toqito.measurements.pretty_good_measurement
===========================================

.. py:module:: toqito.measurements.pretty_good_measurement

.. autoapi-nested-parse::

   Compute the set of pretty good measurements from an ensemble.





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

.. py:function:: pretty_good_measurement(states, probs = None, tol = 1e-08)

   Return the set of pretty good measurements from a set of vectors and corresponding probabilities.

   This computes the "pretty good measurement" (PGM), also known as the
   square-root measurement, which is a widely used measurement for quantum
   state discrimination [@Belavkin_1975_Optimal,Hughston_1993_Complete].

   The PGM is the set of POVMs \((G_1, \ldots, G_n)\) such that

   \[
       G_i = P^{-1/2} \left(p_i \rho_i\right) P^{-1/2} \quad \text{where} \quad
       P = \sum_{i=1}^n p_i \rho_i.
   \]

   !!! See Also
       [pretty_bad_measurement()][toqito.measurements.pretty_bad_measurement.pretty_bad_measurement]

   .. rubric:: Examples

   Consider the collection of trine states.

   \[
       u_0 = |0\rangle, \quad
       u_1 = -\frac{1}{2}\left(|0\rangle + \sqrt{3}|1\rangle\right), \quad \text{and} \quad
       u_2 = -\frac{1}{2}\left(|0\rangle - \sqrt{3}|1\rangle\right).
   \]

   ```python exec="1" source="above"
   from toqito.states import trine
   from toqito.measurements import pretty_good_measurement

   states = trine()
   probs = [1 / 3, 1 / 3, 1 / 3]
   pgm = pretty_good_measurement(states, probs)
   print(pgm)
   ```

   :raises ValueError: If number of vectors does not match number of probabilities.
   :raises ValueError: If probabilities do not sum to 1.

   :param states: A collection of states provided as either vectors or density matrices.
   :param probs: A set of fixed probabilities for each quantum state. If not provided, a uniform distribution is assumed.
   :param tol: A tolerance value for numerical comparisons.

   :returns: A list of POVM operators for the PGM.