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arXiv:2202.06826 · ACM Symposium on Theory of Computing (STOC)

Parallel repetition for all 3-player games over binary alphabet

Uma Girish, Justin Holmgren, Kunal Mittal, Ran Raz, Wei Zhan

Published Jun 9, 2022• 6 citations
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Abstract

We prove that for every 3-player (3-prover) game, with binary questions and answers and value <1, the value of the n-fold parallel repetition of the game decays polynomially fast to 0. That is, for every such game, there exists a constant c>0, such that the value of the n-fold parallel repetition of the game is at most n−c.
DOI
10.1145/3519935.3520071

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We prove that for every 3-player (3-prover) game G\mathcal GG with value less than one, whose query distribution has the support S={(1,0,0),(0,1,0),(0,0,1)}\mathcal S = \{(1,0,0), (0,1,0), (0,0,1)\}S={(1,0,0),(0,1,0),(0,0,1)} of hamming weight one vectors, the value of the nnn-fold parallel repetition G⊗n\mathcal G^{\otimes n}G⊗n decays polynomially fast to zero; that is, there is a constant c=c(G)>0c = c(\mathcal G)>0c=c(G)>0 such that the value of the game G⊗n\mathcal G^{\otimes n}G⊗n is at most n−cn^{-c}n−c. Following the recent work of Girish, Holmgren, Mittal, Raz and Zhan (STOC 2022), our result is the missing piece that implies a similar bound for a much more general class of multiplayer games: For every\textbf{every}every 3-player game G\mathcal GG over binary questions\textit{binary questions}binary questions and arbitrary answer lengths\textit{arbitrary answer lengths}arbitrary answer lengths, with value less than 1, there is a constant c=c(G)>0c = c(\mathcal G)>0c=c(G)>0 such that the value of the game G⊗n\mathcal G^{\otimes n}G⊗n is at most n−cn^{-c}n−c. Our proof technique is new and requires many new ideas. For example, we make use of the Level-kkk inequalities from Boolean Fourier Analysis, which, to the best of our knowledge, have not been explored in this context prior to our work.
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.
In a kkk-XOR game ΨΨΨ over Fp\mathbb{F}_pFp​, a verifier samples challenges (x1,…,xk)∼μ(x_1,\ldots,x_k)\simμ(x1​,…,xk​)∼μ where μμμ is a distribution over Σ1×…×ΣkΣ_1\times\ldots\times Σ_kΣ1​×…×Σk​, sends xix_ixi​ to player iii and receives an answer fi(xi)∈Fpf_i(x_i)\in\mathbb{F}_pfi​(xi​)∈Fp​, and accepts if and only if f1(x1)+…+fk(xk)=t(x1,…,xk)f_1(x_1)+\ldots+f_k(x_k)=t(x_1,\ldots,x_k)f1​(x1​)+…+fk​(xk​)=t(x1​,…,xk​), where t ⁣:Σ1×…×Σk→Fpt\colon Σ_1\times\ldots\times Σ_k\to\mathbb{F}_pt:Σ1​×…×Σk​→Fp​ is a fixed map. We show that if ΨΨΨ has value strictly less than 111, then the value of the nnn-fold repeated game Ψ⊗nΨ^{\otimes n}Ψ⊗n is at most 2−cn2^{-cn}2−cn where c=c(Ψ)>0c = c(Ψ)>0c=c(Ψ)>0 is a fixed constant. This extends a previous result of [Bhangale-Braverman-Khot-Liu-Minzer] showing exponential decay for some 333-XOR games. Our proof uses a relaxation of the value of games studied in the quantum setting, called the commuting value of a game. We show a transformation that maps a kkk-player game ΨΨΨ with value strictly less than 111 to a 2k2k2k-player game Ψ′Ψ'Ψ′ with commuting value less than 111, such that val(Ψ⊗n)≤val((Ψ′)⊗n)\mathsf{val}(Ψ^{\otimes n})\leq \mathsf{val}((Ψ')^{\otimes n})val(Ψ⊗n)≤val((Ψ′)⊗n). We prove that val(Ψ′)<1\mathsf{val}(Ψ')<1val(Ψ′)<1 has a direct proof using nothing more than the Cauchy-Schwarz inequality, which we are able to tensorize to get that val((Ψ′)⊗n)\mathsf{val}((Ψ')^{\otimes n})val((Ψ′)⊗n) vanishes exponentially fast in nnn.
We prove that for any 3-player game G, whose query distribution has the same support as the GHZ game (i.e., all x,y,z ∈ {0,1} satisfying x+y+z = 0 (mod 2)), the value of the n-fold parallel repetition of G decays exponentially fast: val(G^{⊗ n}) ≤ exp(-n^c) for all sufficiently large n, where c > 0 is an absolute constant. We also prove a concentration bound for the parallel repetition of the GHZ game: For any constant ε > 0, the probability that the players win at least a (3/4+ε) fraction of the n coordinates is at most exp(-n^c), where c = 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)}. Our key technical tool is the notion of algebraic spreadness adapted from the breakthrough work of Kelley and Meka (FOCS '23) on sets free of 3-term progressions.
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 nnn-fold parallel repetition is exponentially small in nnn. 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.
We give a proof of the cubic parallel-repetition bound for finite two-player entangled games using quantum conditional relative entropy. For a game with entangled value 1−ε<11-\varepsilon<11−ε<1 and answer alphabets A,B\mathcal A,\mathcal BA,B, its nnn-fold repeated value is at most exp⁡[−Ω(ε3n/L)],L:=max⁡{1,log⁡(∣A∣∣B∣)}, \exp[-Ω(\varepsilon^3n/L)],\qquad L:=\max\{1,\log(|\mathcal A||\mathcal B|)\}, exp[−Ω(ε3n/L)],L:=max{1,log(∣A∣∣B∣)}, for every n≥1n\ge1n≥1. This recovers Song's cubic rate. We also prove a threshold bound: the probability of winning a fraction at least 1−ε+δ1-\varepsilon+δ1−ε+δ is at most exp⁡[−Ω(δ5n/(ε2L))]\exp[-Ω(δ^5n/(\varepsilon^2L))]exp[−Ω(δ5n/(ε2L))] for 0<δ≤ε0<δ\le\varepsilon0<δ≤ε. Our information charge is the decrease in relative entropy when a question reference is discarded. Our comparison states preserve the original conditional question law. A chain rule bounds the coordinate charges using the cost of conditioning on checked wins and retaining checked answers. Our main lemma converts small charges into nearby pure-state descriptions and local corrections, each depending only on one player's question. Together with Song's sampler, these enable approximate local preparation of the conditioned workspace.