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Consequences and Limits of Nonlocal Strategies
, 2010
"... Thispaperinvestigatesthepowersandlimitationsofquantum entanglementinthecontext of cooperative games of incomplete information. We give several examples of such nonlocal games where strategies that make use of entanglement outperform all possible classical strategies. One implication ofthese examples ..."
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Cited by 120 (20 self)
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Thispaperinvestigatesthepowersandlimitationsofquantum entanglementinthecontext of cooperative games of incomplete information. We give several examples of such nonlocal games where strategies that make use of entanglement outperform all possible classical strategies. One implication ofthese examplesis that entanglement canprofoundly affectthesoundness property of twoprover interactive proof systems. We then establish limits on the probability with which strategies making use of entanglement can win restricted types of nonlocal games. These upperbounds mayberegardedasgeneralizationsof Tsirelsontypeinequalities, which place bounds on the extent to which quantum information can allow for the violation of Bell inequalities. We also investigate the amount of entanglement required by optimal and nearly optimal quantum strategies forsome games.
Parallel Repetition of Entangled Games ∗
"... We consider oneround games between a classical verifier and two provers. 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 provers? Classically, th ..."
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Cited by 16 (3 self)
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We consider oneround games between a classical verifier and two provers. 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 provers? Classically, this question, open for many years, has 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 provers 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 twoprover oneround interactive proofs. One of the main components in our proof is an orthogonalization lemma for operators, which might be of independent interest. Twoprover games play a major role both in theoretical computer science, where they led to many breakthroughs such as the discovery of tight inapproximability results, and in quantum physics, where they first arose in the context of Bell inequalities. In such games, a referee chooses a pair of questions
Entanglementresistant twoprover interactive proof systems and nonadaptive private information retrieval systems
, 2007
"... Abstract. We show that every language in NP is recognized by a twoprover interactive proof system with the following properties. The proof system is entanglementresistant (i.e., its soundness is robust against provers who have prior shared entanglement), it has two provers and one round of interac ..."
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Cited by 16 (1 self)
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Abstract. We show that every language in NP is recognized by a twoprover interactive proof system with the following properties. The proof system is entanglementresistant (i.e., its soundness is robust against provers who have prior shared entanglement), it has two provers and one round of interaction, the provers ’ answers are single bits, and the completenesssoundness gap is constant (formally, NP ⊆ ⊕MIP ∗ 1−ε,1/2+ε[2], for any ε such that 0 < ε < 1/4). Our result is based on the “oracularizing ” property of a particular private information retrieval scheme (PIR), and it suggests that investigating related properties of other PIRs might bear further fruit. 1
Coherent state exchange in multiprover quantum interactive proof system
 Chicago Journal of Theoretical Computer Science
"... Abstract: We show that any number of parties can coherently exchange any one pure quantum state for another, without communication, given prior shared entanglement. Two applications of this fact to the study of multiprover quantum interactive proof systems are given. First, we prove that there exi ..."
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Cited by 13 (1 self)
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Abstract: We show that any number of parties can coherently exchange any one pure quantum state for another, without communication, given prior shared entanglement. Two applications of this fact to the study of multiprover quantum interactive proof systems are given. First, we prove that there exists a oneround twoprover quantum interactive proof system for which no finite amount of shared entanglement allows the provers to implement an optimal strategy. More specifically, for every fixed input string, there exists a sequence of strategies for the provers, with each strategy requiring more entanglement than the last, for which the probability for the provers to convince the verifier to accept approaches one. It is not possible, however, for the provers to convince the verifier to accept with certainty with a finite amount of shared entanglement. The second application is a simple proof that multiprover quantum interactive proofs can be transformed to have nearperfect completeness by the addition of one round of communication.
Parallel approximation of noninteractive zerosum quantum games
, 2008
"... This paper studies a simple class of zerosum games played by two competing quantum players: each player sends a mixed quantum state to a referee, who performs a joint measurement on the two states to determine the players ’ payoffs. We prove that an equilibrium point of any such game can be approxi ..."
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Cited by 13 (3 self)
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This paper studies a simple class of zerosum games played by two competing quantum players: each player sends a mixed quantum state to a referee, who performs a joint measurement on the two states to determine the players ’ payoffs. We prove that an equilibrium point of any such game can be approximated by means of an efficient parallel algorithm, which implies that oneturn quantum refereed games, wherein the referee is specified by a quantum circuit, can be simulated in polynomial space. 1
NearOptimal and Explicit Bell Inequality Violations
"... Bell inequality violations correspond to behavior of entangled quantum systems that cannot be simulated classically. We give two new twoplayer games with Bell inequality violations that are stronger, fully explicit, and arguably simpler than earlier work. The first game is based on the Hidden Match ..."
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Cited by 12 (3 self)
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Bell inequality violations correspond to behavior of entangled quantum systems that cannot be simulated classically. We give two new twoplayer games with Bell inequality violations that are stronger, fully explicit, and arguably simpler than earlier work. The first game is based on the Hidden Matching problem of quantum communication complexity, introduced by BarYossef, Jayram, and Kerenidis. This game can be won with probability 1 by a quantum strategy using a maximally entangled state with local dimension n (e.g., log n EPRpairs), while we show that the winning probability of any classical strategy differs from 1 2 by at most O(log n/ √ n). The second game is based on the integrality gap for Unique Games by Khot and Vishnoi and the quantum rounding procedure of Kempe, Regev, and Toner. Here ndimensional entanglement allows to win the game with probability 1/(logn) 2, while the best winning probability without entanglement is 1/n. This nearlinear ratio (“Bell inequality violation”) is nearoptimal, both in terms of the local dimension of the entangled state, and in terms of the number of possible outputs of the two players.
Grothendiecktype inequalities in combinatorial optimization
 COMM. PURE APPL. MATH
, 2011
"... We survey connections of the Grothendieck inequality and its variants to combinatorial optimization and computational complexity. ..."
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Cited by 9 (3 self)
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We survey connections of the Grothendieck inequality and its variants to combinatorial optimization and computational complexity.
Why philosophers should care about computational complexity
 In Computability: Gödel, Turing, Church, and beyond (eds
, 2012
"... One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed casethat onewouldbe wrong. In particular, I arguethat computational complexity theory—the field that ..."
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Cited by 9 (0 self)
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One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed casethat onewouldbe wrong. In particular, I arguethat computational complexity theory—the field that studies the resources (such as time, space, and randomness) needed to solve computational problems—leads to new perspectives on the nature of mathematical knowledge, the strong AI debate, computationalism, the problem of logical omniscience, Hume’s problem of induction, Goodman’s grue riddle, the foundations of quantum mechanics, economic rationality, closed timelike curves, and several other topics of philosophical interest. I end by discussing