def is_positive_definite(mat: np.ndarray,
rtol: float = 1e-05,
atol: float = 1e-08) -> bool:
r"""
Check if matrix is positive definite (PD) [WikPD]_.
Examples
==========
Consider the following matrix
.. math::
A = \begin{pmatrix}
2 & -1 & 0 \\
-1 & 2 & -1 \\
0 & -1 & 2
\end{pmatrix}
our function indicates that this is indeed a positive definite matrix.
>>> from toqito.matrix_props import is_positive_definite
>>> import numpy as np
>>> A = np.array([[2, -1, 0], [-1, 2, -1], [0, -1, 2]])
>>> is_positive_definite(A)
True
Alternatively, the following example matrix :math:`B` defined as
.. math::
B = \begin{pmatrix}
-1 & -1 \\
-1 & -1
\end{pmatrix}
is not positive definite.
>>> from toqito.matrix_props import is_positive_definite
>>> import numpy as np
>>> B = np.array([[-1, -1], [-1, -1]])
>>> is_positive_definite(B)
False
See Also
========
is_positive_semidefinite
References
==========
.. [WikPD] Wikipedia: Definiteness of a matrix.
https://en.wikipedia.org/wiki/Definiteness_of_a_matrix
:param mat: Matrix to check.
:param rtol: The relative tolerance parameter (default 1e-05).
:param atol: The absolute tolerance parameter (default 1e-08).
:return: Return :code:`True` if matrix is positive definite, and :code:`False` otherwise.
"""
if not is_hermitian(mat, rtol=rtol, atol=atol):
return False
evals, _ = np.linalg.eigh(mat)
return all(x > -abs(atol) for x in evals)
def test_is_hermitian():
"""Test if matrix is Hermitian."""
mat = np.array([[2, 2 + 1j, 4], [2 - 1j, 3, 1j], [4, -1j, 1]])
np.testing.assert_equal(is_hermitian(mat), True)
def test_is_non_hermitian():
"""Test non-Hermitian matrix."""
mat = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
np.testing.assert_equal(is_hermitian(mat), False)
def is_herm_preserving(
phi: Union[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)
def test_is_hermitian_not_square():
"""Input must be a square matrix."""
mat = np.array([[-1, 1, 1], [1, 2, 3]])
np.testing.assert_equal(is_hermitian(mat), False)