qbraid_algorithms.bells_inequality

Bell’s Inequality Experiment

Bell’s Inequality Experiment

This module provides an implementation of Bell’s inequality experiment, a fundamental test of quantum mechanics that demonstrates Non-local correlations. The experiment creates an entangled Bell state, applies different measurement bases to each qubit, and measures correlations. When Bell’s inequality is violated, it proves that quantum correlations cannot be explained by classical local hidden variable theories.

FORMULATION

This implementation tests Bell’s inequality using three Bell singlet states prepared between qubit pairs.

  1. State Preparation: Each qubit pair is prepared in the Bell singlet state:

    \(|\Psi^-\rangle = 1/\sqrt{2}(|01\rangle - |10\rangle)\)

  2. Measurement Settings: Three different measurement configurations:

    • Circuit AB: Alice measures at 0°, Bob measures at 60° (\(\pi/3\))

    • Circuit AC: Alice measures at 0°, Charlie measures at 120° (\(2\pi/3\))

    • Circuit BC: Bob measures at 60° (\(\pi/3\)), Charlie measures at 120° (\(2\pi/3\))

  3. Bell’s Inequality: The CHSH inequality for Bell singlet states:

    \(|E(0°, 60°) - E(0°, 120°) + E(60°, 120°)| \leq 2\)

    where \(E(\theta_A, \theta_B) = -\cos(\theta_A - \theta_B)\) is the correlation function.

  4. Quantum Prediction: For Bell singlet states, quantum mechanics predicts:

    \(|E(0°, 60°) - E(0°, 120°) + E(60°, 120°)| = |-\cos(60°) + \cos(120°) - \cos(60°)| = 3 > 2\)

    This violates Bell’s inequality, demonstrating non-local quantum correlations.

Functions

generate_program()

Load the Bell's inequality circuit as a pyqasm module.