \ 5 & 6 & 7 & 8 \ 9 & 10 & 11 & 12 \ 13 & 14 & 15 & 16

end{pmatrix}.

]

Applying the dephasing channel to (rho) we have that

[

Phi(rho) = begin{pmatrix}

1 & 0 & 0 & 0 \ 0 & 6 & 0 & 0 \ 0 & 0 & 11 & 0 \ 0 & 0 & 0 & 16

end{pmatrix}.

]

This can be observed in |toqito⟩ as follows.

```python exec=”1” source=”above” import numpy as np from toqito.channels import dephasing from toqito.channel_ops import apply_channel

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

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

We may also consider setting the parameter p = 0.5.

```python exec=”1” source=”above” import numpy as np from toqito.channels import dephasing from toqito.channel_ops import apply_channel

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

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

Parameters:

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

• param_p (float) – Default is 0.

Returns:

The Choi matrix of the dephasing channel.

Return type:

numpy.ndarray