toqito.state_props.common_quantum_overlap

Computes the common quantum overlap quantum states.

Module Contents

toqito.state_props.common_quantum_overlap.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|phirangle| = 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:

`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))) `

Parameters:

• states (list[numpy.ndarray]) – A list of quantum states represented as numpy arrays. States can be pure states

• states

Returns:

The common quantum overlap value.

Return type:

float