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118
Toward a general theory of quantum games
 In Proceedings of 39th ACM STOC
, 2006
"... Abstract We study properties of quantum strategies, which are complete specifications of a givenparty's actions in any multipleround interaction involving the exchange of quantum information with one or more other parties. In particular, we focus on a representation of quantumstrategies that g ..."
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Cited by 44 (14 self)
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Abstract We study properties of quantum strategies, which are complete specifications of a givenparty's actions in any multipleround interaction involving the exchange of quantum information with one or more other parties. In particular, we focus on a representation of quantumstrategies that generalizes the ChoiJamiol/kowski representation of quantum operations. This new representation associates with each strategy a positive semidefinite operator acting onlyon the tensor product of its input and output spaces. Various facts about such representations are established, and two applications are discussed: the first is a new and conceptually simpleproof of Kitaev's lower bound for strong coinflipping, and the second is a proof of the exact characterization QRG = EXP of the class of problems having quantum refereed games. 1 Introduction The theory of games provides a general structure within which both cooperation and competitionamong independent entities may be modeled, and provides powerful tools for analyzing these models. Applications of this theory have fundamental importance in many areas of science.This paper considers games in which the players may exchange and process quantum information. We focus on competitive games, and within this context the types of games we consider arevery general. For instance, they allow multiple rounds of interaction among the players involved, and place no restrictions on players ' strategies beyond those imposed by the theory of quantuminformation. While classical games can be viewed as a special case of quantum games, it is important tostress that there are fundamental differences between general quantum games and classical games. For example, the two most standard representations of classical games, namely the normal formand extensive form representations, are not directly applicable to general quantum games. This is due to the nature of quantum information, which admits a continuum of pure (meaning extremal)
The quantum moment problem and bounds on entangled multiprover games
 In Proceedings of 23rd IEEE Conference on Computational Complexity
, 2008
"... We study the quantum moment problem: Given a conditional probability distribution together with some polynomial constraints, does there exist a quantum state ρ and a collection of measurement operators such that (i) the probability of obtaining a particular outcome when a particular measurement is p ..."
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Cited by 35 (2 self)
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We study the quantum moment problem: Given a conditional probability distribution together with some polynomial constraints, does there exist a quantum state ρ and a collection of measurement operators such that (i) the probability of obtaining a particular outcome when a particular measurement is performed on ρ is specified by the conditional probability distribution, and (ii) the measurement operators satisfy the constraints. For example, the constraints might specify that some measurement operators must commute. We show that if an instance of the quantum moment problem is unsatisfiable, then there exists a certificate of a particular form proving this. Our proof is based on a recent result in algebraic geometry, the noncommutative Positivstellensatz of Helton and McCullough [Trans. Amer. Math. Soc., 356(9):3721, 2004]. A special case of the quantum moment problem is to compute the value of oneround multiprover games with entangled provers. Under the conjecture that the provers need only share states in finitedimensional Hilbert spaces, we prove that a hierarchy of semidefinite programs similar to the one given by Navascués, Pironio and Acín [Phys. Rev. Lett., 98:010401, 2007] converges to the entangled value of the game. It follows that the class of languages recognized by a multiprover interactive proof system where the provers share entanglement is recursive.
A direct product theorem for discrepancy
 In Proceedings of the 23rd IEEE Conference on Computational Complexity. IEEE
, 2008
"... Discrepancy is a versatile bound in communication complexity which can be used to show lower bounds in the distributional, randomized, quantum, and even unbounded error models of communication. We show an optimal product theorem for discrepancy, namely that for any two Boolean functions f, g, disc(f ..."
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Cited by 33 (9 self)
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Discrepancy is a versatile bound in communication complexity which can be used to show lower bounds in the distributional, randomized, quantum, and even unbounded error models of communication. We show an optimal product theorem for discrepancy, namely that for any two Boolean functions f, g, disc(f ⊕ g) = Θ(disc(f)disc(g)). As a consequence we obtain a strong direct product theorem for distributional complexity, and direct sum theorems for worstcase complexity, for bounds shown by the discrepancy method. Our results resolve an open problem of Shaltiel (2003) who showed a weaker product theorem for discrepancy with respect to the uniform distribution, disc U ⊗k(f ⊗k) = O(discU(f)) k/3. The main tool for our results is semidefinite programming, in particular a recent characterization of discrepancy in terms of a semidefinite programming quantity by Linial and Shraibman (2006). 1
Unique games with entangled provers are easy
 In Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
, 2008
"... We consider oneround games between a classical verifier and two provers who share entanglement. We show that when the constraints enforced by the verifier are ‘unique ’ constraints (i.e., permutations), the value of the game can be well approximated by a semidefinite program. Essentially the only a ..."
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Cited by 29 (8 self)
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We consider oneround games between a classical verifier and two provers who share entanglement. We show that when the constraints enforced by the verifier are ‘unique ’ constraints (i.e., permutations), the value of the game can be well approximated by a semidefinite program. Essentially the only algorithm known previously was for the special case of binary answers, as follows from the work of Tsirelson in 1980. Among other things, our result implies that the variant of the unique games conjecture where we allow the provers to share entanglement is false. Our proof is based on a novel ‘quantum rounding technique’, showing how to take a solution to an SDP and transform it to a strategy for entangled provers. Using our approximation by a semidefinite program we also show a parallel repetition theorem for unique entangled games. 1
Entangled games are hard to approximate
 SIAM J. Comput
"... All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. ..."
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Cited by 26 (3 self)
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All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.
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 26 (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.
Multiparty pseudotelepathy
 Proceedings of the 8th International Workshop on Algorithms and Data Structures, Volume 2748 of Lecture Notes in Computer Science
, 2003
"... Quantum information processing is at the crossroads of physics, mathematics and computer science. It is concerned with that we can and cannot do with quantum information that goes beyond the abilities of classical information processing devices. Communication complexity is an area of classical compu ..."
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Cited by 23 (6 self)
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Quantum information processing is at the crossroads of physics, mathematics and computer science. It is concerned with that we can and cannot do with quantum information that goes beyond the abilities of classical information processing devices. Communication complexity is an area of classical computer science that aims at quantifying the amount of communication necessary to solve distributed computational problems. Quantum communication complexity uses quantum mechanics to reduce the amount of communication that would be classically required. Pseudotelepathy is a surprising application of quantum information processing to communication complexity. Thanks to entanglement, perhaps the most nonclassical manifestation of quantum mechanics, two or more quantum players can accomplish a distributed task with no need for communication whatsoever, which would be an impossible feat for classical players. After a detailed overview of the principle and purpose of pseudotelepathy, we present a survey of recent and nosorecent work on the subject. In particular, we describe and analyse all the pseudotelepathy games currently known to the authors.
Some applications of hypercontractive inequalities in quantum information
, 2012
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Strong parallel repetition theorem for quantum XOR proof systems
 In Proceedings of the 22nd Annual Conference on Computational Complexity
, 2007
"... William Slofstra ∗ Sarvagya Upadhyay ∗ 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. Such proof systems, called XOR proof systems, have previously been s ..."
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Cited by 18 (0 self)
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William Slofstra ∗ Sarvagya Upadhyay ∗ 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. Such proof systems, called XOR proof systems, have previously been shown to characterize MIP ( = NEXP) in the case of classical provers but to reside in EXP in the case of quantum provers (who are allowed to share a priori entanglement). We show that, in the quantum case, a perfect parallel repetition theorem holds for such proof systems 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, it does not hold. The theorem is proved by analyzing an XOR operation on XOR proof systems. Using semidefinite programming techniques, we show that this operation satisfies a certain additivity property, which we then relate to parallel repetitions of XOR games. 1 Introduction and summary of results