def is_unital(
phi: Union[np.ndarray, List[List[np.ndarray]]],
rtol: float = 1e-05,
atol: float = 1e-08,
) -> bool:
r"""
Determine whether the given channel is unital [WatUnital18]_.
A map :math:`\Phi \in \text{T} \left(\mathcal{X}, \mathcal{Y} \right)` is *unital* if it
holds that
.. math::
\Phi(\mathbb{I}_{\mathcal{X}}) = \mathbb{I}_{\mathcal{Y}}.
Examples
==========
Consider the channel whose Choi matrix is the swap operator. This channel is an example of a
unital channel.
>>> from toqito.perms import swap_operator
>>> from toqito.channel_props import is_unital
>>>
>>> choi = swap_operator(3)
>>> is_unital(choi)
True
Alternatively, the channel whose Choi matrix is the depolarizing channel is an example of a
non-unital channel.
>>> from toqito.channels import depolarizing
>>> from toqito.channel_props import is_unital
>>>
>>> choi = depolarizing(4)
>>> is_unital(choi)
False
References
==========
.. [WatUnital18] Watrous, John.
"The theory of quantum information."
Chapter: Unital channels and majorization
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: :code:`True` if the channel is unital, and :code:`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)
dim = int(np.sqrt(phi.shape[0]))
# Channel is unital if :code:`mat` is the identity matrix.
mat = apply_channel(np.identity(dim), phi)
return is_identity(mat, rtol=rtol, atol=atol)
def is_quantum_channel(
phi: Union[np.ndarray, List[List[np.ndarray]]],
rtol: float = 1e-05,
atol: float = 1e-08,
) -> bool:
r"""
Determine whether the given input is a quantum channel [WatQC18]_.
A map :math:`\Phi \in \text{T} \left(\mathcal{X}, \mathcal{Y} \right)` is a *quantum
channel* for some choice of complex Euclidean spaces :math:`\mathcal{X}`
and :math:`\mathcal{Y}`, if it holds that:
1. :math:`\Phi` is completely positive.
2. :math:`\Phi` is trace preserving.
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}.
References
==========
.. [WatQC18] Watrous, John.
"The Theory of Quantum Information."
Section: "2.2.1 Definitions and basic notions concerning channels".
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: :code:`True` if the channel is a quantum channel, and :code:`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)
# A valid quantum channel is a superoperator that is both completely
# positive and trace-preserving.
return is_completely_positive(phi, rtol, atol) and is_trace_preserving(
phi, rtol, atol)
def test_kraus_to_choi_dephasing_channel():
"""Kraus operators for dephasing channel should yield the proper Choi matrix."""
kraus_1 = np.array([[1, 0], [0, 0]])
kraus_2 = np.array([[1, 0], [0, 0]])
kraus_3 = np.array([[0, 0], [0, 1]])
kraus_4 = np.array([[0, 0], [0, 1]])
kraus_ops = [[kraus_1, kraus_2], [kraus_3, kraus_4]]
choi_res = kraus_to_choi(kraus_ops)
expected_choi_res = dephasing(2)
bool_mat = np.isclose(choi_res, expected_choi_res)
np.testing.assert_equal(np.all(bool_mat), True)
def test_kraus_to_choi_depolarizing_channel():
"""Kraus operators for depolarizing channel should yield the proper Choi matrix."""
kraus_1 = np.array([[1 / np.sqrt(2), 0], [0, 0]])
kraus_2 = np.array([[1 / np.sqrt(2), 0], [0, 0]])
kraus_3 = np.array([[0, 0], [1 / np.sqrt(2), 0]])
kraus_4 = np.array([[0, 0], [1 / np.sqrt(2), 0]])
kraus_5 = np.array([[0, 1 / np.sqrt(2)], [0, 0]])
kraus_6 = np.array([[0, 1 / np.sqrt(2)], [0, 0]])
kraus_7 = np.array([[0, 0], [0, 1 / np.sqrt(2)]])
kraus_8 = np.array([[0, 0], [0, 1 / np.sqrt(2)]])
kraus_ops = [
[kraus_1, kraus_2],
[kraus_3, kraus_4],
[kraus_5, kraus_6],
[kraus_7, kraus_8],
]
choi_res = kraus_to_choi(kraus_ops)
expected_choi_res = depolarizing(2)
bool_mat = np.isclose(choi_res, expected_choi_res)
np.testing.assert_equal(np.all(bool_mat), True)
def test_kraus_to_choi_max_ent_2():
"""Choi matrix of the transpose map is the swap operator."""
kraus_1 = np.array([[1, 0], [0, 0]])
kraus_2 = np.array([[1, 0], [0, 0]]).conj().T
kraus_3 = np.array([[0, 1], [0, 0]])
kraus_4 = np.array([[0, 1], [0, 0]]).conj().T
kraus_5 = np.array([[0, 0], [1, 0]])
kraus_6 = np.array([[0, 0], [1, 0]]).conj().T
kraus_7 = np.array([[0, 0], [0, 1]])
kraus_8 = np.array([[0, 0], [0, 1]]).conj().T
kraus_ops = [
[kraus_1, kraus_2],
[kraus_3, kraus_4],
[kraus_5, kraus_6],
[kraus_7, kraus_8],
]
choi_res = kraus_to_choi(kraus_ops)
expected_choi_res = np.array([[1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0],
[0, 0, 0, 1]])
bool_mat = np.isclose(choi_res, expected_choi_res)
np.testing.assert_equal(np.all(bool_mat), True)
def test_kraus_to_choi_swap_operator_non_unique():
"""As Kraus operators are non-unique, these also should yield the swap operator."""
kraus_1 = np.array([[0, 1 / np.sqrt(2)], [1 / np.sqrt(2), 0]])
kraus_2 = np.array([[0, 1 / np.sqrt(2)], [1 / np.sqrt(2), 0]]).conj().T
kraus_3 = np.array([[1, 0], [0, 0]])
kraus_4 = np.array([[1, 0], [0, 0]]).conj().T
kraus_5 = np.array([[0, 0], [0, 1]])
kraus_6 = np.array([[0, 0], [0, 1]]).conj().T
kraus_7 = np.array([[0, 1 / np.sqrt(2)], [-1 / np.sqrt(2), 0]])
kraus_8 = np.array([[0, 1 / np.sqrt(2)], [-1 / np.sqrt(2), 0]]).conj().T
kraus_ops = [
[kraus_1, kraus_2],
[kraus_3, kraus_4],
[kraus_5, kraus_6],
[kraus_7, kraus_8],
]
choi_res = kraus_to_choi(kraus_ops)
expected_choi_res = np.array([[1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0],
[0, 0, 0, 1]])
bool_mat = np.isclose(choi_res, expected_choi_res)
np.testing.assert_equal(np.all(bool_mat), True)
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 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 is_positive(phi: Union[np.ndarray, List[List[np.ndarray]]],
tol: float = 1e-05) -> bool:
r"""
Determine whether the given channel is positive [WatPM18]_.
A map :math:`\Phi \in \text{T} \left(\mathcal{X}, \mathcal{Y} \right)` is
*positive* if it holds that
.. math::
\Phi(P) \in \text{Pos}(\mathcal{Y})
for every positive semidefinite operator :math:`P \in \text{Pos}(\mathcal{X})`.
Alternatively, a channel is 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_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_positive(kraus_ops)
False
We can also specify the input as a Choi matrix. For instance, consider the
Choi matrix corresponding to the :math:`4`-dimensional completely
depolarizing channel and may verify that this channel is positive.
>>> from toqito.channels import depolarizing
>>> from toqito.channel_props import is_positive
>>> is_positive(depolarizing(4))
True
References
==========
.. [WatPM18] 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 tol: The tolerance parameter to determine positivity.
:return: True if the channel is 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)
return is_psd(phi, tol)
def choi_rank(phi: Union[np.ndarray, List[List[np.ndarray]]]) -> int:
r"""
Calculate the rank of the Choi representation of a quantum channel [WatChoiRank18]_.
Examples
==========
The transpose map can be written either in Choi representation (as a
SWAP operator) or in Kraus representation. If we choose the latter, it
will be given by the following matrices:
.. math::
\begin{equation}
\begin{aligned}
\frac{1}{\sqrt{2}}
\begin{pmatrix}
0 & i \\ -i & 0
\end{pmatrix}, &\quad
\frac{1}{\sqrt{2}}
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}, \\
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix}, &\quad
\begin{pmatrix}
0 & 0 \\
0 & 1
\end{pmatrix}.
\end{aligned}
\end{equation}
and can be generated in :code:`toqito` with the following list:
>>> import numpy as np
>>> kraus_1 = np.array([[1, 0], [0, 0]])
>>> kraus_2 = np.array([[1, 0], [0, 0]]).conj().T
>>> kraus_3 = np.array([[0, 1], [0, 0]])
>>> kraus_4 = np.array([[0, 1], [0, 0]]).conj().T
>>> kraus_5 = np.array([[0, 0], [1, 0]])
>>> kraus_6 = np.array([[0, 0], [1, 0]]).conj().T
>>> kraus_7 = np.array([[0, 0], [0, 1]])
>>> kraus_8 = np.array([[0, 0], [0, 1]]).conj().T
>>> kraus_ops = [
>>> [kraus_1, kraus_2],
>>> [kraus_3, kraus_4],
>>> [kraus_5, kraus_6],
>>> [kraus_7, kraus_8],
>>> ]
To calculate its Choi rank, we proceed in the following way:
>>> from toqito.channel_props import choi_rank
>>> choi_rank(kraus_ops)
4
We can the verify the associated Choi representation (the SWAP gate)
gets the same Choi rank:
>>> choi_matrix = np.array([[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]])
>>> choi_rank(choi_matrix)
4
References
==========
.. [WatChoiRank18] Watrous, John.
"The Theory of Quantum Information."
Section: "2.2 Quantum Channels".
Cambridge University Press, 2018.
:param phi: Either a Choi matrix or a list of Kraus operators
:return: The Choi rank of the provided channel representation.
"""
if isinstance(phi, list):
phi = kraus_to_choi(phi)
elif not isinstance(phi, np.ndarray):
raise ValueError("Not a valid Choi matrix.")
return np.linalg.matrix_rank(phi)
