We use a recently discovered constrained de Finetti reduction (aka "Post-Selection Lemma") to study the parallel repetition of multi-player non-local games under nosignalling strategies. Since the technique allows us to reduce general strategies to independent plays, we obtain parallel repetition (corresponding to winning all rounds) in the same way as exponential concentration of the probability to win a fraction larger than the value of the game. Our proof technique leads us naturally to a relaxation of no-signalling (NS) strategies, which we dub sub-no-signalling (SNOS). While for two players the two concepts coincide, they differ for three or more players. Our results are most complete and satisfying for arbitrary number of sub-no-signalling players, where we get universal parallel repetition and concentration for any game, while the no-signalling case is obtained as a corollary, but only for games with "full support". A multi-player non-local game is played between cooperating but non-communicating players. Each player receives an input from some input alphabet and has to produce an output in some output alphabet. The common goal of the players is to satisfy some pre-defined predicate on their inputs and outputs. For that, they may agree on a strategy before the game starts, but are then not allowed to communicate anymore. Such games are especially relevant in theoretical physics in the context of the foundations of quantum mechanics and quantum information, and in computer science where they arise in multi-prover interactive proof systems. Indeed, they may provide an intuitive and quantitative understanding of the role played by various degrees of correlations in global systems which are composed of several local subsystems. These games also arise in complexity theory, under the formulation of multi-provers with some shared resources producing a protocol that should convince a referee, or in cryptography as attacks from malicious parties having a more or less restricted physical power. The value of a game is the maximum winning probability of the players, over all allowed joint strategies, using possibly some prescribed correlation resource such as shared randomness, quantum entanglement or no-signalling correlations. It has been a subject of considerable study how the availability of different resources affects the values of certain games [3, 4, 12, 32, 36] . In this context, a natural question is how the value of a game behaves when n independent instances of the game are played simultaneously, i.e. each player gets n independent inputs and has to provide n outputs such that each game instance is won (or a large fraction of them). This
An interesting fact about the latter work REF is the use of a notion of sub-no-signalling strategies which seems related to our notion of quantum/classical substrategies.
Parallel repetition and concentration for (sub-)no-signalling games via a flexible constrained de Finetti reduction
Chicago Journal of Theoretical Computer Science, Vol. 2016, No. 11 (2016)
Mathematics Computer Science Physics