17 research outputs found
A parallel repetition theorem for all entangled games
The behavior of games repeated in parallel, when played with quantumly
entangled players, has received much attention in recent years. Quantum
analogues of Raz's classical parallel repetition theorem have been proved for
many special classes of games. However, for general entangled games no parallel
repetition theorem was known. We prove that the entangled value of a two-player
game repeated times in parallel is at most for a
constant depending on , provided that the entangled value of is
less than 1. In particular, this gives the first proof that the entangled value
of a parallel repeated game must converge to 0 for all games whose entangled
value is less than 1. Central to our proof is a combination of both classical
and quantum correlated sampling.Comment: To appear in the 43rd International Colloquium on Automata,
Languages, and Programming (ICALP
Information Value of Two-Prover Games
We introduce a generalization of the standard framework for studying the difficulty of two-prover games. Specifically, we study the model where Alice and Bob are allowed to communicate (with information constraints) - in contrast to the usual two-prover game where they are not allowed to communicate after receiving their respective input. We study the trade-off between the information cost of the protocol and the achieved value of the game after the protocol.
In particular, we show the connection of this trade-off and the amortized behavior of the game (i.e. repeated value of the game).
We show that if one can win the game with at least (1 - epsilon)-probability by communicating at most epsilon bits of information,
then one can win n copies with probability at least 2^{-O(epsilon n)}. This gives an intuitive explanation why Raz\u27s counter-example to strong parallel repetition [Raz2008] (the odd cycle game) is a counter-example to strong parallel repetition - one can win the odd-cycle game on a cycle of length by communicating O(m^{-2})-bits where m is the number of vertices.
Conversely, for projection games, we show that if one can win n copies with probability larger than (1-epsilon)^n,
then one can win one copy with at least (1 - O(epsilon))-probability by communicating O(epsilon) bits of information.
By showing the equivalence between information value and amortized value, we give an alternative direction for further works in studying amortized behavior of the two-prover games.
The main technical tool is the "Chi-Squared Lemma" which bounds the information cost of the protocol in terms of Chi-Squared distance,
instead of usual divergence. This avoids the square loss from using Pinsker\u27s Inequality
Parallel Repetition of Entangled Games with Exponential Decay via the Superposed Information Cost
In a two-player game, two cooperating but non communicating players, Alice
and Bob, receive inputs taken from a probability distribution. Each of them
produces an output and they win the game if they satisfy some predicate on
their inputs/outputs. The entangled value of a game is the
maximum probability that Alice and Bob can win the game if they are allowed to
share an entangled state prior to receiving their inputs.
The -fold parallel repetition of consists of instances of
where the players receive all the inputs at the same time and produce all
the outputs at the same time. They win if they win each instance of .
In this paper we show that for any game such that , decreases exponentially in . First, for
any game on the uniform distribution, we show that , where and are the sizes of the input
and output sets. From this result, we show that for any entangled game ,
where is the input distribution of and
. This implies parallel
repetition with exponential decay as long as for
general games. To prove this parallel repetition, we introduce the concept of
\emph{Superposed Information Cost} for entangled games which is inspired from
the information cost used in communication complexity.Comment: In the first version of this paper we presented a different, stronger
Corollary 1 but due to an error in the proof we had to modify it in the
second version. This third version is a minor update. We correct some typos
and re-introduce a proof accidentally commented out in the second versio
Parallel Repetition of Free Entangled Games: Simplification and Improvements
In a two-player game, two cooperating but non communicating players, Alice
and Bob, receive inputs taken from a probability distribution. Each of them
produces an output and they win the game if they satisfy some predicate on
their inputs/outputs. The entangled value of a game is the
maximum probability that Alice and Bob can win the game if they are allowed to
share an entangled state prior to receiving their inputs.
The -fold parallel repetition of consists of instances of
where Alice and Bob receive all the inputs at the same time and must
produce all the outputs at the same time. They win if they win each
instance of . Recently, there has been a series of works showing parallel
repetition with exponential decay for projection games [DSV13], games on the
uniform distribution [CS14] and for free games, i.e. games on a product
distribution [JPY13].
This article is meant to be a follow up of [CS14], where we improve and
simplify several parts of our previous paper. Our main result is that for any
free game with value , we have where is the size of
the output set of the game. This result improves on both the results in [JPY13]
and [CS14]. The framework we use can also be extended to free projection games.
We show that for a free projection game with value
, we have .Comment: 17 pages, this paper is a follow up and supersedes our previous paper
'Parallel Repetition of Entangled Games with Exponential Decay via the
Superposed Information Cost' [CS14, arXiv:1310.7787] v2: updated GS
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