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arXiv:1410.4397

Parallel Repetition of Free Entangled Games: Simplification and Improvements

André Chailloux, Scarpa, Giannicola

Published Oct 16, 2014• 7 citations
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Abstract

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 $ω^*(G^n) \le (1 - \varepsilon)^{Ω(n)}$.

DOI
10.48550/arxiv.1410.4397

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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 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.
$ \newcommand{\vbar}{\bar{v}} $ Consider a game where a referee chooses (x,y)(x,y)(x,y) according to a publicly known distribution, sends xxx to Alice, and yyy to Bob. Without communicating with each other, Alice responds with a value aaa and Bob responds with a value bbb. Alice and Bob jointly win if a publicly known predicate Q(x,y,a,b)Q(x,y,a,b)Q(x,y,a,b) is satisfied. Assume that the maximum probability that Alice and Bob can win in such a game is v<1v<1v<1. Raz (SIAM J. Comput. 27, 1998) shows that if the game is repeated nnn times in parallel, then the probability that Alice and Bob win all games simultaneously is at most $\vbar^{\tfrac{n}{\log(s)}}$, where sss is the maximal number of possible responses from Alice and Bob in the initial game, and $\vbar<1$ is a constant depending only on vvv. In this work, we simplify Raz's proof in various ways and thus shorten it significantly. Further we study the case where Alice and Bob are not restricted to local computations and can use any strategy which does not imply communication between them.
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.
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.
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.
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 - εc)k), for some universal constant c ≥ 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 to the following approximate scenario: two isolated parties, given classical descriptions of arbitrary bipartite states |ψ〉, |φ〉 respectively such that |ψ〉 ≈ |φ〉, are able to locally generate a joint entangled state|Ψ〉 ≈ |ψ〉 ≈ |φ〉 using an initial entangled state that is independent of their inputs.