state_props.common_quantum_overlap¶

Computes the common quantum overlap quantum states.

Functions¶

common_quantum_overlap(states)

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

Module Contents¶

state_props.common_quantum_overlap.common_quantum_overlap(states)¶

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

For more information, see [1].

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.

Examples

Consider the Bell states:

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)

3.3910878549647805e-09

For maximally mixed states in any dimension:

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)

0.9999999999999998

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:

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)))

0.29289321883356556

References

Parameters:: states (list[numpy.ndarray]) – 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).
Returns:: The common quantum overlap value.
Return type:: float