measurement_props.is_povm

Determine if a list of matrices are POVM elements.

Module Contents

Functions

is_povm(mat_list)

Determine if a list of matrices constitute a valid set of POVMs [1].
measurement_props.is_povm.is_povm(mat_list)

Determine if a list of matrices constitute a valid set of POVMs [1].

A valid set of measurements are defined by a set of positive semidefinite operators

\[\{P_a : a \in \Gamma\} \subset \text{Pos}(\mathcal{X}),\]

indexed by the alphabet \(\Gamma\) of measurement outcomes satisfying the constraint that

\[\sum_{a \in \Gamma} P_a = I_{\mathcal{X}}.\]

Examples

Consider the following matrices:

\[\begin{split}M_0 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \quad \text{and} \quad M_1 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}.\end{split}\]

Our function indicates that this set of operators constitute a set of POVMs.

>>> 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 random_povm function from 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.rand 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

\[\begin{split}M_0 = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \quad \text{and} \quad M_1 = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix},\end{split}\]

do 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

[1] (1,2)

Wikipedia. POVM. https://en.wikipedia.org/wiki/POVM.

Parameters:

mat_list (list[numpy.ndarray]) – A list of matrices.

Returns:

Return True if set of matrices constitutes a set of measurements, and False otherwise.

Return type:

bool