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

.. py:module:: toqito.state_props.common_quantum_overlap

.. autoapi-nested-parse::

   Computes the common quantum overlap quantum states.





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

.. py:function:: common_quantum_overlap(states)

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

   For more information, see [@Campos_2024_Epistemic].

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

   \[
       \omega_Q[n] = n(1 - A_Q[n])
   \]

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

   \[
       \omega_Q = 1 - \sqrt{1 - p^2}
   \]

   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:

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

   For maximally mixed states in any dimension:

   ```python exec="1" source="above"
   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]
   print(common_quantum_overlap(states))
   ```

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

   \[
       \omega_Q = 1 - \sqrt{1 - \cos(\theta)^2}
   \]

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

   ```python exec="1" source="above"
   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)])]
   print(common_quantum_overlap(states)) # Should approximate (1-sqrt(1-cos²(π/4)))
   ```

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

   :returns: The common quantum overlap value.