Image: CERN – https://home.cern/news/news/physics/fifty-years-bells-theorem
We run a Bell test in the CHSH formulation on two real quantum platforms: the goal was not to demonstrate anything new, but to use a well-known experiment as a direct physical benchmark to understand how much a real quantum processor can preserve entangled correlations. This test could fit the purpose because it produces a number that is easy to interpret, that is the Bell parameter: if the value remains less than or equal to 2, the result is compatible with a local classical description. If it exceeds 2, a violation of Bell’s inequality is observed, and, in particular, the theoretical quantum maximum is 2*√2, approximately 2.82843. An important aspect of this experiment is that the measurement basis is chosen using block randomization, that is instead of first measuring all the data for one given configuration and then moving on to the next, the different measurement settings are randomly mixed throughout the execution. This has been done to reduce the imperfection of the real hardware due to calibrations, noise, temporal drift, and operating conditions, thus making a random choice of measurement basis would reduce the risk. Further, this point wanted to reproduce the spirit of Alain Aspect’s experiments on the violation of Bell inequalities. In that case, the choice of measurement basis was critical: changing the orientation of the analyzers during the experiment served to avoid the particles from being interpreted as already “prepared” with respect to a fixed measurement configuration. In Aspect’s case, the change of basis therefore had a very deep foundational meaning, connected to locality and the separation between measurement choices.
In our case, much more humbly, no locality loophole is being closed: the qubits are on the same device, the experiment is digitally programmed, and the measurements are not spatially separated as in an optical Bell test and the methodological idea is: not allowing a fixed measurement configuration to dominate an ordered part of the experiment.
For each experiment, 100,000 shots were used because the quantities observed in the Bell test are derived from measured probabilities and increasing the number of shots reduces the statistical error on the observed frequencies and as a consequence on the calculated correlations.
The result (Bell parameter) on the first real hardware technology was 2.45294 ± 0.00499, well above the classical limit of 2. The deviation from the ideal quantum value was 0.37549, corresponding to about 86.7% of the theoretical maximum, that is the apparent loss of visibility was about 13.3% and the statistical significance with respect to the classical limit was approximately 90.7 sigma.
On the second real hardware technology, the result was also clearly quantum, as the measured value was 2.40736 ± 0.00505, still above the classical limit and this time the deviation from the ideal value was 0.42107, corresponding to about 85.1% of the theoretical maximum with an apparent loss of visibility of about 14.9%, with a statistical significance of approximately 80.7 sigma with respect to the classical limit.
It is useful to interpret the deviation from the ideal value not as a pure measure of a single type of “noise” but as an aggregate indicator of experimental degradation: gate errors, readout errors, decoherence, drift, and calibration imperfections all contribute to reducing the observed value, giving a better view of what measured experimentally with random circuits in previous pieces of work. In particular, it should be noted that the closer the result is to the ideal maximum, the better the device is preserving quantum correlations; the closer it moves toward the classical limit, the more noise has degraded the experiment.
Here below a sample of the used circuits and result of the 100000 runs on the first real hardware.
RESULTS ON REAL HARDWARE
E(a,b) = 0.62446 ± 0.00247 N = 100000
E(a,b’) = 0.59396 ± 0.00254 N = 100000
E(a’,b) = 0.60250 ± 0.00252 N = 100000
E(a’,b’) = -0.63202 ± 0.00245 N = 100000
S CHSH = 2.45294 ± 0.00499