monogamy of non-signalling correlations

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monogamy of non-signalling correlations. Aram Harrow (MIT) Simons Institute, 27 Feb 2014. based on joint work with Fernando Brandão (UCL) arXiv:1210.6367 + εunpublished. “correlations” (multipartite conditional probability distributions). b. a. y. x. why study boxes?. - PowerPoint PPT Presentation

Text of monogamy of non-signalling correlations

monogamy of non-signalling correlations

monogamy of non-signalling correlationsAram Harrow (MIT)Simons Institute, 27 Feb 2014based on joint work with Fernando Brando (UCL)arXiv:1210.6367 + unpublished

correlations(multipartite conditional probability distributions)localp(x,y|a,b) = qA(x|a) qB(y|b)LHV (local hidden variable)p(x,y|a,b) = r (r) qA(x|a,r) qB(y|b,r)quantump(x,y|a,b) = h| Aax Bby |iwith x Aax = y Bby = Inon-signallingy p(x,y|a,b) = y p(x,y|a,b)x p(x,y|a,b) = x p(x,y|a,b)axby

why study boxes?

Foundational: considering theories more generalthan quantum mechanics (e.g. Bells Theorem)Operational: behavior of quantum states underlocal measurement (e.g. this work)Computational: corresponds to constraint-satisfaction problems and multi-prover proof systems.why non-signalling?

Foundational: minimal assumption for plausible theoryOperational: yields well-defined partial tracep(x|a) := y p(x,y|a,b) for any choice of bComputational: yields efficient linear programthe dual picture: gamesComplexity:classical (local or LHV) value is NP-hardquantum value has unknown complexitynon-signalling value in P due to linear programmingNon-local games:Inputs chosen according to (a,b)Payoff function is V(x,y|a,b)The value of a game using strategy p isx,y,a,b p(x,y|a,b) (a,b) V(x,y|a,b).monogamyLHV correlations can be infinitely shared.This is an alternate definition.ApplicationsNon-shareability secrecycan be certified by Bell tests

Gives a hierarchy of approximations for LHV correlationsrunning in time poly(|X| |Y|k |A| |B|k)

de Finetti theorems (i.e. k-extendable states separable)p(x,y|a,b) is k-extendable if there exists a NS boxq(x,y1,,yk|a,b1,,bk) with q(x,yi|a,bi) = p(x,yi|a,bi) for each iresultsTheorem 1: If p is k-extendable and is a distribution on A, then there exists qLHV such that

Theorem 2: If p(x1,,xk|a1,,ak) is symmetric, 0