"""Is channel Hermiticity-preserving."""
from __future__ import annotations
import numpy as np
from toqito.channel_ops import kraus_to_choi
from toqito.matrix_props import is_hermitian
[docs]
def is_herm_preserving(
phi: np.ndarray | list[list[np.ndarray]],
rtol: float = 1e-05,
atol: float = 1e-08,
) -> bool:
r"""
Determine whether the given channel is Hermitian-preserving [WatH18]_.
A map :math:`\Phi \in \text{T} \left(\mathcal{X}, \mathcal{Y} \right)` is
*Hermitian-preserving* if it holds that
.. math::
\Phi(H) \in \text{Herm}(\mathcal{Y})
for every Hermitian operator :math:`H \in \text{Herm}(\mathcal{X})`.
Examples
==========
The map :math:`\Phi` defined as
.. math::
\Phi(X) = X - U X U^*
is Hermitian-preserving, where
.. math::
U = \frac{1}{\sqrt{2}}
\begin{pmatrix}
1 & 1 \\
-1 & 1
\end{pmatrix}.
>>> import numpy as np
>>> from toqito.channel_props import is_herm_preserving
>>> unitary_mat = np.array([[1, 1], [-1, 1]]) / np.sqrt(2)
>>> kraus_ops = [[np.identity(2), np.identity(2)], [unitary_mat, -unitary_mat]]
>>> is_herm_preserving(kraus_ops)
True
We may also verify whether the corresponding Choi matrix of a given map is
Hermitian-preserving. The swap operator is the Choi matrix of the transpose map, which is
Hermitian-preserving as can be seen as follows:
>>> import numpy as np
>>> from toqito.perms import swap_operator
>>> from toqito.channel_props import is_herm_preserving
>>> unitary_mat = np.array([[1, 1], [-1, 1]]) / np.sqrt(2)
>>> choi_mat = swap_operator(3)
>>> is_herm_preserving(choi_mat)
True
References
==========
.. [WatH18] Watrous, John.
"The theory of quantum information."
Section: "Linear maps of square operators".
Cambridge University Press, 2018.
:param phi: The channel provided as either a Choi matrix or a list of Kraus operators.
:param rtol: The relative tolerance parameter (default 1e-05).
:param atol: The absolute tolerance parameter (default 1e-08).
:return: True if the channel is Hermitian-preserving, and False otherwise.
"""
# If the variable `phi` is provided as a list, we assume this is a list
# of Kraus operators.
if isinstance(phi, list):
phi = kraus_to_choi(phi)
# Phi is Hermiticity-preserving if and only if its Choi matrix is Hermitian.
if phi.shape[0] != phi.shape[1]:
return False
return is_hermitian(phi, rtol=rtol, atol=atol)