Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1411.1397

Strong parallel repetition for free entangled games, with any number of players

Kai-Min Chung, Xiaodi Wu, Henry Yuen

Published Nov 5, 2014• 1 citation
arXivHTMLPDF

Abstract

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 $G$ be a $k$-player free game, with entangled value $\mathrm{val}^*(G) = 1 - ε$. We show that the entangled value of the $n$-fold repetition of $G$, $\mathrm{val}^*(G^{\otimes n})$, is at most $(1 - ε)^{Ω(n/k^2)}$. In the traditional setting of $k=2$ players, our parallel repetition theorem is optimal in terms of its dependence on $ε$ and $n$. For an arbitrary number of players, our result is nearly optimal: for all $k$, we exhibit a $k$-player free game $G$ and $n > 1$ such that $\mathrm{val}^*(G^{\otimes n}) \geq \mathrm{val}^*(G)^{n/k}$. Hence, exponent of the repeated game value cannot be improved beyond $Ω(n/k)$. 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 $1 - ε$, rather than $1 - ε^2$; and (3) there is no dependence of the repeated game value on the input and output alphabets of $G$. 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.

DOI
10.48550/arxiv.1411.1397

Related papers

Parallel Repetition for Entangled k-player Games via Fast Quantum Search

Kai-Min Chung, Xiaodi Wu, Henry Yuen

We present two parallel repetition theorems for the entangled value of multi-player, one-round free games (games where the inputs come from a product distribution). Our first theorem shows that for a $k$-player free game $G$ with entangled value $\mathrm{val}^*(G) = 1 - ε$, the $n$-fold repetition of $G$ has entangled value $\mathrm{val}^*(G^{\otimes n})$ at most $(1 - ε^{3/2})^{Ω(n/sk^4)}$, where $s$ is the answer length of any player. In contrast, the best known parallel repetition theorem for the classical value of two-player free games is $\mathrm{val}(G^{\otimes n}) \leq (1 - ε^2)^{Ω(n/s)}$, due to Barak, et al. (RANDOM 2009). This suggests the possibility of a separation between the behavior of entangled and classical free games under parallel repetition. Our second theorem handles the broader class of free games $G$ where the players can output (possibly entangled) quantum states. For such games, the repeated entangled value is upper bounded by $(1 - ε^2)^{Ω(n/sk^2)}$. We also show that the dependence of the exponent on $k$ is necessary: we exhibit a $k$-player free game $G$ and $n \geq 1$ such that $\mathrm{val}^*(G^{\otimes n}) \geq \mathrm{val}^*(G)^{n/k}$. Our analysis exploits the novel connection between communication protocols and quantum parallel repetition, first explored by Chailloux and Scarpa (ICALP 2014). We demonstrate that better communication protocols yield better parallel repetition theorems: our first theorem crucially uses a quantum search protocol by Aaronson and Ambainis, which gives a quadratic speed-up for distributed search problems. Finally, our results apply to a broader class of games than were previously considered before; in particular, we obtain the first parallel repetition theorem for entangled games involving more than two players, and for games involving quantum outputs.

arXiv:1501.00033•Leibniz international proceedings in informatics•Dec 26, 2014•3 citations

Parallel Repetition of Free Entangled Games: Simplification and Improvements

André Chailloux, Scarpa, Giannicola

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 $ω^*(G)$ of a game $G$ 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 $n$-fold parallel repetition $G^n$ of $G$ consists of $n$ instances of $G$ where Alice and Bob receive all the inputs at the same time and must produce all the outputs at the same time. They win $G^n$ if they win each instance of $G$. 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 $G$ with value $ω^*(G)=1-\varepsilon$, we have $ω^*(G^n) \le (1 - \varepsilon^2)^{Ω(\frac{n}{\log(l)})}$ where $l$ 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 $G$ with value $ω^*(G)=1-\varepsilon$, we have $ω^*(G^n) \le (1 - \varepsilon)^{Ω(n)}$.

arXiv:1410.4397•Oct 16, 2014•7 citations

On the Parallel Repetition of Multi-Player Games: The No-Signaling Case

Harry Buhrman, Serge Fehr, Christian Schaffner

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 $n$-fold parallel repetition is exponentially small in $n$. 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.

arXiv:1312.7455•Leibniz international proceedings in informatics•Dec 28, 2013•5 citations

A Parallel Repetition Theorem for Entangled Two-Player One-Round Games under Product Distributions

Rahul Jain, Attila Pereszlényi, Penghui Yao

We show a parallel repetition theorem for the entangled value ω*(G) of any two-player one-round game G where the questions (x, y) ∈ X × Y to Alice and Bob are drawn from a product distribution on X × Y. We show that for the k-fold product Gkof the game G (which represents the game G played in parallel k times independently) ω*(Gk) = (1 - (1 - ω*(G))3)Ω(k/Iog(|A|·|B|)where A and B represent the sets from which the answers of Alice and Bob are drawn. The arguments we use are information theoretic and are broadly on similar lines as that of Raz [1] and Holenstein [2] for classical games. The additional quantum ingredients we need, to deal with entangled games, are inspired by the work of Jain, Radhakrishnan, and Sen [3], where quantum information theoretic arguments were used to achieve message compression in quantum communication protocols.

arXiv:1311.6309•IEEE Conference on Computational Complexity (CCC)•Jun 1, 2014•34 citations

A Parallel Repetition Theorem for All Entangled Games

Henry Yuen

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.

arXiv:1604.04340•Leibniz international proceedings in informatics•Jan 1, 2016•12 citations

A parallel repetition theorem for entangled projection games

Irit Dinur, David Steurer, Thomas Vidick

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-eps < 1, the value of the $k$-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 $k$ 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.

arXiv:1310.4113•Oct 15, 2013•5 citations
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.
We present two parallel repetition theorems for the entangled value of multi-player, one-round free games (games where the inputs come from a product distribution). Our first theorem shows that for a kkk-player free game GGG with entangled value val∗(G)=1−ε\mathrm{val}^*(G) = 1 - εval∗(G)=1−ε, the nnn-fold repetition of GGG has entangled value val∗(G⊗n)\mathrm{val}^*(G^{\otimes n})val∗(G⊗n) at most (1−ε3/2)Ω(n/sk4)(1 - ε^{3/2})^{Ω(n/sk^4)}(1−ε3/2)Ω(n/sk4), where sss is the answer length of any player. In contrast, the best known parallel repetition theorem for the classical value of two-player free games is val(G⊗n)≤(1−ε2)Ω(n/s)\mathrm{val}(G^{\otimes n}) \leq (1 - ε^2)^{Ω(n/s)}val(G⊗n)≤(1−ε2)Ω(n/s), due to Barak, et al. (RANDOM 2009). This suggests the possibility of a separation between the behavior of entangled and classical free games under parallel repetition. Our second theorem handles the broader class of free games GGG where the players can output (possibly entangled) quantum states. For such games, the repeated entangled value is upper bounded by (1−ε2)Ω(n/sk2)(1 - ε^2)^{Ω(n/sk^2)}(1−ε2)Ω(n/sk2). We also show that the dependence of the exponent on kkk is necessary: we exhibit a kkk-player free game GGG and n≥1n \geq 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. Our analysis exploits the novel connection between communication protocols and quantum parallel repetition, first explored by Chailloux and Scarpa (ICALP 2014). We demonstrate that better communication protocols yield better parallel repetition theorems: our first theorem crucially uses a quantum search protocol by Aaronson and Ambainis, which gives a quadratic speed-up for distributed search problems. Finally, our results apply to a broader class of games than were previously considered before; in particular, we obtain the first parallel repetition theorem for entangled games involving more than two players, and for games involving quantum outputs.
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 ω∗(G)ω^*(G)ω∗(G) of a game GGG 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 nnn-fold parallel repetition GnG^nGn of GGG consists of nnn instances of GGG where Alice and Bob receive all the inputs at the same time and must produce all the outputs at the same time. They win GnG^nGn if they win each instance of GGG. 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 GGG with value ω∗(G)=1−εω^*(G)=1-\varepsilonω∗(G)=1−ε, we have ω∗(Gn)≤(1−ε2)Ω(nlog⁡(l))ω^*(G^n) \le (1 - \varepsilon^2)^{Ω(\frac{n}{\log(l)})}ω∗(Gn)≤(1−ε2)Ω(log(l)n​) where lll 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 GGG with value ω∗(G)=1−εω^*(G)=1-\varepsilonω∗(G)=1−ε, we have ω∗(Gn)≤(1−ε)Ω(n)ω^*(G^n) \le (1 - \varepsilon)^{Ω(n)}ω∗(Gn)≤(1−ε)Ω(n).
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 show a parallel repetition theorem for the entangled value ω*(G) of any two-player one-round game G where the questions (x, y) ∈ X × Y to Alice and Bob are drawn from a product distribution on X × Y. We show that for the k-fold product Gkof the game G (which represents the game G played in parallel k times independently) ω*(Gk) = (1 - (1 - ω*(G))3)Ω(k/Iog(|A|·|B|)where A and B represent the sets from which the answers of Alice and Bob are drawn. The arguments we use are information theoretic and are broadly on similar lines as that of Raz [1] and Holenstein [2] for classical games. The additional quantum ingredients we need, to deal with entangled games, are inspired by the work of Jain, Radhakrishnan, and Sen [3], where quantum information theoretic arguments were used to achieve message compression in quantum communication protocols.
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 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.