Local simultaneous state discrimination (LSSD) is a recently introduced problem in quantum information processing. Its classical version is a non-local game played by non-communicating players against a referee. Based on a known probability distribution, the referee generates one input for each of the players and keeps one secret value. The players have to guess the referee's value and win if they all do so. For this game, we investigate the advantage of no-signalling strategies over classical ones. We show numerically that for three players and binary values, no-signalling strategies cannot provide any improvement over classical ones. For a certain LSSD game based on a binary symmetric channel, we show that no-signalling strategies are strictly better when multiple simultaneous instances of the game are played. Good classical strategies for this game can be defined by codes, and good no-signalling strategies by list-decoding schemes. We expand this example game to a class of games defined by an arbitrary channel, and extend the idea of using codes and list decoding to define strategies for multiple simultaneous instances of these games. Finally, we give an expression for the limit of the exponent of the classical winning probability, and show that no-signalling strategies based on list-decoding schemes achieve this limit.
Local simultaneous state discrimination (LSSD) is a recently introduced problem in quantum information processing. Its classical version is a non-local game played by non-communicating players against a referee. Based on a known probability distribution, the referee generates one input for each of the players and keeps one secret value. The players have to guess the referee's value and win if they all do so. For this game, we investigate the advantage of no-signalling strategies over classical ones. We show numerically that for three players and binary values, no-signalling strategies cannot provide any improvement over classical ones. For a certain LSSD game based on a binary symmetric channel, we show that no-signalling strategies are strictly better when multiple simultaneous instances of the game are played. Good classical strategies for this game can be defined by codes, and good no-signalling strategies by list-decoding schemes. We expand this example game to a class of games defined by an arbitrary channel, and extend the idea of using codes and list decoding to define strategies for multiple simultaneous instances of these games. Finally, we give an expression for the limit of the exponent of the classical winning probability, and show that no-signalling strategies based on list-decoding schemes achieve this limit.
Quantum state discrimination is one of the most fundamental problems in quantum information theory, with applications ranging from channel coding to metrology and cryptography. In this work, we introduce a variant of this task: local simultaneous state discrimination (LSSD). While previously studied distributed variants of state discrimination allowed some communication between the parties to come up with a joint answer, in LSSD they cannot communicate and have to simultaneously answer correctly. We illustrate by multiple examples that this problem significantly differs from single-party state discrimination, even when the states are completely classical. We show that an additional entangled resource can increase the optimal success probability in LSSD, and stronger-than-quantum no-signaling resources can allow for an even higher success probability. We also show that finding the optimal strategy in (classical) three-party LSSD is NP-hard. Furthermore, we provide an example of symmetric LSSD for which the optimal strategy is not symmetric, and prove a sufficient condition for the existence of an optimal symmetric strategy. While interesting in its own right, the LSSD problem also arises in quantum cryptography. In particular, we explore the connections between the problem of unclonable encryption and LSSD. We give an explicit cloning-indistinguishable attack that succeeds with probability $1/2+\ensuremath{\mu}/16$ where $\ensuremath{\mu}$ is related to the largest eigenvalue of the resulting quantum ciphertext states.
We consider the natural extension of two-player nonlocal games to an arbitrary number of players. An important question for such nonlocal games is their behavior under parallel repetition. For two-player nonlocal games, it is known that both the classical and the non-signaling value of any game converges to zero exponentially fast under parallel repetition, given that the game is non-trivial to start with (i.e., has classical/non-signaling value <1). Very recent results show similar behavior for the quantum value of a two-player game under parallel repetition. For nonlocal games with three or more players, very little is known up to present on their behavior under parallel repetition; this is true for the classical, the quantum and the non-signaling value. In this work, we show a parallel repetition theorem for the non-signaling value of a large class of multi-player games, for an arbitrary number of players. Our result applies to all multi-player games for which all possible combinations of questions have positive probability; this class in particular includes all free games, in which the questions to the players are chosen independently. Specifically, we prove that if the original game has a non-signaling value smaller than 1, then the non-signaling value of the n-fold parallel repetition is exponentially small in n. Our parallel repetition theorem for multi-player games is weaker than the known parallel repetition results for two-player games in that the rate at which the non-signaling value of the game decreases not only depends on the non-signaling value of the original game (and the number of possible responses), but on the complete description of the game. Nevertheless, we feel that our result is a first step towards a better understanding of the parallel repetition of nonlocal games with more than two players.
$ \newcommand{\vbar}{\bar{v}} $ Consider a game where a referee chooses (x,y) according to a publicly known distribution, sends x to Alice, and y to Bob. Without communicating with each other, Alice responds with a value a and Bob responds with a value b. Alice and Bob jointly win if a publicly known predicate Q(x,y,a,b) is satisfied. Assume that the maximum probability that Alice and Bob can win in such a game is v<1. Raz (SIAM J. Comput. 27, 1998) shows that if the game is repeated n times in parallel, then the probability that Alice and Bob win all games simultaneously is at most $\vbar^{\tfrac{n}{\log(s)}}$, where s is the maximal number of possible responses from Alice and Bob in the initial game, and $\vbar<1$ is a constant depending only on v. In this work, we simplify Raz's proof in various ways and thus shorten it significantly. Further we study the case where Alice and Bob are not restricted to local computations and can use any strategy which does not imply communication between them.
In the context of multiplayer games, the parallel repetition problem can be phrased as follows: given a game G with optimal winning probability 1 - α and its repeated version Gn(in which n games are played together, in parallel), can the players use strategies that are substantially better than ones in which each game is played independently? This question is relevant in physics for the study of correlations and plays an important role in computer science in the context of complexity and cryptography. In this paper, the case of multiplayer non-signaling games is considered, i.e., the only restriction on the players is that they are not allowed to communicate during the game. For complete-support games (games where all possible combinations of questions have non-zero probability to be asked) with any number of players, we prove a threshold theorem stating that the probability that non-signaling players win more than a fraction 1-α+β of the n games is exponentially small in nβ2for every 0 ≤ β ≤ α. For games with incomplete support, we derive a similar statement for a slightly modified form of repetition. The result is proved using a new technique based on a recent de Finetti theorem, which allows us to avoid central technical difficulties that arise in standard proofs of parallel repetition theorems.
In the context of multiplayer games, the parallel repetition problem can be phrased as follows: given a game G with optimal winning probability 1−α and its repeated version Gn (in which n games are played together, in parallel), can the players use strategies that are substantially better than ones in which each game is played independently? This question is relevant in physics for the study of correlations and plays an important role in computer science in the context of complexity and cryptography. In this work the case of multiplayer non-signalling games is considered, i.e., the only restriction on the players is that they are not allowed to communicate during the game. For complete-support games (games where all possible combinations of questions have non-zero probability to be asked) with any number of players we prove a threshold theorem stating that the probability that non-signalling players win more than a fraction 1−α+β of the n games is exponentially small in nβ2, for every 0≤β≤α. For games with incomplete support we derive a similar statement, for a slightly modified form of repetition. The result is proved using a new technique, based on a recent de Finetti theorem, which allows us to avoid central technical difficulties that arise in standard proofs of parallel repetition theorems.
Consider a game where a refereed chooses (x,y) according to a publiclyknown distribution PXY, sends x to Alice, and y to Bob. Withoutcommunicating with each other, Alice responds with a value "a" and Bobresponds with a value "b". Alice and Bob jointly win if a publiclyknown predicate Q(x,y,a,b) holds. Let such a game be given and assume that the maximum probabilitythat Alice and Bob can win is v<1. Raz (SIAM J. Comput. 27, 1998)shows that if the game is repeated n times in parallel, then the probability that Alice and Bob win all games simultaneously is at most v'(n/log(s)), where s is the maximal number of possible responses from Alice and Bob in the initial game, and v' is a constant depending only on v. In this work, we simplify Raz's proof in various ways and thus shorten it significantly. Further we study the case where Alice and Bob are not restricted to local computations and can use any strategy which does not imply communication among them.