Clauser-Horne-Shimony-Holt game, Tsirelson's bound, and causal locality

Source abstract

Starting with heuristic assumptions about locality and causality, Bell established a numerical upper bound on statistical correlations between spatially separated systems and showed that quantum systems could violate it. This upper bound, also known as the Bell or Clauser-Horne-Shimony-Holt (CHSH) inequality, can be derived from an experimental setup known as the CHSH game. Tsirelson used the CHSH formulation to derive a higher, maximal bound on quantum correlations. Later, Popescu and Rohrlich (PR) studied correlations beyond quantum theory that were constrained only by the no-signaling principle, a condition that disallows superluminal communication between observers. They showed that such correlations could theoretically violate the Tsirelson bound. Is there some conceptually simple or transparent principle that explains the large gap between the Tsirelson bound and the PR bound? Or do we have to appeal to the technical axiomatic structure of quantum theory? In this paper, we reformulate the CHSH game in terms of indivisible stochastic processes. Using Barandes's stochastic-quantum correspondence and its associated definition of causal locality, we present a proof of the Tsirelson bound. In particular, we show that, unlike the no-signaling principle alone, the postulates defining causally local, indivisible stochastic processes are precisely strong enough to allow for violations of the Bell inequality up to, but not beyond, the Tsirelson bound.