A parallel repetition theorem for entangled two-player one-round games under product distributions

Jain, Rahul; Pereszlényi, Attila; Yao, Penghui

doi:10.48550/arxiv.1311.6309

Cited by 2 publications

(2 citation statements)

References 16 publications

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“…All currently known results deal with limited classes of two-player games and no general proof is known. The latest results are given in [16][17][18], where different assumptions on the probability distribution over the questions of the game are considered.…”

Section: Related Workmentioning

confidence: 99%

Non-Signaling Parallel Repetition Using de Finetti Reductions

Arnon-Friedman

Renner

Vidick

2016

IEEE Trans. Inform. Theory

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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 G n (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 nβ 2 , 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.

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Section: Related Workmentioning

confidence: 99%

Non-Signaling Parallel Repetition Using de Finetti Reductions

Arnon-Friedman

Renner

Vidick

2016

IEEE Trans. Inform. Theory

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“…Very recently Jain et al [JPY13] extended the result to arbitrary product distributions on the questions, while also removing the dependence on the number of questions: they obtained the bound…”

Section: Additional Related Workmentioning

confidence: 99%

A parallel repetition theorem for entangled projection games

2015

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We study the behavior of the entangled value of two-player one-round projection games under parallel repetition. We show that for any projection game G of entangled value 1 − ε < 1, the value of the k-fold repetition of G goes to zero as O((1 − ε c ) k ), for some universal constant c ≥ 1. If furthermore the constraint graph of G is expanding we obtain the optimal c = 1. Previously exponential decay of the entangled value under parallel repetition was only known for the case of XOR and unique games. To prove the theorem we extend an analytical framework introduced by Dinur and Steurer for the study of the classical value of projection games under parallel repetition. Our proof, as theirs, relies on the introduction of a simple relaxation of the entangled value that is perfectly multiplicative. The main technical component of the proof consists in showing that the relaxed value remains tightly connected to the entangled value, thereby establishing the parallel repetition theorem. More generally, we obtain results on the behavior of the entangled value under products of arbitrary (not necessarily identical) projection games.Relating our relaxed value to the entangled value is done by giving an algorithm for converting a relaxed variant of quantum strategies that we call "vector quantum strategy" to a quantum strategy. The algorithm is considerably simpler in case the bipartite distribution of questions in the game has good expansion properties. When this is not the case, the algorithm relies on a quantum analogue of Holenstein's correlated sampling lemma which may be of independent interest. Our "quantum correlated sampling lemma" generalizes results of van Dam and Hayden on universal embezzlement to the following approximate scenario: two non-communicating parties, given classical descriptions of bipartite states |ψ , |ϕ respectively such that |ψ ≈ |ϕ , are able to locally generate a joint entangled state |Ψ ≈ |ψ ≈ |ϕ using an initial entangled state that is independent of their inputs.

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A parallel repetition theorem for entangled two-player one-round games under product distributions

Cited by 2 publications

References 16 publications

Non-Signaling Parallel Repetition Using de Finetti Reductions

Non-Signaling Parallel Repetition Using de Finetti Reductions

A parallel repetition theorem for entangled projection games

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