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
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.,
- for any other setting (
SS), the measurements should be the same (i.e.,
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
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%.
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?