toqito.measurement_props.is_povm
================================

.. py:module:: toqito.measurement_props.is_povm

.. autoapi-nested-parse::

   Determine if a list of matrices are POVM elements.





Module Contents
---------------

.. py:function:: is_povm(mat_list)

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

   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

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

   .. rubric:: Examples

   Consider the following matrices:

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

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

   ```python exec="1" source="above"
   import numpy as np
   from toqito.measurement_props import is_povm

   meas_1 = np.array([[1, 0], [0, 0]])
   meas_2 = np.array([[0, 0], [0, 1]])
   meas = [meas_1, meas_2]

   print(is_povm(meas))
   ```

   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.

   ```python exec="1" source="above"
   import numpy as np
   from toqito.rand import random_povm
   from toqito.measurement_props import is_povm

   dim, num_inputs, num_outputs = 2, 2, 2
   measurements = random_povm(dim, num_inputs, num_outputs)

   print(is_povm([measurements[:, :, 0, 0], measurements[:, :, 0, 1]]))
   ```

   Alternatively, the following matrices

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

   do not constitute a POVM set.

   ```python exec="1" source="above"
   import numpy as np
   from toqito.measurement_props import is_povm

   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]

   print(is_povm(non_meas))
   ```

   :param mat_list: A list of matrices.

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