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 test_dephasing_partially_dephasing():
"""The partially dephasing channel for `p = 0.5`."""
test_input_mat = np.array([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12],
[13, 14, 15, 16]])
expected_res = np.array([[17.5, 0, 0, 0], [0, 20, 0, 0], [0, 0, 22.5, 0],
[0, 0, 0, 25]])
res = apply_channel(test_input_mat, dephasing(4, 0.5))
bool_mat = np.isclose(expected_res, res)
np.testing.assert_equal(np.all(bool_mat), True)
def test_dephasing_completely_dephasing():
"""The completely dephasing channel kills everything off diagonal."""
test_input_mat = np.array([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12],
[13, 14, 15, 16]])
expected_res = np.array([[1, 0, 0, 0], [0, 6, 0, 0], [0, 0, 11, 0],
[0, 0, 0, 16]])
res = apply_channel(test_input_mat, dephasing(4))
bool_mat = np.isclose(expected_res, res)
np.testing.assert_equal(np.all(bool_mat), True)
def test_depolarizing_complete_depolarizing():
"""Maps every density matrix to the maximally-mixed state."""
test_input_mat = np.array(
[[1 / 2, 0, 0, 1 / 2], [0, 0, 0, 0], [0, 0, 0, 0], [1 / 2, 0, 0, 1 / 2]]
)
expected_res = (
1 / 4 * np.array([[1 / 2, 0, 0, 1 / 2], [0, 0, 0, 0], [0, 0, 0, 0], [1 / 2, 0, 0, 1 / 2]])
)
res = apply_channel(test_input_mat, depolarizing(4))
bool_mat = np.isclose(expected_res, res)
np.testing.assert_equal(np.all(bool_mat), True)
def test_apply_channel_choi():
"""
The swap operator is the Choi matrix of the transpose map.
The following test is a (non-ideal, but illustrative) way of computing
the transpose of a matrix.
"""
test_input_mat = np.array([[1, 4, 7], [2, 5, 8], [3, 6, 9]])
expected_res = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
res = apply_channel(test_input_mat, swap_operator(3))
bool_mat = np.isclose(res, expected_res)
np.testing.assert_equal(np.all(bool_mat), True)
def test_depolarizing_partially_depolarizing():
"""The partially depolarizing channel for `p = 0.5`."""
test_input_mat = np.array([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]])
expected_res = np.array(
[
[17.125, 0.25, 0.375, 0.5],
[0.625, 17.75, 0.875, 1],
[1.125, 1.25, 18.375, 1.5],
[1.625, 1.75, 1.875, 19],
]
)
res = apply_channel(test_input_mat, depolarizing(4, 0.5))
bool_mat = np.isclose(expected_res, res)
np.testing.assert_equal(np.all(bool_mat), True)
def test_apply_channel_kraus():
"""
Apply Kraus map.
The following test computes PHI(X) where X = [[1, 2], [3, 4]] and
where PHI is the superoperator defined by:
Phi(X) = [[1,5],[1,0],[0,2]] X [[0,1][2,3][4,5]].conj().T -
[[1,0],[0,0],[0,1]] X [[0,0][1,1],[0,0]].conj().T
"""
test_input_mat = np.array([[1, 2], [3, 4]])
kraus_1 = np.array([[1, 5], [1, 0], [0, 2]])
kraus_2 = np.array([[0, 1], [2, 3], [4, 5]])
kraus_3 = np.array([[-1, 0], [0, 0], [0, -1]])
kraus_4 = np.array([[0, 0], [1, 1], [0, 0]])
expected_res = np.array([[22, 95, 174], [2, 8, 14], [8, 29, 64]])
res = apply_channel(test_input_mat,
[[kraus_1, kraus_2], [kraus_3, kraus_4]])
bool_mat = np.isclose(res, expected_res)
np.testing.assert_equal(np.all(bool_mat), True)
def partial_channel(
rho: np.ndarray,
phi_map: Union[np.ndarray, List[List[np.ndarray]]],
sys: int = 2,
dim: Union[List[int], np.ndarray] = None,
) -> np.ndarray:
r"""Apply channel to a subsystem of an operator [WatPMap18]_.
Applies the operator
.. math::
\left(\mathbb{I} \otimes \Phi \right) \left(\rho \right).
In other words, it is the result of applying the channel :math:`\Phi` to the
second subsystem of :math:`\rho`, which is assumed to act on two
subsystems of equal dimension.
The input :code:`phi_map` should be provided as a Choi matrix.
This function is adapted from the QETLAB package.
Examples
==========
>>> from toqito.channel_ops import partial_channel
>>> from toqito.channels import depolarizing
>>> rho = np.array([[0.3101, -0.0220-0.0219*1j, -0.0671-0.0030*1j, -0.0170-0.0694*1j],
>>> [-0.0220+0.0219*1j, 0.1008, -0.0775+0.0492*1j, -0.0613+0.0529*1j],
>>> [-0.0671+0.0030*1j, -0.0775-0.0492*1j, 0.1361, 0.0602 + 0.0062*1j],
>>> [-0.0170+0.0694*1j, -0.0613-0.0529*1j, 0.0602-0.0062*1j, 0.4530]])
>>> phi_x = partial_channel(rho, depolarizing(2))
[[ 0.20545+0.j 0. +0.j -0.0642 +0.02495j 0. +0.j ]
[ 0. +0.j 0.20545+0.j 0. +0.j -0.0642 +0.02495j]
[-0.0642 -0.02495j 0. +0.j 0.29455+0.j 0. +0.j ]
[ 0. +0.j -0.0642 -0.02495j 0. +0.j 0.29455+0.j ]]
>>> from toqito.channel_ops import partial_channel
>>> from toqito.channels import depolarizing
>>> rho = np.array([[0.3101, -0.0220-0.0219*1j, -0.0671-0.0030*1j, -0.0170-0.0694*1j],
>>> [-0.0220+0.0219*1j, 0.1008, -0.0775+0.0492*1j, -0.0613+0.0529*1j],
>>> [-0.0671+0.0030*1j, -0.0775-0.0492*1j, 0.1361, 0.0602 + 0.0062*1j],
>>> [-0.0170+0.0694*1j, -0.0613-0.0529*1j, 0.0602-0.0062*1j, 0.4530]])
>>> phi_x = partial_channel(rho, depolarizing(2), 1)
[[0.2231+0.j 0.0191-0.00785j 0. +0.j 0. +0.j ]
[0.0191+0.00785j 0.2769+0.j 0. +0.j 0. +0.j ]
[0. +0.j 0. +0.j 0.2231+0.j 0.0191-0.00785j]
[0. +0.j 0. +0.j 0.0191+0.00785j 0.2769+0.j ]]
References
==========
.. [WatPMap18] Watrous, John.
The theory of quantum information.
Cambridge University Press, 2018.
:param rho: A matrix.
:param phi_map: The map to partially apply.
: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: The partial map :code:`phi_map` applied to matrix :code:`rho`.
"""
if dim is None:
dim = np.round(np.sqrt(list(rho.shape))).conj().T * np.ones(2)
if isinstance(dim, list):
dim = np.array(dim)
# Force dim to be a row vector.
if dim.ndim == 1:
dim = dim.T.flatten()
dim = np.array([dim, dim])
prod_dim_r1 = int(np.prod(dim[0, :sys - 1]))
prod_dim_c1 = int(np.prod(dim[1, :sys - 1]))
prod_dim_r2 = int(np.prod(dim[0, sys:]))
prod_dim_c2 = int(np.prod(dim[1, sys:]))
if isinstance(phi_map, list):
# Compute the Kraus operators on the full system.
s_phi_1, s_phi_2 = len(phi_map), len(phi_map[0])
# Map is completely positive.
if s_phi_2 == 1 or s_phi_1 == 1 and s_phi_2 > 2:
phi = []
for i, _ in enumerate(phi_map):
phi.append(
np.kron(
np.kron(np.identity(prod_dim_r1), phi_map[i]),
np.identity(prod_dim_r2),
))
phi_x = apply_channel(rho, phi)
else:
phi_1 = []
for i, _ in enumerate(phi_map):
phi_1.append(
np.kron(
np.kron(np.identity(prod_dim_r1), phi_map[i][0]),
np.identity(prod_dim_r2),
))
phi_2 = []
for i, _ in enumerate(phi_map):
phi_2.append(
np.kron(
np.kron(np.identity(prod_dim_c1), phi_map[i][1]),
np.identity(prod_dim_c2),
))
phi_x = [list(l) for l in zip(phi_1, phi_2)]
phi_x = apply_channel(rho, phi_x)
return phi_x
# The `phi_map` variable is provided as a Choi matrix.
if isinstance(phi_map, np.ndarray):
dim_phi = phi_map.shape
dim = np.array([
[
prod_dim_r2,
prod_dim_r2,
int(dim[0, sys - 1]),
int(dim_phi[0] / dim[0, sys - 1]),
prod_dim_r1,
prod_dim_r1,
],
[
prod_dim_c2,
prod_dim_c2,
int(dim[1, sys - 1]),
int(dim_phi[1] / dim[1, sys - 1]),
prod_dim_c1,
prod_dim_c1,
],
])
psi_r1 = max_entangled(prod_dim_r2, False, False)
psi_c1 = max_entangled(prod_dim_c2, False, False)
psi_r2 = max_entangled(prod_dim_r1, False, False)
psi_c2 = max_entangled(prod_dim_c1, False, False)
phi_map = permute_systems(
np.kron(np.kron(psi_r1 * psi_c1.conj().T, phi_map),
psi_r2 * psi_c2.conj().T),
[1, 3, 5, 2, 4, 6],
dim,
)
phi_x = apply_channel(rho, phi_map)
return phi_x
raise ValueError("The `phi_map` variable is assumed to be provided as "
"either a Choi matrix or a list of Kraus operators.")
def test_apply_channel_invalid_input():
"""Invalid input for apply map."""
with np.testing.assert_raises(ValueError):
apply_channel(np.array([[1, 2], [3, 4]]), 2)