What Can Be Computed without Communications?

Abstract

This paper addresses the following 2-player problem. Alice (resp., Bob) receives a boolean x (resp., y) as input, and must return a boolean a (resp., b) as output. A game between Alice and Bob is defined by a pair (δ,f) of boolean functions. The objective of Alice and Bob playing game (δ,f) is, for every inputs x and y, to output values a and b, respectively, satisfying δ(a,b) = f(x,y), in absence of any communication between the two players.It is known that, for xor-games, that is, games equivalent, up to individual reversible transformations, to a game (δ,f) with δ(a,b) = a ⊕ b, the ability for the players to use entangled quantum bits (qbits) helps: there exist a distributed protocol for the chsh game, using quantum correlations, for which the probability that the two players produce a successful output is higher than the maximum probability of success of any classical distributed protocol for that game, even when using shared randomness.

In this paper, we show that, apart from xor-games, quantum correlations does not help, in the sense that, for every such game, there exists a classical protocol (using shared randomness) whose probability of success is at least as large as the one of any protocol using quantum correlations. This result holds for both worst case and average case analysis. It is achieved by considering a model stronger than quantum correlations, the non-signaling model, for which we show that, if the game is not an xor-game, then shared randomness is a sufficient resource for the design of optimal protocols. These results provide an invitation to revisit the theory of distributed checking, a.k.a. distributed verification. Indeed, the literature dealing with this theory is mostly focusing on decision functions δ equivalent to the and-operator. This paper demonstrates that such a decision function may not well be suited for taking benefit of the computational power of quantum correlations.