Tight Limits on Nonlocality from Nontrivial Communication Complexity; a.k.a. Reliable Computation with Asymmetric Gate Noise

TL;DR

This paper investigates the limits of nonlocal correlations in quantum mechanics by analyzing their impact on communication complexity, providing bounds and evidence that certain nonlocal probabilities cannot be achieved without collapsing communication complexity.

Contribution

It introduces new bounds on nonlocal correlations related to communication complexity and connects these bounds to reliable classical computation with noisy gates.

Findings

A nonlocal game where communication complexity collapses if winning probability exceeds quantum limits.

Evidence that achieving a 0.91 winning probability in CHSH game is likely impossible with current proof strategies.

New insights into reliable classical computation with noisy XOR and AND gates.

Abstract

It has long been known that the existence of certain superquantum nonlocal correlations would cause communication complexity to collapse. The absurdity of a world in which any nonlocal binary function could be evaluated with a constant amount of communication in turn provides a tantalizing way to distinguish quantum mechanics from incorrect theories of physics; the statement "communication complexity is nontrivial" has even been conjectured to be a concise information-theoretic axiom for characterizing quantum mechanics. We directly address the viability of that perspective with two results. First, we exhibit a nonlocal game such that communication complexity collapses in any physical theory whose maximal winning probability exceeds the quantum value. Second, we consider the venerable CHSH game that initiated this line of inquiry. In that case, the quantum value is about 0.85 but it is known that a winning probability of approximately 0.91 would collapse communication complexity. We provide evidence that the 0.91 result is the best possible using a large class of proof strategies, suggesting that the communication complexity axiom is insufficient for characterizing CHSH correlations. Both results build on new insights about reliable classical computation. The first exploits our formalization of an equivalence between amplification and reliable computation, while the second follows from an upper bound on the threshold for reliable computation with formulas of noisy XOR and AND gates.