state_props.common_quantum_overlap
==================================

.. py:module:: state_props.common_quantum_overlap

.. autoapi-nested-parse::

   Computes the common quantum overlap quantum states.



Functions
---------

.. autoapisummary::

   state_props.common_quantum_overlap.common_quantum_overlap


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

.. py:function:: common_quantum_overlap(states)

   Calculate the common quantum overlap of a collection of quantum states.

   For more information, see :cite:`Campos_2024_Epistemic`.

   The common quantum overlap :math:`\omega_Q[n]` quantifies the "overlap" between :math:`n` quantum states
   based on their antidistinguishability properties. It is related to the
   antidistinguishability probability :math:`A_Q[n]` by the formula:

   .. math::
       \omega_Q[n] = n(1 - A_Q[n])

   For two pure states with inner product :math:`|\langle\psi|\phi\rangle| = p`, the common quantum overlap is:

   .. math::
       \omega_Q = 1 - \sqrt{1 - p^2}

   The common quantum overlap is a key concept in analyzing epistemic models of quantum
   mechanics and understanding quantum state preparation contextuality.

   .. rubric:: Examples

   Consider the Bell states:

   .. jupyter-execute::

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

   For maximally mixed states in any dimension:

   .. jupyter-execute::

       import numpy as np
       from toqito.state_props import common_quantum_overlap
       dim = 2
       states = [np.eye(dim) / dim, np.eye(dim) / dim, np.eye(dim) / dim]
       common_quantum_overlap(states)

   The common quantum overlap :math:`\omega_Q` for two pure states
   with inner product :math:`|\langle \psi | \phi \rangle| = \cos(\theta)` is given by:

   .. math::
       \omega_Q = 1 - \sqrt{1 - \cos(\theta)^2}

   where :math:`\theta` represents the angle between the two states in Hilbert space.
   For two pure states with a known inner product:

   .. jupyter-execute::

       import numpy as np
       from toqito.state_props import common_quantum_overlap
       theta = np.pi/4
       states = [np.array([1, 0]), np.array([np.cos(theta), np.sin(theta)])]
       common_quantum_overlap(states) # Should approximate (1-sqrt(1-cos²(π/4)))

   .. rubric:: References

   .. bibliography::
       :filter: docname in docnames

   :param states: A list of quantum states represented as numpy arrays. States can be
                 pure states (represented as state vectors) or mixed states
                 (represented as density matrices).
   :return: The common quantum overlap value.