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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.
Quantum ArthurMerlin games
 Computational Complexity
"... Abstract This paper studies quantum ArthurMerlin games, whichare a restricted form of quantum interactive proof system in which the verifier's messages are given by unbiased coinflips. The following results are proved. ffl For onemessage quantum ArthurMerlin games, whichcorrespond to the co ..."
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Cited by 71 (4 self)
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Abstract This paper studies quantum ArthurMerlin games, whichare a restricted form of quantum interactive proof system in which the verifier's messages are given by unbiased coinflips. The following results are proved. ffl For onemessage quantum ArthurMerlin games, whichcorrespond to the complexity class QMA, completeness and soundness errors can be reduced exponentially without increasing the length of Merlin's message. Previous constructions for reducing error required a polynomial increase in the length of Merlin's message. Applications of this fact include a proof that logarithmic length quantum certificates yield no increase in powerover BQP and a simple proof that QMA ` PP. ffl In the case of three or more messages, quantum ArthurMerlin games are equivalent in power to ordinary quantum interactive proof systems. In fact, for any languagehaving a quantum interactive proof system there exists a threemessage quantum ArthurMerlin game in whichArthur's only message consists of just a single coinflip that achieves perfect completeness and soundness errorexponentially close to 1/2. ffl Any language having a twomessage quantum ArthurMerlin game is contained in BP \Delta PP. This gives somesuggestion that three messages are stronger than two in
Quantum MerlinArthur Proof Systems: Are Multiple Merlins More Helpful to Arthur?
, 2008
"... This paper introduces quantum “multipleMerlin”Arthur proof systems in which Arthur receives multiple quantum proofs that are unentangled with each other. Although classical multiproof systems are obviously equivalent to classical singleproof systems (i.e., usual MerlinArthur proof systems), it ..."
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Cited by 41 (8 self)
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This paper introduces quantum “multipleMerlin”Arthur proof systems in which Arthur receives multiple quantum proofs that are unentangled with each other. Although classical multiproof systems are obviously equivalent to classical singleproof systems (i.e., usual MerlinArthur proof systems), it is unclear whether or not quantum multiproof systems collapse to quantum singleproof systems (i.e., usual quantum MerlinArthur proof systems). This paper presents a necessary and sufficient condition under which the number of quantum proofs is reducible to two. It is also proved that, in the case of perfect soundness, using multiple quantum proofs
The power of unentanglement
, 2008
"... The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than o ..."
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Cited by 27 (3 self)
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The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Can we show any upper bound on QMA(k), besides the trivial NEXP? Does QMA(k) = QMA(2) for k ≥ 2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. • We give a protocol by which a verifier can be convinced that a 3Sat formula of size n is satisfiable, with constant soundness, given Õ ( √ n) unentangled quantum witnesses with O (log n) qubits each. Our protocol relies on Dinur’s version of the PCP Theorem and is inherently nonrelativizing. • We show that assuming the famous Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k) = QMA(2) for all k ≥ 2. • We give evidence that QMA(2) ⊆ PSPACE, by showing that this would follow from “strong amplification ” of QMA(2) protocols. • We prove the nonexistence of “perfect disentanglers” for simulating multiple Merlins with one.
PERFECT PARALLEL REPETITION THEOREM FOR QUANTUM XOR PROOF SYSTEMS
, 2008
"... Abstract. We consider a class of twoprover interactive proof systems where each prover returns a single bit to the verifier and the verifier’s verdict is a function of the XOR of the two bits received. We show that, when the provers are allowed to coordinate their behavior using a shared entangled ..."
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Cited by 27 (5 self)
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Abstract. We consider a class of twoprover interactive proof systems where each prover returns a single bit to the verifier and the verifier’s verdict is a function of the XOR of the two bits received. We show that, when the provers are allowed to coordinate their behavior using a shared entangled quantum state, a perfect parallel repetition theorem holds in the following sense. The prover’s optimal success probability for simultaneously playing a collection of XOR proof systems is exactly the product of the individual optimal success probabilities. This property is remarkable in view of the fact that, in the classical case (where the provers can only utilize classical information), it does not hold. The theorem is proved by analyzing parities of XOR proof systems using semidefinite programming techniques, which we then relate to parallel repetitions of XOR games via Fourier analysis.
Entanglement in interactive proof systems with binary answers
 In Proceedings of STACS 2006
, 2006
"... If two classical provers share an entangled state, the resulting interactive proof system is significantly weakened [6]. We show that for the case where the verifier computes the XOR of two binary answers, the resulting proof system is in fact no more powerful than a system based on a single quantum ..."
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Cited by 17 (1 self)
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If two classical provers share an entangled state, the resulting interactive proof system is significantly weakened [6]. We show that for the case where the verifier computes the XOR of two binary answers, the resulting proof system is in fact no more powerful than a system based on a single quantum prover: ⊕MIP ∗ [2] ⊆ QIP(2). This also implies that ⊕MIP ∗ [2] ⊆ EXP which was previously shown using a different method [7]. This contrasts with an interactive proof system where the two provers do not share entanglement. In that case, ⊕MIP[2] = NEXP for certain soundness and completeness parameters [6]. 1
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