:py:mod:`measurements.pretty_bad_measurement`
=============================================

.. py:module:: measurements.pretty_bad_measurement

.. autoapi-nested-parse::

   Compute the set of pretty bad measurements from an ensemble.



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


Functions
~~~~~~~~~

.. autoapisummary::

   measurements.pretty_bad_measurement.pretty_bad_measurement



.. py:function:: pretty_bad_measurement(states, probs = None)

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

   This computes the "pretty bad measurement" as defined in :cite:`Hughston_1993_Complete` and is an analogous idea to
   the "pretty good measurement" from :cite:`McIrvin_2024_Pretty`. The "pretty bad measurement" is useful in the
   context of state exclusion where the pretty good measurement is often used for minimum-error quantum state
   discrimination.

   The pretty bad measurement (PBM) is defined in terms of an offset of the pretty good measurement (PGM). Recall that
   the PGM is defined as a set of POVMs :math:`(G_1, \ldots, G_n)` such that

   .. math::
       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.

   By proxy, the corresponding PBM is defined as a set of POVMs :math:`(B_1, \ldots, B_n)` where

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

   .. seealso:: :obj:`pretty_good_measurement`

   .. rubric:: Examples

   Consider the collection of trine states.

   .. math::
       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).

   >>> from toqito.states import trine
   >>> from toqito.measurements import pretty_bad_measurement
   >>>
   >>> states = trine()
   >>> probs = [1 / 3, 1 / 3, 1 / 3]
   >>> pgm = pretty_bad_measurement(states, probs)
   >>> pgm
   [array([[0.16666667, 0.        ],
          [0.        , 0.5       ]]), array([[ 0.41666667, -0.14433757],
          [-0.14433757,  0.25      ]]), array([[0.41666667, 0.14433757],
          [0.14433757, 0.25      ]])]

   .. rubric:: References

   .. bibliography::
       :filter: docname in docnames

   :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 either states provided as either vectors or density matrices.
   :param probs: A set of probabilities.