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arXiv:1112.3563

Rank-one Quantum Games

Tom Cooney, Marius Junge, Carlos Palazuelos, David Pérez-Garcı́a

Published Dec 15, 2011• 4 citations
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Abstract

In this work we study rank-one quantum games. In particular, we focus on the study of the computability of the entangled value $ω^*$. We show that the value $ω^*$ can be efficiently approximated up to a multiplicative factor of 4. We also study the behavior of $ω^*$ under the parallel repetition of rank-one quantum games, showing that it does not verify a perfect parallel repetition theorem. To obtain these results, we first connect rank-one games with the mathematical theory of operator spaces. We also reprove with these new tools essentially known results about the entangled value of rank-one games with one-way communication $ω_{qow}$. In particular, we show that $ω_{qow}$ can be computed efficiently and it satisfies a perfect parallel repetition theorem.

DOI
10.48550/arxiv.1112.3563

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In this work we study rank-one quantum games. In particular, we focus on the study of the computability of the entangled value ω∗ω^*ω∗. We show that the value ω∗ω^*ω∗ can be efficiently approximated up to a multiplicative factor of 4. We also study the behavior of ω∗ω^*ω∗ under the parallel repetition of rank-one quantum games, showing that it does not verify a perfect parallel repetition theorem. To obtain these results, we first connect rank-one games with the mathematical theory of operator spaces. We also reprove with these new tools essentially known results about the entangled value of rank-one games with one-way communication ωqowω_{qow}ωqow​. In particular, we show that ωqowω_{qow}ωqow​ can be computed efficiently and it satisfies a perfect parallel repetition theorem.
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 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 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 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.
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 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.