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In the CHSH inequality, we construct an experiment whereby two observers each receive a particle and measure two given properties of their particles, for which the outcomes are ±1. We then consider the expectation value of some quantity defined by:

$\langle S \rangle = \langle AB - Ab + aB + ab \rangle$,

where A and a correspond to the measurement results for the two properties measured by the first observer, B and b for the second.

My confusion stems from the fact that each observer only measures one property at a time. Thus, it follows that the values for A,a/B,b must be the values that each observer would have gotten if they had measured the corresponding property. This will be equivalent to the value that each observer actually does measure if we assume realism, which is valid as this is one of the two assumptions we make for the CHSH inequality (the other being locality). However, when this experiment is actually carried out in the lab, each observer only measures one property, thus, for each iteration, there only exist 2 terms that can be plugged into S. Is this a problem and if so how is it reconciled?

EDIT: Resolved in comments below but idk how to resolve a post without a formal answer.

David
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    e.g. the Wiki article on Bell's theorem repeatedly states that experiments can't measure this quantity in a single trial and we look at averages over repeated trials. Are you not satisfied with these statements? – ACuriousMind May 21 '23 at 11:34
  • Does usage of averages over repeated trials in place of single trial measurements assume realism? – David May 21 '23 at 11:39
  • Ah right, makes sense now. I was troubled that we were assuming realism both in the gedankenexperiment and the real thing, but I realise that this isn't a problem because we're checking for a failure case rather than a success. – David May 21 '23 at 11:45
  • see also https://physics.stackexchange.com/a/699048/58382. The idea is simply that you want to verify that correlations are stronger than those allowed by local realism. Thus you measure correlations between the parties in different bases, and compare the results, which is what $S$ amounts to – glS May 22 '23 at 09:03

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