def is_completely_positive(
phi: Union[np.ndarray, List[List[np.ndarray]]],
rtol: float = 1e-05,
atol: float = 1e-08,
) -> bool:
r"""
Determine whether the given channel is completely positive [WatCP18]_.
A map :math:`\Phi \in \text{T} \left(\mathcal{X}, \mathcal{Y} \right)` is *completely
positive* if it holds that
.. math::
\Phi \otimes \mathbb{I}_{\text{L}(\mathcal{Z})}
is a positive map for every complex Euclidean space :math:`\mathcal{Z}`.
Alternatively, a channel is completely positive if the corresponding Choi matrix of the
channel is both Hermitian-preserving and positive semidefinite.
Examples
==========
We can specify the input as a list of Kraus operators. Consider the map :math:`\Phi` defined as
.. math::
\Phi(X) = X - U X U^*
where
.. math::
U = \frac{1}{\sqrt{2}}
\begin{pmatrix}
1 & 1 \\
-1 & 1
\end{pmatrix}.
This map is not completely positive, as we can verify as follows.
>>> from toqito.channel_props import is_completely_positive
>>> import numpy as np
>>> unitary_mat = np.array([[1, 1], [-1, 1]]) / np.sqrt(2)
>>> kraus_ops = [[np.identity(2), np.identity(2)], [unitary_mat, -unitary_mat]]
>>> is_completely_positive(kraus_ops)
False
We can also specify the input as a Choi matrix. For instance, consider the Choi matrix
corresponding to the :math:`2`-dimensional completely depolarizing channel
.. math::
\Omega =
\frac{1}{2}
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}.
We may verify that this channel is completely positive
>>> from toqito.channels import depolarizing
>>> from toqito.channel_props import is_completely_positive
>>> is_completely_positive(depolarizing(2))
True
References
==========
.. [WatCP18] 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 completely positive, 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)
# Use Choi's theorem to determine whether :code:`phi` is completely positive.
return is_herm_preserving(phi, rtol, atol) and is_positive_semidefinite(phi, rtol, atol)
def test_is_herm_preserving_non_square():
"""Verify non-square Choi matrix returns False."""
non_square_mat = np.array([[1, 2, 3], [4, 5, 6]])
np.testing.assert_equal(is_herm_preserving(non_square_mat), False)
def test_is_herm_preserving_kraus_true():
"""Verify Hermitian-preserving channel as Kraus ops as True."""
unitary_mat = np.array([[1, 1], [-1, 1]]) / np.sqrt(2)
kraus_ops = [[np.identity(2), np.identity(2)], [unitary_mat, -unitary_mat]]
np.testing.assert_equal(is_herm_preserving(kraus_ops), True)
def test_is_herm_preserving_choi_true():
"""Swap operator is Choi matrix of the (Herm-preserving) transpose map."""
choi_mat = swap_operator(3)
np.testing.assert_equal(is_herm_preserving(choi_mat), True)