def is_projection(mat: np.ndarray) -> bool:
r"""
Check if matrix is a projection matrix [WikProj]_.
A matrix is a projection matrix if it is positive semidefinite (PSD) and if
.. math::
\begin{equation}
X^2 = X
\end{equation}
where :math:`X` is the matrix in question.
Examples
==========
Consider the following matrix
.. math::
A = \begin{pmatrix}
0 & 1 \\
0 & 1
\end{pmatrix}
our function indicates that this is indeed a projection matrix.
>>> from toqito.matrix_props import is_projection
>>> import numpy as np
>>> A = np.array([[0, 1], [0, 1]])
>>> is_projection(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_projection
>>> import numpy as np
>>> B = np.array([[-1, -1], [-1, -1]])
>>> is_projection(B)
False
References
==========
.. [WikProj] Wikipedia: Projection matrix.
https://en.wikipedia.org/wiki/Projection_matrix
:param mat: Matrix to check.
:return: Return True if matrix is a projection matrix, and False otherwise.
"""
if not is_psd(mat):
return False
return np.allclose(np.linalg.matrix_power(mat, 2), mat)
def is_ensemble(states: List[np.ndarray]) -> bool:
r"""
Determine if a set of states constitute an ensemble [WatEns18]_.
An ensemble of quantum states is defined by a function
.. math::
\eta : \Gamma \rightarrow \text{Pos}(\mathcal{X})
that satisfies
.. math::
\text{Tr}\left( \sum_{a \in \Gamma} \eta(a) \right) = 1.
Examples
==========
Consider the following set of matrices
.. math::
\eta = \left\{ \rho_0, \rho_1 \right\}
where
.. math::
\rho_0 = \frac{1}{2} \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \quad
\rho_1 = \frac{1}{2} \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}.
The set :math:`\eta` constitutes a valid ensemble
>>> from toqito.state_props import is_ensemble
>>> import numpy as np
>>> rho_0 = np.array([[0.5, 0], [0, 0]])
>>> rho_1 = np.array([[0, 0], [0, 0.5]])
>>> states = [rho_0, rho_1]
>>> is_ensemble(states)
True
References
==========
.. [WatEns18] Watrous, John.
"The theory of quantum information."
Section: "Ensemble of quantum states".
Cambridge University Press, 2018.
:param states: The list of states to check.
:return: True if states form an ensemble and False otherwise.
"""
trace_sum = 0
for state in states:
trace_sum += np.trace(state)
# Constraint: All states in ensemble must be positive semidefinite.
if not is_psd(state):
return False
# Constraint: The sum of the traces of all states within the ensemble must
# be equal to 1.
return np.allclose(trace_sum, 1)
def is_density(mat: np.ndarray) -> bool:
r"""
Check if matrix is a density matrix [WikDensity]_.
A matrix is a density matrix if its trace is equal to one and it has the
property of being positive semidefinite (PSD).
Examples
==========
Consider the Bell state
.. math::
u = \frac{1}{\sqrt{2}} |00 \rangle + \frac{1}{\sqrt{2}} |11 \rangle.
Constructing the matrix :math:`\rho = u u^*` defined as
.. math::
\rho = \frac{1}{2} \begin{pmatrix}
1 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
1 & 0 & 0 & 1
\end{pmatrix}
our function indicates that this is indeed a density operator as the trace
of :math:`\rho` is equal to :math:`1` and the matrix is positive
semidefinite:
>>> from toqito.matrix_props import is_density
>>> from toqito.states import bell
>>> import numpy as np
>>> rho = bell(0) * bell(0).conj().T
>>> is_density(rho)
True
Alternatively, the following example matrix :math:`\sigma` defined as
.. math::
\sigma = \frac{1}{2} \begin{pmatrix}
1 & 2 \\
3 & 1
\end{pmatrix}
does satisfy :math:`\text{Tr}(\sigma) = 1`, however fails to be positive
semidefinite, and is therefore not a density operator. This can be
illustrated using `toqito` as follows.
>>> from toqito.matrix_props import is_density
>>> from toqito.states import bell
>>> import numpy as np
>>> sigma = 1/2 * np.array([[1, 2], [3, 1]])
>>> is_density(sigma)
False
References
==========
.. [WikDensity] Wikipedia: Density matrix
https://en.wikipedia.org/wiki/Density_matrix
:param mat: Matrix to check.
:return: Return `True` if matrix is a density matrix, and `False`
otherwise.
"""
return is_psd(mat) and np.isclose(np.trace(mat), 1)
def is_povm(mat_list: List[np.ndarray]) -> bool:
r"""
Determine if a list of matrices constitute a valid set of POVMs [WikPOVM]_.
A valid set of measurements are defined by a set of positive semidefinite operators
.. math::
\{P_a : a \in \Gamma\} \subset \text{Pos}(\mathcal{X}),
indexed by the alphabet :math:`\Gamma` of measurement outcomes satisfying the constraint that
.. math::
\sum_{a \in \Gamma} P_a = I_{\mathcal{X}}.
Examples
==========
Consider the following matrices
.. math::
M_0 =
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix}, \qquad
M_1 =
\begin{pmatrix}
0 & 0 \\
0 & 1
\end{pmatrix}
our function indicates that this set of operators constitute a POVM.
>>> from toqito.measurement_props import is_povm
>>> import numpy as np
>>> meas_1 = np.array([[1, 0], [0, 0]])
>>> meas_2 = np.array([[0, 0], [0, 1]])
>>> meas = [meas_1, meas_2]
>>> is_povm(meas)
True
We may also use the :code:`random_povm` function from :code:`toqito`, and can verify that a
randomly generated set satisfies the criteria for being a POVM set.
>>> from toqito.measurement_props import is_povm
>>> from toqito.random import random_povm
>>> import numpy as np
>>> dim, num_inputs, num_outputs = 2, 2, 2
>>> measurements = random_povm(dim, num_inputs, num_outputs)
>>> is_povm([measurements[:, :, 0, 0], measurements[:, :, 0, 1]])
True
Alternatively, the following matrices
.. math::
M_0 =
\begin{pmatrix}
1 & 2 \\
3 & 4
\end{pmatrix}, \qquad
M_1 =
\begin{pmatrix}
5 & 6 \\
7 & 8
\end{pmatrix}
does not constitute a POVM set.
>>> from toqito.measurement_props import is_povm
>>> import numpy as np
>>> non_meas_1 = np.array([[1, 2], [3, 4]])
>>> non_meas_2 = np.array([[5, 6], [7, 8]])
>>> non_meas = [non_meas_1, non_meas_2]
>>> is_povm(non_meas)
False
References
==========
.. [WikPOVM] Wikipedia: POVM
https://en.wikipedia.org/wiki/POVM
:param mat_list: A list of matrices.
:return: Return :code:`True` if set of matrices constitutes a set of
measurements, and :code:`False` otherwise.
"""
dim = mat_list[0].shape[0]
mat_sum = np.zeros((dim, dim), dtype=complex)
for mat in mat_list:
# Each measurement in the set must be positive semidefinite.
if not is_psd(mat):
return False
mat_sum += mat
# Summing all the measurements from the set must be equal to the identity.
if not np.allclose(np.identity(dim), mat_sum):
return False
return True
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 test_is_psd():
"""Test that positive semidefinite matrix returns True."""
mat = np.array([[1, -1], [-1, 1]])
np.testing.assert_equal(is_psd(mat), True)
def test_is_not_psd():
"""Test that non-positive semidefinite matrix returns False."""
mat = np.array([[-1, -1], [-1, -1]])
np.testing.assert_equal(is_psd(mat), False)
def is_ppt(mat: np.ndarray,
sys: int = 2,
dim: Union[int, List[int]] = None,
tol: float = None) -> bool:
r"""
Determine whether or not a matrix has positive partial transpose [WikPPT]_.
Yields either :code:`True` or :code:`False`, indicating that :code:`mat` does or does not have
positive partial transpose (within numerical error). The variable :code:`mat` is assumed to act
on bipartite space.
For shared systems of :math:`2 \otimes 2` or :math:`2 \otimes 3`, the PPT criterion serves as a
method to determine whether a given state is entangled or separable. Therefore, for systems of
this size, the return value :code:`True` would indicate that the state is separable and a value
of :code:`False` would indicate the state is entangled.
Examples
==========
Consider the following matrix
.. math::
X =
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\end{pmatrix}.
This matrix trivially satisfies the PPT criterion as can be seen using the
:code:`toqito` package.
>>> from toqito.state_props import is_ppt
>>> import numpy as np
>>> mat = np.identity(9)
>>> is_ppt(mat)
True
Consider the following Bell state:
.. math::
u = \frac{1}{\sqrt{2}}\left( |01 \rangle + |10 \rangle \right)
For the density matrix :math:`\rho = u u^*`, as this is an entangled state
of dimension :math:`2`, it will violate the PPT criterion, which can be seen
using the :code:`toqito` package.
>>> from toqito.states import bell
>>> from toqito.state_props import is_ppt
>>> rho = bell(2) * bell(2).conj().T
>>> is_ppt(rho)
False
References
==========
.. [WikPPT] Quantiki: Positive partial transpose
https://www.quantiki.org/wiki/positive-partial-transpose
:param mat: A square matrix.
:param sys: Scalar or vector indicating which subsystems the transpose
should be applied on.
:param dim: The dimension is a vector containing the dimensions of the
subsystems on which :code:`mat` acts.
:param tol: Tolerance with which to check whether `mat` is PPT.
:return: Returns :code:`True` if :code:`mat` is PPT and :code:`False` if
not.
"""
eps = np.finfo(float).eps
sqrt_rho_dims = np.round(np.sqrt(list(mat.shape)))
if dim is None:
dim = np.array([[sqrt_rho_dims[0], sqrt_rho_dims[0]],
[sqrt_rho_dims[1], sqrt_rho_dims[1]]])
if tol is None:
tol = np.sqrt(eps)
return is_psd(partial_transpose(mat, sys, dim), tol)