"""Is channel trace-preserving."""
from __future__ import annotations
import numpy as np
from toqito.matrix_props import is_identity
from toqito.channels import partial_trace
[docs]
def is_trace_preserving(
phi: np.ndarray | list[list[np.ndarray]],
rtol: float = 1e-05,
atol: float = 1e-08,
sys: int | list[int] = 2,
dim: list[int] | np.ndarray = None,
) -> bool:
r"""
Determine whether the given channel is trace-preserving [WatTP18]_.
A map :math:`\Phi \in \text{T} \left(\mathcal{X}, \mathcal{Y} \right)` is
*trace-preserving* if it holds that
.. math::
\text{Tr} \left( \Phi(X) \right) = \text{Tr}\left( X \right)
for every operator :math:`X \in \text{L}(\mathcal{X})`.
Given the corresponding Choi matrix of the channel, a neccessary and sufficient condition is
.. math::
\text{Tr}_{\mathcal{Y}} \left( J(\Phi) \right) = \mathbb{I}_{\mathcal{X}}
In case :code:`sys` is not specified, the default convention is that the Choi matrix
is the result of applying the map to the second subsystem of the standard maximally
entangled (unnormalized) state.
The dimensions of the subsystems are given by the vector :code:`dim`. By default,
both subsystems have equal dimension.
Alternatively, given a list of Kraus operators, a neccessary and sufficient condition is
.. math::
\sum_{a \in \Sigma} A_a^* B_a = \mathbb{I}_{\mathcal{X}}
Examples
==========
The map :math:`\Phi` defined as
.. math::
\Phi(X) = X - U X U^*
is not trace-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_trace_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_trace_preserving(kraus_ops)
False
As another example, the depolarizing channel is trace-preserving.
>>> from toqito.channels import depolarizing
>>> from toqito.channel_props import is_trace_preserving
>>> choi_mat = depolarizing(2)
>>> is_trace_preserving(choi_mat)
True
References
==========
.. [WatTP18] 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).
:param sys: Scalar or vector specifying the size of the subsystems.
:param dim: Dimension of the subsystems. If :code:`None`, all dimensions are assumed to be
equal.
:return: True if the channel is trace-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_l = [A for A, _ in phi]
phi_r = [B for _, B in phi]
k_l = np.concatenate(phi_l, axis=0)
k_r = np.concatenate(phi_r, axis=0)
mat = k_l.conj().T @ k_r
else:
mat = partial_trace(input_mat=phi, sys=sys, dim=dim)
return is_identity(mat, rtol=rtol, atol=atol)