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arXiv:2008.05059

A Parallel Repetition Theorem for the GHZ Game

Justin Holmgren, Ran Raz

Published Aug 12, 2020• 3 citations
arXivHTMLPDF

Abstract

We prove that parallel repetition of the (3-player) GHZ game reduces the value of the game polynomially fast to 0. That is, the value of the GHZ game repeated in parallel $t$ times is at most $t^{-Ω(1)}$. Previously, only a bound of $\approx \frac{1}{α(t)}$, where $α$ is the inverse Ackermann function, was known. The GHZ game was recently identified by Dinur, Harsha, Venkat and Yuen as a multi-player game where all existing techniques for proving strong bounds on the value of the parallel repetition of the game fail. Indeed, to prove our result we use a completely new proof technique. Dinur, Harsha, Venkat and Yuen speculated that progress on bounding the value of the parallel repetition of the GHZ game may lead to further progress on the general question of parallel repetition of multi-player games. They suggested that the strong correlations present in the GHZ question distribution represent the "hardest instance" of the multi-player parallel repetition problem. Another motivation for studying the parallel repetition of the GHZ game comes from the field of quantum information. The GHZ game, first introduced by Greenberger, Horne and Zeilinger, is a central game in the study of quantum entanglement and has been studied in numerous works. For example, it is used for testing quantum entanglement and for device-independent quantum cryptography. In such applications a game is typically repeated to reduce the probability of error, and hence bounds on the value of the parallel repetition of the game may be useful.

DOI
10.48550/arxiv.2008.05059

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We prove that parallel repetition of the (3-player) GHZ game reduces the value of the game polynomially fast to 0. That is, the value of the GHZ game repeated in parallel ttt times is at most t−Ω(1)t^{-Ω(1)}t−Ω(1). Previously, only a bound of ≈1α(t)\approx \frac{1}{α(t)}≈α(t)1​, where ααα is the inverse Ackermann function, was known. The GHZ game was recently identified by Dinur, Harsha, Venkat and Yuen as a multi-player game where all existing techniques for proving strong bounds on the value of the parallel repetition of the game fail. Indeed, to prove our result we use a completely new proof technique. Dinur, Harsha, Venkat and Yuen speculated that progress on bounding the value of the parallel repetition of the GHZ game may lead to further progress on the general question of parallel repetition of multi-player games. They suggested that the strong correlations present in the GHZ question distribution represent the "hardest instance" of the multi-player parallel repetition problem. Another motivation for studying the parallel repetition of the GHZ game comes from the field of quantum information. The GHZ game, first introduced by Greenberger, Horne and Zeilinger, is a central game in the study of quantum entanglement and has been studied in numerous works. For example, it is used for testing quantum entanglement and for device-independent quantum cryptography. In such applications a game is typically repeated to reduce the probability of error, and hence bounds on the value of the parallel repetition of the game may be useful.
We give a new proof of the fact that the parallel repetition of the (3-player) GHZ game reduces the value of the game to zero polynomially quickly. That is, we show that the value of the n-fold GHZ game is at most n^{-Ω(1)}. This was first established by Holmgren and Raz [Holmgren and Raz, 2020]. We present a new proof of this theorem that we believe to be simpler and more direct. Unlike most previous works on parallel repetition, our proof makes no use of information theory, and relies on the use of Fourier analysis. The GHZ game [Greenberger et al., 1989] has played a foundational role in the understanding of quantum information theory, due in part to the fact that quantum strategies can win the GHZ game with probability 1. It is possible that improved parallel repetition bounds may find applications in this setting. Recently, Dinur, Harsha, Venkat, and Yuen [Dinur et al., 2017] highlighted the GHZ game as a simple three-player game, which is in some sense maximally far from the class of multi-player games whose behavior under parallel repetition is well understood. Dinur et al. conjectured that parallel repetition decreases the value of the GHZ game exponentially quickly, and speculated that progress on proving this would shed light on parallel repetition for general multi-player (multi-prover) games.
We give a new proof of the fact that the parallel repetition of the (3-player) GHZ game reduces the value of the game to zero polynomially quickly. That is, we show that the value of the nnn-fold GHZ game is at most n−Ω(1)n^{-Ω(1)}n−Ω(1). This was first established by Holmgren and Raz [HR20]. We present a new proof of this theorem that we believe to be simpler and more direct. Unlike most previous works on parallel repetition, our proof makes no use of information theory, and relies on the use of Fourier analysis. The GHZ game [GHZ89] has played a foundational role in the understanding of quantum information theory, due in part to the fact that quantum strategies can win the GHZ game with probability 1. It is possible that improved parallel repetition bounds may find applications in this setting. Recently, Dinur, Harsha, Venkat, and Yuen [DHVY17] highlighted the GHZ game as a simple three-player game, which is in some sense maximally far from the class of multi-player games whose behavior under parallel repetition is well understood. Dinur et al. conjectured that parallel repetition decreases the value of the GHZ game exponentially quickly, and speculated that progress on proving this would shed light on parallel repetition for general multi-player (multi-prover) games.
We investigate the value of parallel repetition of one-round games with any number of players k≥2k\ge 2k≥2. It has been an open question whether an analogue of Raz's Parallel Repetition Theorem holds for games with more than two players, i.e., whether the value of the repeated game decays exponentially with the number of repetitions. Verbitsky has shown, via a reduction to the density Hales-Jewett theorem, that the value of the repeated game must approach zero, as the number of repetitions increases. However, the rate of decay obtained in this way is extremely slow, and it is an open question whether the true rate is exponential as is the case for all two-player games. Exponential decay bounds are known for several special cases of multi-player games, e.g., free games and anchored games. In this work, we identify a certain expansion property of the base game and show all games with this property satisfy an exponential decay parallel repetition bound. Free games and anchored games satisfy this expansion property, and thus our parallel repetition theorem reproduces all earlier exponential-decay bounds for multiplayer games. More generally, our parallel repetition bound applies to all multiplayer games that are connected in a certain sense. We also describe a very simple game, called the GHZ game, that does not satisfy this connectivity property, and for which we do not know an exponential decay bound. We suspect that progress on bounding the value of this the parallel repetition of the GHZ game will lead to further progress on the general question.
We investigate the value of parallel repetition of one-round games with any number of players k>=2. It has been an open question whether an analogue of Raz's Parallel Repetition Theorem holds for games with more than two players, i.e., whether the value of the repeated game decays exponentially with the number of repetitions. Verbitsky has shown, via a reduction to the density Hales-Jewett theorem, that the value of the repeated game must approach zero, as the number of repetitions increases. However, the rate of decay obtained in this way is extremely slow, and it is an open question whether the true rate is exponential as is the case for all two-player games. Exponential decay bounds are known for several special cases of multi-player games, e.g., free games and anchored games. In this work, we identify a certain expansion property of the base game and show all games with this property satisfy an exponential decay parallel repetition bound. Free games and anchored games satisfy this expansion property, and thus our parallel repetition theorem reproduces all earlier exponential-decay bounds for multiplayer games. More generally, our parallel repetition bound applies to all multiplayer games that are *connected* in a certain sense. We also describe a very simple game, called the GHZ game, that does not satisfy this connectivity property, and for which we do not know an exponential decay bound. We suspect that progress on bounding the value of this the parallel repetition of the GHZ game will lead to further progress on the general question.
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 G repeated n times in parallel is at most c_G*n^{-1/4}*log(n) for a constant c_G depending on G, provided that the entangled value of G 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.
We prove that for any 3-player game G\mathcal GG, whose query distribution has the same support as the GHZ game (i.e., all x,y,z∈{0,1}x,y,z\in \{0,1\}x,y,z∈{0,1} satisfying x+y+z=0(mod2)x+y+z=0\pmod{2}x+y+z=0(mod2)), the value of the nnn-fold parallel repetition of G\mathcal GG decays exponentially fast: val(G⊗n)≤exp⁡(−nc) \text{val}(\mathcal G^{\otimes n}) \leq \exp(-n^c)val(G⊗n)≤exp(−nc) for all sufficiently large nnn, where c>0c>0c>0 is an absolute constant. We also prove a concentration bound for the parallel repetition of the GHZ game: For any constant ε>0ε>0ε>0, the probability that the players win at least a (34+ε)\left(\frac{3}{4}+ε\right)(43​+ε) fraction of the nnn coordinates is at most exp⁡(−nc)\exp(-n^c)exp(−nc), where c=c(ε)>0c=c(ε)>0c=c(ε)>0 is a constant. In both settings, our work exponentially improves upon the previous best known bounds which were only polynomially small, i.e., of the order n−Ω(1)n^{-Ω(1)}n−Ω(1). Our key technical tool is the notion of \emph{algebraic spreadness} adapted from the breakthrough work of Kelley and Meka (FOCS '23) on sets free of 3-term progressions.