:py:mod:`state_props.is_antidistinguishable`
============================================

.. py:module:: state_props.is_antidistinguishable

.. autoapi-nested-parse::

   Check if set of states are antidistinguishable.



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


Functions
~~~~~~~~~

.. autoapisummary::

   state_props.is_antidistinguishable.is_antidistinguishable



.. py:function:: is_antidistinguishable(states)

   Check whether a collection of vectors are antidistinguishable or not.

   For more information, see :cite: `Heinosaari_2018_Antidistinguishability`.

   The ability to determine whether a set of quantum states are antidistinguishable can be obtained via the state
   exclusion SDP :cite:`Bandyopadhyay_2014_Conclusive` such that we ignore the associated probabilities with which
   the states are chosen from the set of vectors.

   .. rubric:: Examples

   The set of Bell states are an example of antidistinguishable states. Recall that the Bell states are defined as:

   .. math::
       u_1 = \frac{1}{\sqrt{2}} \left(|00\rangle + |11\rangle\right), &\quad
       u_2 = \frac{1}{\sqrt{2}} \left(|00\rangle - |11\rangle\right), \\
       u_3 = \frac{1}{\sqrt{2}} \left(|01\rangle + |10\rangle\right), &\quad
       u_4 = \frac{1}{\sqrt{2}} \left(|01\rangle - |10\rangle\right).

   It can be checked in :code`toqito` that the Bell states are antidistinguishable:

   >>> from toqito.states import bell
   >>> from toqito.state_props import is_antidistinguishable
   >>>
   >>> bell_states = [bell(0), bell(1), bell(2), bell(3)]
   >>> is_antidistinguishable(bell_states)
   True

   Consider the following measurement operators

   .. math::
       M_i = \frac{1}{3}\left(\mathbb{I}_{\mathcal{X} - u_i u_i^*\right)

   for all :math:`1 \leq i \leq 4`. It can be verified that these constitute a valid set of POVMs, that is
   :math:`\sum_{i=1}^4 M_i = \mathbb{I}_{\mathcal{X}}` and :math:`M_i \in \text{Pos}(\mathcal{X})` for all :math:`1
   \leq i \leq 4`. It may also be verified that

   .. math::
       \sum_{i=1}^4 \langle M_i, u_i u_i^* \rangle = 0,

   and hence, the Bell states are antidistinguishable.

   .. rubric:: References

   .. bibliography::
       :filter: docname in docnames

   :param states: A set of vectors consisting of quantum states to determine the antidistinguishability of.
   :return: :code:`True` if the vectors are antidistinguishable; :code:`False` otherwise.