"""Apply channel to an operator."""
from __future__ import annotations
import numpy as np
from toqito.matrix_ops import vec
from toqito.perms import swap
[docs]
def apply_channel(mat: np.ndarray, phi_op: np.ndarray | list[list[np.ndarray]]) -> np.ndarray:
r"""
Apply a quantum channel to an operator [WatAChan18]_.
Specifically, an application of the channel is defined as
.. math::
\Phi(X) = \text{Tr}_{\mathcal{X}} \left(J(\Phi)
\left(\mathbb{I}_{\mathcal{Y}} \otimes X^{T}\right)\right),
where
.. math::
J(\Phi): \text{T}(\mathcal{X}, \mathcal{Y}) \rightarrow
\text{L}(\mathcal{Y} \otimes \mathcal{X})
is the Choi representation of :math:`\Phi`.
We assume the quantum channel given as :code:`phi_op` is provided as either the Choi matrix
of the channel or a set of Kraus operators that define the quantum channel.
This function is adapted from the QETLAB package.
Examples
==========
The swap operator is the Choi matrix of the transpose map. The following is a (non-ideal,
but illustrative) way of computing the transpose of a matrix.
Consider the following matrix
.. math::
X = \begin{pmatrix}
1 & 4 & 7 \\
2 & 5 & 8 \\
3 & 6 & 9
\end{pmatrix}
Applying the swap operator given as
.. math::
\Phi =
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1
\end{pmatrix}
to the matrix :math:`X`, we have the resulting matrix of
.. math::
\Phi(X) = \begin{pmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{pmatrix}
Using :code:`toqito`, we can obtain the above matrices as follows.
>>> from toqito.channel_ops import apply_channel
>>> from toqito.perms import swap_operator
>>> import numpy as np
>>> test_input_mat = np.array([[1, 4, 7], [2, 5, 8], [3, 6, 9]])
>>> apply_channel(test_input_mat, swap_operator(3))
[[1., 2., 3.],
[4., 5., 6.],
[7., 8., 9.]]
References
==========
.. [WatAChan18] Watrous, John.
The theory of quantum information.
Section: Representations and characterizations of channels.
Cambridge University Press, 2018.
:raises ValueError: If matrix is not Choi matrix.
:param mat: A matrix.
:param phi_op: A superoperator. :code:`phi_op` should be provided either as a Choi matrix,
or as a list of numpy arrays with either 1 or 2 columns whose entries are its
Kraus operators.
:return: The result of applying the superoperator :code:`phi_op` to the operator :code:`mat`.
"""
# Both of the following methods of applying the superoperator are much faster than naively
# looping through the Kraus operators or constructing eigenvectors of a Choi matrix.
# The superoperator was given as a list of Kraus operators:
if isinstance(phi_op, list):
s_phi_op = [len(phi_op), len(phi_op[0])]
# Map is completely positive.
if s_phi_op[1] == 1 or (s_phi_op[0] == 1 and s_phi_op[1] > 2):
for i in range(s_phi_op[0]):
phi_op[i][1] = phi_op[i][0].conj().T
else:
for i in range(s_phi_op[0]):
phi_op[i][1] = phi_op[i][1].conj().T
phi_0_list = []
phi_1_list = []
for i in range(s_phi_op[0]):
phi_0_list.append(phi_op[i][0])
phi_1_list.append(phi_op[i][1])
k_1 = np.concatenate(phi_0_list, axis=1)
k_2 = np.concatenate(phi_1_list, axis=0)
a_mat = np.kron(np.identity(len(phi_op)), mat)
return k_1 @ a_mat @ k_2
# The superoperator was given as a Choi matrix:
if isinstance(phi_op, np.ndarray):
mat_size = np.array(list(mat.shape))
phi_size = np.array(list(phi_op.shape)) / mat_size
a_mat = np.kron(vec(mat).T[0], np.identity(int(phi_size[0])))
b_mat = np.reshape(
swap(
phi_op.T,
[1, 2],
[[mat_size[1], phi_size[1]], [mat_size[0], phi_size[0]]],
True,
).T,
(int(phi_size[0] * np.prod(mat_size)), int(phi_size[1])),
)
return a_mat @ b_mat
raise ValueError(
"Invalid: The variable `phi_op` must either be a list of "
"Kraus operators or as a Choi matrix."
)