channels.dephasing

The dephasing channel.

Module Contents

Functions

dephasing(dim[, param_p])

Produce the partially dephasing channel.

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

Produce the partially dephasing channel.

(Section: The Completely Dephasing Channel from [1]).

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

\[\begin{split}\rho = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{pmatrix}.\end{split}\]

Applying the dephasing channel to \(\rho\) we have that

\[\begin{split}\Phi(\rho) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 6 & 0 & 0 \\ 0 & 0 & 11 & 0 \\ 0 & 0 & 0 & 16 \end{pmatrix}.\end{split}\]

This can be observed in toqito as follows.

>>> from toqito.channel_ops import apply_channel
>>> from toqito.channels import dephasing
>>> 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, dephasing(4))
array([[ 1.,  0.,  0.,  0.],
       [ 0.,  6.,  0.,  0.],
       [ 0.,  0., 11.,  0.],
       [ 0.,  0.,  0., 16.]])

We may also consider setting the parameter p = 0.5.

>>> from toqito.channel_ops import apply_channel
>>> from toqito.channels import dephasing
>>> 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, dephasing(4, 0.5))
array([[ 1. ,  1. ,  1.5,  2. ],
       [ 2.5,  6. ,  3.5,  4. ],
       [ 4.5,  5. , 11. ,  6. ],
       [ 6.5,  7. ,  7.5, 16. ]])

References

[1]

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

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