toqito.channels.dephasing
- toqito.channels.dephasing(dim, param_p=0)[source]
Produce the partially dephasing channel [WatDeph18].
The Choi matrix of the completely dephasing channel that acts on
dim-by-dimmatrices.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
toqitoas 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)) [[17.5 0. 0. 0. ] [ 0. 20. 0. 0. ] [ 0. 0. 22.5 0. ] [ 0. 0. 0. 25. ]]
References
[WatDeph18]Watrous, John. “The theory of quantum information.” Section: “The completely dephasing channel”. Cambridge University Press, 2018.
- Parameters:
dim – The dimensionality on which the channel acts.
param_p – Default is 0.
- Returns:
The Choi matrix of the dephasing channel.