toqito.state_props.is_antidistinguishable
=========================================

.. py:module:: toqito.state_props.is_antidistinguishable

.. autoapi-nested-parse::

   Check if set of states are antidistinguishable.





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

.. py:function:: is_antidistinguishable(states)

   Check whether a collection of vectors are antidistinguishable or not.

   For more information, see [@Heinosaari_2018_Antidistinguishability].

   The ability to determine whether a set of quantum states are antidistinguishable can be obtained via the state
   exclusion SDP [@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:

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

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

   ```python exec="1" source="above"
   from toqito.states import bell
   from toqito.state_props import is_antidistinguishable
   bell_states = [bell(0), bell(1), bell(2), bell(3)]
   print(is_antidistinguishable(bell_states))
   ```

   Consider the following measurement operators

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

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

   \[
       \sum_{i=1}^4 \langle M_i, u_i u_i^* \rangle = 0,
   \]

   and hence, the Bell states are antidistinguishable.

   :param states: A set of vectors consisting of quantum states to determine the antidistinguishability of.

   :returns: `True` if the vectors are antidistinguishable; `False` otherwise.