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arXiv:2610.21410

Parallel Repetition for Entangled Games via Conditional Relative Entropy

Rahul Jain

quant-phcs.CCcs.ITSubmitted Oct 15, 2026• Announced Oct 16, 2026• 0 citations
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Abstract

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 $1-\varepsilon<1$ and answer alphabets $\mathcal A,\mathcal B$, its $n$-fold repeated value is at most \[ \exp[-Ω(\varepsilon^3n/L)],\qquad L:=\max\{1,\log(|\mathcal A||\mathcal B|)\}, \] for every $n\ge1$. This recovers Song's cubic rate. We also prove a threshold bound: the probability of winning a fraction at least $1-\varepsilon+δ$ is at most $\exp[-Ω(δ^5n/(\varepsilon^2L))]$ for $0<δ\le\varepsilon$. 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.

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35 pages, version 1

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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(\varepsilon^3n/L)],\qquad
L:=\max{1,\log(|\mathcal A||\mathcal 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.
For every finite two-player game with entangled value v<1, we prove exponential decay for the probability of winning at least a v+\delta fraction of k independent repetitions, uniformly over all finite-dimensional joint strategies. The result allows arbitrary correlated question distributions and holds for every 0\lt \delta\lt 1-v and k\ge1. Its universal rate is proportional to \delta^5/(1+\log(|\mathcal A||\mathcal B|)), where \mathcal A,\mathcal B are the answer alphabets. For each fixed question distribution, we also obtain an explicit cubic rate in \delta.
We present a strong parallel repetition theorem for the entangled value of multi-player, one-round free games (games where the inputs come from a product distribution). Our result is the first parallel repetition theorem for entangled games involving more than two players. Furthermore, our theorem applies to games where the players are allowed to output (possibly entangled) quantum states as answers. More specifically, let GGG be a kkk-player free game, with entangled value val∗(G)=1−ε\mathrm{val}^*(G) = 1 - εval∗(G)=1−ε. We show that the entangled value of the nnn-fold repetition of GGG, val∗(G⊗n)\mathrm{val}^*(G^{\otimes n})val∗(G⊗n), is at most (1−ε)Ω(n/k2)(1 - ε)^{Ω(n/k^2)}(1−ε)Ω(n/k2). In the traditional setting of k=2k=2k=2 players, our parallel repetition theorem is optimal in terms of its dependence on εεε and nnn. For an arbitrary number of players, our result is nearly optimal: for all kkk, we exhibit a kkk-player free game GGG and n>1n > 1n>1 such that val∗(G⊗n)≥val∗(G)n/k\mathrm{val}^*(G^{\otimes n}) \geq \mathrm{val}^*(G)^{n/k}val∗(G⊗n)≥val∗(G)n/k. Hence, exponent of the repeated game value cannot be improved beyond Ω(n/k)Ω(n/k)Ω(n/k). Our parallel repetition theorem improves on the prior results of [Jain, et al. 2014] and [Chailloux, Scarpa 2014] in a number of ways: (1) our theorem applies to a larger class of games (arbitrary number of players, quantum outputs); (2) we demonstrate that strong parallel repetition holds for the entangled value of free games: i.e., the base of the repeated game value is 1−ε1 - ε1−ε, rather than 1−ε21 - ε^21−ε2; and (3) there is no dependence of the repeated game value on the input and output alphabets of GGG. In contrast, it is known that the repeated game value of classical free games must depend on the output size. Thus our results demonstrate a seperation between the behavior of entangled games and classical games.
A two-player game is played by cooperating players who are not allowed to communicate. A referee asks the players questions sampled from some known distribution and decides whether they win or not based on a known predicate of the questions and the players' answers. The parallel repetition of the game is the game in which the referee samples n independent pairs of questions and sends the corresponding questions to the players simultaneously. If the players cannot win the original game with probability better than (1−ϵ)(1-\epsilon)(1−ϵ), what's the best they can do in the repeated game? We improve earlier results of [R. Raz, SIAM J. Comput., 27 (1998), pp. 763–803] and [T. Holenstein, Theory Comput., 5 (2009), pp. 141–172], who showed that the players cannot win all copies in the repeated game with probability better than (1−ϵ/2)Ω(nϵ2/c)(1-\epsilon/2)^{\Omega(n\epsilon^2/c)}(1−ϵ/2)Ω(nϵ2/c) (here c is the length of the answers in the game), in the following ways: (i) We show that the probability of winning all copies is (1−ϵ/2)Ω(ϵn)(1-\epsilon/2)^{\Omega(\epsilon n)}(1−ϵ/2)Ω(ϵn) as long as the game is a “projection game,” the type of game most commonly used in hardness of approximation results. (ii) We prove a concentration bound for parallel repetition (of general games) showing that for any constant 0<δ<ϵ0<\delta <\epsilon0<δ<ϵ, the probability that the players win a (1−ϵ+δ)(1-\epsilon+\delta)(1−ϵ+δ) fraction of the games in the parallel repetition is at most exp⁡(−Ωϵ(δ3n/c))\exp\left(-\Omega_{\epsilon}(\delta^3 n/c)\right)exp(−Ωϵ​(δ3n/c)) (here the constant may depend on ϵ\epsilonϵ); our result has applications to testing Bell inequalities, since it implies that the parallel repetition of the CHSH game can be used to get an experiment that has a very large classical versus quantum gap. Our first bound is independent of the answer length and has a better dependence on ϵ\epsilonϵ. By the recent work of Raz [Proceedings of the 494949th Annual IEEE Symposium on Foundations of Computer Science, 2008, pp. 369–373], this bound is tight. Our bound gives a generic way to improve the soundness of a probabilistically checkable proof (PCP), in a way that is independent of the answer length of the PCP. Using it, for every k, one can convert any q query PCP with answer length c, size scscsc, and soundness (1−ϵ)(1-\epsilon)(1−ϵ) into a two-query PCP with answer length ckckck, size O(ck(2s)k)O(ck (2s)^k)O(ck(2s)k), and soundness (1−ϵ/2q)Ω(ϵk/q)(1-\epsilon/2q)^{\Omega(\epsilon k/q)}(1−ϵ/2q)Ω(ϵk/q). Another consequence of our bound is that the unique games conjecture of Khot [Proceedings of the 343434th Annual ACM Symposium on Theory of Computing, 2002, pp. 767–775] can now be shown to be equivalent to the following a priori weaker conjecture: There is an unbounded increasing function f:R+→R+f:\mathbb{R}^+\rightarrow \mathbb{R}^+f:R+→R+ such that for every ϵ>0\epsilon> 0ϵ>0, there exists an alphabet size M(ϵ)M(\epsilon)M(ϵ) for which it is NP-hard to distinguish a unique game with alphabet size M in which a (1−ϵ2)(1-\epsilon^2)(1−ϵ2) fraction of the constraints can be satisfied from one in which a (1−ϵf(1/ϵ))(1-\epsilon f(1/\epsilon))(1−ϵf(1/ϵ)) fraction of the constraints can be satisfied.
This thesis contributes to the study of parallel repetition theorems and concentration bounds for nonlocal games and quantum interactive proofs. We make the following contributions: \n \n- A lemma that is useful for converting parallel repetition theorems (bounds on the probability of winning all instances of a nonlocal game which is being repeated in parallel) into concentration bounds (bounds on winning a certain fraction of the instances). \n \n- Exponentially-decaying concentration bounds for two-player games on the uniform distribution and k-player free games, against quantum strategies. \n \n- A proof that given a quantum interactive proof system with parameters α (the probability with which the verifier can be convinced to accept when they should accept) and β (the soundness error), as long as α > β, both the soundness error and completeness error can be reduced exponentially by repeating the protocol in parallel and requiring an (α + β)/2 fraction of the repetitions to be won. Our result requires a log-factor more repetitions than are necessary in the classical case.
We consider one-round games between a classical referee and two players. One of the main questions in this area is the parallel repetition question: Is there a way to decrease the maximum winning probability of a game without increasing the number of rounds or the number of players? Classically, efforts to resolve this question, open for many years, have culminated in Raz's celebrated parallel repetition theorem on one hand, and in efficient product testers for PCPs on the other. In the case where players share entanglement, the only previously known results are for special cases of games, and are based on techniques that seem inherently limited. Here we show for the first time that the maximum success probability of entangled games can be reduced through parallel repetition, provided it was not initially 1. Our proof is inspired by a seminal result of Feige and Kilian in the context of classical two-prover one-round interactive proofs. One of the main components in our proof is an orthogonalization lemma for operators, which might be of independent interest.
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 GGG of entangled value 1-eps < 1, the value of the kkk-fold repetition of G goes to zero as O((1-eps^c)^k), for some universal constant c\geq 1. Previously parallel repetition with an exponential decay in kkk was only known for the case of XOR and unique games. To prove the theorem we extend an analytical framework recently 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, rounding 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.