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BACKGROUND

The purpose of Bell's inequalities is to put bounds on certain combinations of classical measurement probabilities. One can then show that quantum entanglement can yield probabilities that add up to values beyond the classical bounds, hence the demonstration of non-locality. One such combination of measurement outcomes is presented in a lecture by Spekkens whereby, for each one of two measurement settings T and S, the outcome can be either red (R) or green (G). The combinations of measurement outcomes for which the measurement probabilities are then collected are that, for

  • the measurement setting TT, the readout should be different (i.e., RG or GR), and
  • for any other setting (TS, ST, and SS), the measurements should be the same (i.e., RR or GG).

The lecture goes on to show that the above requirements on measurement outcomes can only be satisfied 75% of the time by classical systems. One such "optimal strategy" that can be adopted by classical players is that the first party reveals G if measured by T and that the second reveals R if measured by T and vice-versa when measured by S. I.e.,

TT: GR (condition satisfied)
TS: GG (condition satisfied)
ST: RR (condition satisfied)
SS: RG (condition not satisfied)

so the probability of success is 3/4 = 75%.

QUESTION

How are these optimal probabilities elicited in the most general case? In this bi-partite case, it's relatively easy to trace all the scenarios and find out that it should be 75%. However, in a generic scenario, are the bounds of the inequality obtained from brute-force combinatorics or is there a more subtle way to find the contour of classicality for a given combination of desired measurement outcomes?

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  • $\begingroup$ I can not answer your question. But can comment that the classical percentage (75%) is not necessarily correct to begin with and in the end the inequality appears to be violated. 75% is calculated considering TT, TS, ST, SS all equally likely. i.e. 3/4 = 75%. However, the 4 outcomes may not stay equally likely throughout the experiment. In order to honor the conservation laws, the likely hood of TT, TS, ST, SS can change mid experiment as a consequence of previous outcomes and their measurements. This would render the 4 unequally likely and violation of inequality would not be any mystery. $\endgroup$ – kpv Apr 28 '19 at 23:16
  • $\begingroup$ For example, your shoe size is 10. You have room full of shoes with 20% of #10 shoes, so you expect overall 20% shoes will fit you if you were to try out all the shoe pairs one at a time. But as a consequence of trying out so many pairs, suppose your foot starts to swell little by little. Your shoe size is changing during the course of experiment! Now the original prediction can obviously be violated and should not prove non-locality. Just that you did not account for swelling of foot. Bell's inequality is a correct theorem by itself, but its application to entanglement is very likely, faulty. $\endgroup$ – kpv Apr 28 '19 at 23:23
  • $\begingroup$ Physics without mathematics is not possible, and can only be a guess work at best. Mathematics without considering all possible physics can get weird to say the least. $\endgroup$ – kpv Apr 28 '19 at 23:32
  • $\begingroup$ What's the analogy with the swelling foot here? The states which are measured are all the same, just like a foot is expected to remain the same. $\endgroup$ – Tfovid Apr 29 '19 at 8:16
  • $\begingroup$ You are not measuring the states, you are measuring (or rather aligning) spin direction. Just an example - suppose the current spin direction is x, you measure along y, it will align, or anti align along y. This align or anti align from x to y needs an equal and opposite change in the measuring device. Thereby altering angular momentum of measuring device over time, which would start preferring for example - aligning over anti aligning or vice e versa. This shift itself can swing back and forth, ultimately results matching QM predictions. $\endgroup$ – kpv Apr 29 '19 at 13:48
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There is a rather extensive body of literature on this topic, but two good papers giving some formal (subtle, if you will) approaches are https://arxiv.org/abs/1102.0264 and https://arxiv.org/abs/1401.7081.

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  • $\begingroup$ I'm afraid that if I had the mathematical prowess to read those articles I wouldn't have asked the question! :-) While I'm wading through those papers, could you please summarize in basic terms whether the inequality is derived by brute force or by some other more systematic method? In the latter case, what's the gist of it? Is it an optimization problem? A geometric argument? $\endgroup$ – Tfovid Apr 24 '19 at 13:19
  • $\begingroup$ @Tfovid yes, they are very hard papers! The first paper I put in because it is the first in a string of papers working towards a topological approach to not just Bell inequalities, but contextuality more generally. To my knowledge, though, there is not a complete characterization of all possible Bell inequalities (in arbitrary dimensions) and their quantum mechanical violations yet along these lines. The second paper provides a general characterization of Bell inequalities in terms of properties of (in)compatibility graphs, but this is still combinatorial, just with a unifying framework. $\endgroup$ – Will Apr 24 '19 at 20:11

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