toqito.measurements.pretty_bad_measurement
==========================================

.. py:module:: toqito.measurements.pretty_bad_measurement

.. autoapi-nested-parse::

   Compute the set of pretty bad measurements from an ensemble.





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

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

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

   This computes the "pretty bad measurement" (PBM) as defined in
   [@McIrvin_2024_Pretty]. The PBM is an analogue to the "pretty
   good measurement" defined in [@Belavkin_1975_Optimal,Hughston_1993_Complete]
   and is useful for approximating the optimal measurement
   for state exclusion.

   The PBM is defined in terms of the pretty good measurement (PGM).
   Given the PGM operators \((G_1, \ldots, G_n)\), the corresponding PBM
   is the set of POVMs \((B_1, \ldots, B_n)\) where

   \[
       B_i = \frac{1}{n - 1} \left(\mathbb{I} - G_i\right).
   \]

   !!! See Also
       [pretty_good_measurement()][toqito.measurements.pretty_good_measurement.pretty_good_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_bad_measurement

   states = trine()
   probs = [1 / 3, 1 / 3, 1 / 3]
   pbm = pretty_bad_measurement(states, probs)
   print(pbm)
   ```

   :raises ValueError: If number of states 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 PBM.