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Wiring of No-Signaling Boxes Expands the Hypercontractivity Ribbon

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Wiring of No-Signaling Boxes Expands the Hypercontractivity Ribbon Wiring of No-Signaling Boxes Expands the Hypercontractivity Ribbon Salman A. Beigi Institute for Research in Fundamental Sciences (IPM) Tehran, Iran January 12, 2015 Joint work with Amin Gohari arXiv:1409.3665 Closed sets of nonlocal correlations Closed sets of nonlocal correlations Foundations of Physics, Vol. 24, No. 3, 1994 Quantum Nonlocality as an Axiom Sandu Popescu t and Daniel Rohrlich 2 Received July 2, 1993: revised July 19, 1993 In the conventional approach to quantum and nonlocality mechanics, &determinism is a theorem. We is an axiom nonlocality consider inverting the an axiom and logical order, mak#1g indeterminism a theorem. correlations, preserving Nonlocal "superquantum" relativistic causality, more strongly can violate the CHSH than any quantum correlations. inequality What is the quantum principle? J. Wheeler named after the legendary it the "Merlin principle" magician who, when again and pursued, could change again. The more we his form changes: pursue the quantum from discreteness, principle, the more it to indeterminism, to worlds, and so on. sums over paths, to many By comparison, the relativity Relativity theory and principle is easy to grasp. quantum theory underlie not always know all of physics, but how to reconcile them. we do quantum Here, we take nonlocality principle, and we ask as the together what nonlocality and imply. It is a pleasure relativistic causality to dedicate this Rohrlich, who has contributed paper to Professor Fritz and much to the juncture relativity theory, including of quantum theory electrodynamics, its most spectacular and who success, quantum has written both on the logical structure quantum paradoxes tll of physical theory, t2~ and Bell t31 proved that some predictions of quantum reproduced by any theory mechanics cannot be of local physical variables. within nonrelativistic Although Bell worked quantum theory, the relativistic: a definition of local local variable can be variable is ward influenced only by events light cone, not by events in its back- outside, and can influence events in its i Universit6 Libre de Bruxelles, Campus Bruxelles, Plaine, C.P. 225, Boulevard Belgium. du Triomphe, B-1050 2 School of Physics and Astronomy, TeI-Aviv University, Ramat-Aviv, Tel-Aviv 69978 Israel. 379 825/24/3-4 0015-9018/94/0300-0379507.00/0 ~-, 1994 Plenum PublishingCorporation Closed sets of nonlocal correlations Foundations of Physics, Vol. 24, No. 3, 1994 week ending 30 JUNE 2006 Quantum PHYSICAL REVIEW LETTERS Nonlocality as an Axiom PRL 96, 250401 (2006) Sandu Popescu 3 t and Daniel Rohrlich 1 2 1 Alain Tapp, and Falk Unger 4 ´thot, Limit on Nonlocality inReceived Any WorldJuly 2, 1993: inWhich CommunicationAndre´ Allan Me Complexity Is´bec Not H3C Trivial 3J7, Canada 2,3 Noahrevised Linden, July 19, 1993 ´al, Que 1 Harry Buhrman, Gilles Brassard, In the deconventional Montre´al, C.P. 6128, Succursale Centre-Ville, Montre ´ approach to quantum 1De´partement IRO, Universiteand nonlocality mechanics, &determinism 2 is a theorem. We is an axiom nonlocality consider inverting the ILLC, Universiteit van Amsterdam,an axiom Plantage and Muidergracht 24, 1018 TV Amsterdam,logical order, The mak#1g Netherlands 3 indeterminism a theorem. Centrum voor Wiskundecorrelations, en Informatica preserving (CWI), Post Office Box 94079, 1090Nonlocal GB Amsterdam, "superquantum" The Netherlands relativistic causality, 4Department of Mathematics,more strongly University of Bristol, University Walk,can violate Bristol, the BS8 CHSH 1TW, United Kingdom (Receivedthan any 2 Marchquantum 2006; correlations. published 27 June 2006) inequality Bell proved that quantum entanglement enables two spacelike separated parties to exhibit classically impossibleWhat correlations. Even though these correlations are stronger than anything classically achievable, is the quantum principle? they cannot be harnessed to make instantaneousJ. Wheeler (faster than named light) communication possible. Yet, Popescu after the legendary it the "Merlin principle" and Rohrlich have shown thatmagician even stronger who, correlations when can be defined, under which instantaneous again and pursued, could change communicationagain. remains The impossible. more we This raises the question: Why are the correlationshis achievable form by changes: pursue the quantum quantum mechanicsfrom discreteness, not maximal among those that preserve causality?principle, We give athe partial more answer it to this to indeterminism, to questionworlds, byand showing so on. that slightly stronger correlations wouldsums result inover a world paths, in which to communicationmany By comparison, the relativity complexityRelativity becomestheory trivial.and principlePACS is numbers: easy to 03.65.Ud, grasp. 03.67.Hk, 03.67.Mn quantum theory underlie not always know all of physics, but DOI: 10.1103/PhysRevLett.96.250401how to reconcile them. we do quantum Here, we take nonlocality principle, and we ask as the together what nonlocalitythat and can be observed are purely random from a local imply. It is a pleasure relativistic causality Rohrlich, to dedicate this perspective.paper In other words, NLBs are nonlocal, yet they who has contributed to :Professor they cannot Fritz make an effect precede its cause in Entanglementand can be harnessed to accomplishmuch amazing to the junctureare causal relativity theory, including of quantum theory informationelectrodynamics, processing feats. The first proof that genuinelyits most spectacularthe context of special relativity. We are interested in the and who success, quantum nonclassical behavior could be produced byhas quantum- written both on question of how well the correlation of NLBs can be the logical structure quantum paradoxesby devices tll that follow the laws of physics. mechanical devices was given by Bell,of physical who proved theory, that t2~ approximated and Bell t31 proved that entanglement enables two spacelike separatedsome predictions parties to of quantum Although originally presented differently, the CHSH reproduced by any theory mechanics cannot be exhibit correlations that are stronger thanof anything local physical allowed variables.inequality can be recast in terms of imperfect NLBs. The within nonrelativistic Although Bell worked by classical physics [1]. Later, Clauser,quantum Horne, theory, Shimony, the availability of shared entanglement allows Alice and Bob relativistic: a local definition of local variable and Holt (CHSH), inspired by thevariable work of can Bell, be proposed influenced to approximate NLBs withis success probability ward light cone, only by events in p2 another inequality [2], whichnot was easierby events to translate outside, into a its back- 2 85:4%: and can influence events 2 } incos its 4 feasible experiment to test local hidden-variable theories. 8 Their proposali Universit6 fits nicely into the more modern framework Libre de Bruxelles, Campus of nonlocalBruxelles, boxes, introduced by Popescu andPlaine, Rohrlich C.P. 225, BoulevardThis can be used to test local hidden-variable theories Belgium. du Triomphe, B-1050 [ [3], Eq.2 School (7)]. of Physics because it follows also from CHSH that no local realistic (NLB)and is Astronomy, an imaginary TeI-Aviv device that has an A nonlocal box University, Ramat-Aviv, (classical)Tel-Aviv 69978 theory Israel. can succeed with probability greater than input-output port at Alice’s location and another379 one at 3=4 if Alice and Bob are spacelike separated. Later, 825/24/3-4 Bob’s, even though Alice and Bob canx beinto0015-9018/94/0300-0379507.00/0 spacelike her input sepa- port, she Tsirelson [6] proved the optimality of the CHSH inequal- a, locally~-, } at rated. Whenever Alice feeds a bit 1994 Plenumity, which Publishing translatesCorporation into saying that quantum mechanics gets a uniformly distributed random output bit does not allow for a success probability greater than uncorrelated with anything else, including her own input the game of simulating NLBs. See also Ref. [7] for an bit. The same applies to Bob, whose input and output bits b, respectively. The ‘‘magic’’ appears in the information-theoretic proof of the same result. we call y and There are two questions of interest in this Letter: form of a correlation between the pair(sum of outputs modulo and two, the denoted OR AND of (1) Considering that perfect NLBs would not violate cau- pair of inputs: the exclusive sality, why do the laws of quantum mechanics only allow ‘‘ ’’) of the outputs is always. Much equal liketo the the logical correlations that us to implement NLBs better than anything classically a b x y the inputs: ^ possible, yet not perfectly? (2) Why do they provide us can be established by use of quantum entanglement, this with an approximation of NLBs that succeeds with proba- device is atemporal: Alice gets her output as soon as she bility } rather than something better? feeds in her input, regardless of if and when Bob feeds in Before we can pursue this line of thoughtcommunication further, we complex- his input, and vice versa. Also inspired by entanglement, device: the correlation appears only as a need to review briefly the field of this is a one-shot ity [8–11]. Assume Alice and Bob wishx, known to compute to Alice some only, f x; y of input result of the first pair of inputs fed in by Alice and Bob. Of Boolean function y, known to Bob only. Their concern is to course, they can have more thanresource one NLB[4] of at a their different disposal, nature and input which is then seen as a minimize the amount of communication required between
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