channels.dephasing
The dephasing channel.
Module Contents
Functions
|
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)) [[ 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)) [[ 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