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Should the measured value always be about $2\sqrt{2}$ or is it just a maximum value and it is possible to measure something like $S = \{1.90, 1.8, 2.0, 2.4, 2.6, 1.8\}$ (with average equal 2) to violate the CHSH inequality?

EDIT: By $S$ I mean $S$ received from statistically significant number of measurements.

dk14
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1 Answers1

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The CHSH inequality is a statement about a linear combination of expectation values of different products of observables. It is not a statement about the outcome of a single measurement. Thus, violation of the CHSH inequality can only be inferred in a statistical sense after performing many repetitions of the experiment.

See: What exactly does $S$ represent in the CHSH inequality $-2\leq S\leq 2$?

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StephenJ
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  • Really, you never disprove local realism. Instead, as you do more repetitions of the measurement, you can make local realism less plausible. It's like if you flip a coin 4 times and it comes out heads each time, you might suspect it is not a fair coin. If you got heads 100 times in a row, then the probability that a fair coin could produce such an outcome is tiny, but still not zero. If you are interested in a more deterministic test of local realism, GHZ might be a better choice than CHSH. – StephenJ Oct 01 '14 at 01:49
  • Regarding your comment about different sets of experiments, I think it is a standard problem of doing statistical inference from experiments, rather than a problem specific to CHSH. If you think your experiment has been thrown out of alignment or something in between the sets of tests then maybe you can throw one set away. Otherwise, you should aggregate them all. – StephenJ Oct 01 '14 at 01:52
  • Suppose two experimenters each do a million repetitions of the CHSH experiment. Experimenter A gets a result consistent with local realism (S < 2). Experimenter B gets a result that couldn't be produced by any local realistic theory with probability greater than 0.001%. Should you then believe in local realism? I would say not. According to QM it is perfectly possible for a CHSH experiment to behave local-realistically, e.g due to environmentally induced decoherence. However, if local realism is true, one should never be able to violate CHSH in any experiment with good statistics. – StephenJ Oct 01 '14 at 02:06