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60
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 genera ..."
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Cited by 20 (10 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)
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 18 (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.
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 16 (7 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
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 16 (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.
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 15 (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
On the unique games conjecture
 In FOCS
, 2005
"... This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. 1 ..."
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Cited by 14 (0 self)
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This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. 1
Quantum information and the PCP theorem
 In FOCS
, 2005
"... We show how to encode 2n (classical) bits a1,...,a2 n by a single quantum state Ψ 〉 of size O(n) qubits, such that: for any constant k and any i1,...,ik ∈ {1,...,2 n}, the values of the bits ai1,...,aik can be retrieved from Ψ 〉 by a oneround ArthurMerlin interactive protocol of size polynomial ..."
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Cited by 13 (3 self)
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We show how to encode 2n (classical) bits a1,...,a2 n by a single quantum state Ψ 〉 of size O(n) qubits, such that: for any constant k and any i1,...,ik ∈ {1,...,2 n}, the values of the bits ai1,...,aik can be retrieved from Ψ 〉 by a oneround ArthurMerlin interactive protocol of size polynomial in n. This shows how to go around HolevoNayak’s Theorem, using ArthurMerlin proofs. We use the new representation to prove the following results: 1. Interactive proofs with quantum advice: We show that the class QIP/qpoly contains all languages. That is, for any language L (even nonrecursive), the membership x ∈ L (for x of length n) can be proved by a polynomialsize quantum interactive proof, where the verifier is a polynomialsize quantum circuit with working space initiated with some quantum state ΨL,n〉 (depending only on L and n). Moreover, the interactive proof that we give is of only one round, and the messages communicated are classical. 2. PCP with only one query: We show that the membership x ∈ SAT (for x of length n) can be proved by a logarithmicsize quantum state Ψ〉, together with a polynomialsize classical proof consisting of blocks of length polylog(n) bits each, such that after measuring the state Ψ 〉 the verifier only needs to read one block of the classical proof. While the first result is a straight forward consequence of the new representation, the second requires an additional machinery of quantum lowdegreetest that may be interesting in its own right.
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 10 (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
The unique game conjecture with entangled provers is false
, 2007
"... 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 9 (3 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. 1
On the relationship between convex bodies related to correlation experiments with dichotomic observables
 Journal of Physics A: Mathematical and General
, 2006
"... In this paper we explore further the connections between convex bodies related to quantum correlation experiments with dichotomic variables and related bodies studied in combinatorial optimization, especially cut polyhedra. Such a relationship was established in Avis, Imai, Ito and Sasaki (2005 J. P ..."
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Cited by 7 (4 self)
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In this paper we explore further the connections between convex bodies related to quantum correlation experiments with dichotomic variables and related bodies studied in combinatorial optimization, especially cut polyhedra. Such a relationship was established in Avis, Imai, Ito and Sasaki (2005 J. Phys. A: Math. Gen. 38 10971–87) with respect to Bell inequalities. We show that several well known bodies related to cut polyhedra are equivalent to bodies such as those defined by Tsirelson (1993 Hadronic J. S. 8 329–45) to represent hidden deterministic behaviors, quantum behaviors, and nosignalling behaviors. Among other things, our results allow a unique representation of these bodies, give a necessary condition for vertices of the nosignalling polytope, and give a method for bounding the quantum violation of Bell inequalities by means of a body that contains the set of quantum behaviors. Optimization over this latter body may be performed efficiently by semidefinite programming. In the second part of the paper we apply these results to the study of classical correlation functions. We provide a complete list of tight inequalities for the two party case with (m, n) dichotomic observables when m = 4, n = 4 and when min{m, n} ≤ 3, and give a new general family of correlation inequalities. PACS classification numbers: 03.65.Ud, 02.40.Ft, 02.10.Ud