"""The dephasing channel."""
import numpy as np
from toqito.states import max_entangled
[docs]
def dephasing(dim: int, param_p: float = 0) -> np.ndarray:
r"""
Produce the partially dephasing channel [WatDeph18]_.
The Choi matrix of the completely dephasing channel that acts on :code:`dim`-by-:code:`dim`
matrices.
Let :math:`\Sigma` be an alphabet and let :math:`\mathcal{X} = \mathbb{C}^{\Sigma}`. The map
:math:`\Delta \in \text{T}(\mathcal{X})` defined as
.. math::
\Delta(X) = \sum_{a \in \Sigma} X(a, a) E_{a,a}
for every :math:`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
.. math::
\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 :math:`\rho` we have that
.. math::
\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 :code:`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 :code:`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.
:param dim: The dimensionality on which the channel acts.
:param param_p: Default is 0.
:return: The Choi matrix of the dephasing channel.
"""
# Compute the Choi matrix of the dephasing channel.
# Gives a sparse non-normalized state.
psi = max_entangled(dim=dim, is_sparse=False, is_normalized=False)
return (1 - param_p) * np.diag(np.diag(psi * psi.conj().T)) + param_p * (psi * psi.conj().T)