Rank-one quantum games

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

doi:10.1007/s00037-014-0096-x

Cited by 26 publications

(36 citation statements)

References 51 publications

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“…49) where the last term is bounded by using the fact that Π init U 2 U 1 |ψ is a valid initial state for the game specified by V , and can therefore not lead to a higher acceptance probability than ω * (V ). Putting everything together, one has 50) as desired.…”

Section: It Always Holds Thatmentioning

confidence: 99%

Quantum Proofs

Vidick

Watrous

2016

FNT in Theoretical Computer Science

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Quantum information and computation provide a fascinating twist on the notion of proofs in computational complexity theory. For instance, one may consider a quantum computational analogue of the complexity class \class{NP}, known as QMA, in which a quantum state plays the role of a proof (also called a certificate or witness), and is checked by a polynomial-time quantum computation. For some problems, the fact that a quantum proof state could be a superposition over exponentially many classical states appears to offer computational advantages over classical proof strings. In the interactive proof system setting, one may consider a verifier and one or more provers that exchange and process quantum information rather than classical information during an interaction for a given input string, giving rise to quantum complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit some properties from their classical counterparts, they also possess distinct and uniquely quantum features that lead to an interesting landscape of complexity classes based on variants of this model. In this survey we provide an overview of many of the known results concerning quantum proofs, computational models based on this concept, and properties of the complexity classes they define. In particular, we discuss non-interactive proofs and the complexity class QMA, single-prover quantum interactive proof systems and the complexity class QIP, statistical zero-knowledge quantum interactive proof systems and the complexity class \class{QSZK}, and multiprover interactive proof systems and the complexity classes QMIP, QMIP*, and MIP*.Comment: Survey published by NOW publisher

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Section: It Always Holds Thatmentioning

confidence: 99%

Quantum Proofs

Vidick

Watrous

2016

FNT in Theoretical Computer Science

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“…It plays an important role in recent results on device-independent quantum key distribution [HR09,MPA11] and related cryptographic primitives [TFKW13]. The latter work considers parallel repetition of a game with quantum messages, a setting which is also the focus of [CJPP11]. The approach of [CJPP11] builds upon [JPPG + 10], who relate the (classical) value of a two-player one-round game to the norm of the game when viewed as a tensor on the…”

Section: Additional Related Workmentioning

confidence: 99%

“…The latter work considers parallel repetition of a game with quantum messages, a setting which is also the focus of [CJPP11]. The approach of [CJPP11] builds upon [JPPG + 10], who relate the (classical) value of a two-player one-round game to the norm of the game when viewed as a tensor on the…”

Section: Additional Related Workmentioning

confidence: 99%

“…entangled) value. To the best of our knowledge this connection has not led to an alternative approach to proving parallel repetition for general classes of games, although partial results were obtained in [CJPP11] for the special case of the entangled value of rank-one quantum games. After the completion of this work two new results established an exponential parallel repetition theorem for two-player one-round games with entangled players in which the distribution on questions is a product distribution.…”

Section: Additional Related Workmentioning

confidence: 99%

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A parallel repetition theorem for entangled projection games

2015

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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 − ε < 1, the value of the k-fold repetition of G goes to zero as O((1 − ε c ) k ), for some universal constant c ≥ 1. If furthermore the constraint graph of G is expanding we obtain the optimal c = 1. Previously exponential decay of the entangled value under parallel repetition was only known for the case of XOR and unique games. To prove the theorem we extend an analytical framework 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, the algorithm 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 non-communicating parties, given classical descriptions of 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.

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“…Some generalizations of nonlocal games in which quantum information is exchanged in some way between the players and the referee include ones studied in [3,12,17,23,9,20,13]. In this paper we consider two such generalizations: quantum-classical games and extended nonlocal games.…”

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confidence: 99%

Untitled

Russo¹,

Watrous²

2018

Chicago J. of Theoretical Comp. Sci.

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Several variants of nonlocal games have been considered in the study of quantum entanglement and nonlocality. This paper concerns two of these variants, called quantumclassical games and extended nonlocal games. We give a construction of an extended nonlocal game from any quantum-classical game that allows one to translate certain facts concerning quantum-classical games to extended nonlocal games. In particular, based on work of Regev and Vidick, we conclude that there exist extended nonlocal games for which no finite-dimensional entangled strategy can be optimal. While this conclusion is a direct consequence of recent work of Slofstra, who proved a stronger, analogous result for ordinary (non-extended) nonlocal games, the proof based on our construction is considerably simpler, and the construction itself might potentially have other applications in the study of entanglement and nonlocality.

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Rank-one quantum games

Cited by 26 publications

References 51 publications

Quantum Proofs

Quantum Proofs

A parallel repetition theorem for entangled projection games

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