We illustrate how to test the version of Bell´s inequality proposed by David Mermin using Qiskit. We do it on an ideal simulated quantum computer, a simulated quantum computer with noise and a real quantum computer from the IBM Quantum Experience.
See the references for a clear explanation of this version of Bell's inequality.
EPR vs Quantum MECHANICS
The 2022 Nobel Prize in Physics was awarded to Alain Aspect, John Clauser and Anton Zeilinger
"for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science"
We model, using a quantum computer, the measurement of the spin projections of two spin-1/2 entangled particles A and B which have total spin equal to zero along 3 axes rotated 0 (Axis 1), 120 (Axis 2) y 240 degrees (Axis 3) on the XZ plane
A 0 corresponds to spin up and 1 to spin down along the particular axis used for that measurement
Measurement A <---------------------------- 2 spin 1/2 particles with total S = 0 -------------------> Measurement B
A1B1 means we measure along axis1 for particle A and axis 1 for particle B, A1B2 means we measure along axis1 for particle A and axis 2 for particle B and so on.
Assuming the spins are determined (by hidden variables) prior to the measurements, the EPR prediction would be
A B Anti-Correlations Fraction of Anti-Correlated Measurements
123 123 [A1B1,A1B2,A1B3,A2B1,A2B2,A2B3,A3B1,A3B2,A3B3]
000 111 [1, 1, 1, 1, 1, 1, 1, 1, 1] 9 / 9
001 110 [1, 1, 0, 1, 1, 0, 0, 0, 1] 5 / 9
010 101 [1, 0, 1, 0, 1, 0, 1, 0, 1] 5 / 9
011 100 [1, 0, 0, 0, 1, 1, 0, 1, 1] 5 / 9
100 011 [1, 0, 0, 0, 1, 1, 0, 1, 1] 5 / 9
101 010 [1, 0, 1, 0, 1, 0, 1, 0, 1] 5 / 9
110 001 [1, 1, 0, 1, 1, 0, 0, 0, 1] 5 / 9
111 000 [1, 1, 1, 1, 1, 1, 1, 1, 1] 9 / 9
According to EPR the fraction of anti-correlated measurements is > 5/9 (Bell's inequality: Anti-Corr(A,B) > 56%) but the experiments produce 50% anti-correlated measurements.
Quantum Mechanics violates Bell's inequality and therefore the Einstein-Podolsky-Rosen argument is not valid and Quantum Mechanics is a non-local theory.
References
Mermin, N.D.: Bringing home the atomic world: Quantum mysteries for anybody. American Journal of Physics 49, 940-943 (1981).
Answering Mermin’s Challenge with the Relativity Principle by Mark Stuckey https://www.physicsforums.com/insights/answering-mermins-challenge-with-wilczeks-challenge
Youtube video by Brian Greene: https://www.youtube.com/watch?v=UZiwtfrisTQ