Local orthogonality: a multipartite principle for (quantum) correlations
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1 Locl orthogonlity: multiprtite principle for (quntum) correltions Antonio Acín ICREA Professor t ICFO-Institut de Ciencies Fotoniques, Brcelon Cusl Structure in Quntum Theory, Bensque, Spin, June 2013
2 ICFO Collbortors Remigiusz Augusik Jontn Bohr Brsk Rfel Chves Tobis Fritz Anthony Leverrier Belén Sínz rxiv:
3 Box-World Scenrio N distnt prties performing m different mesurements of r outcomes. x 1 = 1,,m x i = 1,,m x N = 1,,m 1 = 1,,r i = 1,,r N = 1,,r p(,, N x1,, x 1 N )
4 Physicl Correltions Physicl principles trnslte into limits on correltions. No-signlling correltions: correltions comptible with the no-signlling principle, i.e. the impossibility of instntneous communiction. k 1,, N p( 1,, N x 1,, x N ) p( 1,, k x 1,, x k ) x 1 = 1,,m x 2 = 1,,m p( 1 x1) 1 = 1,,r 2 = 1,,r
5 Physicl Correltions Clssicl correltions: correltions estblished by clssicl mens. p,,,,,, 1 N x1 xn p D 1 x1 D N xn These re the stndrd EPR correltions. Independently of fundmentl issues, these re the correltions chievble by clssicl resources. Bell inequlities define the limits on these correltions.
6 Physicl Correltions i i i i i i i i i i i x x x x M M M M ' ' 1 Quntum correltions: correltions estblished by quntum mens. N N x x N N M M x x p 1 1 tr ),,,, ( 1 1
7 Why quntum correltions? NS Q L Q: Why re these correltions not possible in Nture? A: They re incomptible with quntum lws. Tht is, there is no quntum stte nd mesurements ble to reproduce them. Wht would their existence imply opertionlly? Informtion principles hve been proposed s the mechnism to bound quntum correltions. Exmples: non-trivil communiction complexity, informtion cuslity, mcroscopic loclity.
8 Guess Your Neighbour s Input (GYNI) x = 0,1 y = 0,1 = 0,1 b = 0,1 Alice nd Bob receive two rndom bits, x nd y. Their gol is to compute the bit the other prty received. Clerly, winning too often would imply signlling. P ok = 1 4 p p p p Optiml clssicl strtegy: the prties give their input s output P ok = 1/2. This vlue is universl, s violting it would imply signlling between the prties. Tht is, quntum nd supr-quntum non-signlling correltions do not improve it.
9 Guess Your Neighbour s Input (GYNI) x = 0,1 y = 0,1 z = 0,1 = 0,1 b = 0,1 c = 0,1 Alice hs to guess the bit received by Bob, who hs to guess the one received by Chrlie, who hs to guess Alice s bit. P ok = 1 (p p p p p p p p ) Optiml clssicl strtegy: the prties give their input s output P ok = 1/4. This vlue is universl, s violting it would imply signlling between the prties. Tht is, quntum nd supr-quntum non-signlling correltions do not improve it.
10 Guess Your Neighbour s Input (GYNI) P ok = 1 (p p p p p p p p ) Promise: the sum of the inputs is zero, ie x y z = 0. P ok = 1 (p p p p ) 4 Intuition: it should be the sme s Alice s bit does not provide ny informtion bout Bob s, nd the sme pplies for ll the prties. Optiml clssicl strtegy: the prties give their input s output P ok = 1/4. This limit is gin vlid for prties hving ccess to correlted quntum prticles. Yet, it is possible to get lrger probbility without violting the no-signlling principle! Why?!
11 Guess Your Neighbour s Input (GYNI) p p p p Q L First tight tsk with no quntum violtion. Almeid et l, PRL 10 The no-signlling principle is intrinsiclly biprtite.
12 Locl orthogonlity: multiprtite principle
13 Locl orthogonlity Locl orthogonlity: different outcomes of the sme mesurement by one of the observers define orthogonl events, independently of the rest of mesurements. Event Input Output e 1 x 1 x i x N 1 i N e 2 x 1 x i x N 1 i N N events re orthogonl if they re pirwise orthogonl. Opertionlly: the sum of probbilities of pirwise orthogonl events is bounded by 1. p e i 1 e i
14 LO s distributed guessing problem () In stndrd guessing problem, vlue to be guessed is encoded by function f nd the gol is to mke guess bout the encoded vlue. (b) In Distributed Guessing Problem (DGP) string of bit is encoded on string of N bits tht re distributed mong distnt prties, who hve to mke guess. ~
15 LO s distributed guessing problem The figure of merit is the probbility of mking right guess. If the initil bit string cn tke S vlues, this probbility is lower bounded by 1/ S. There exist functions for which the optiml guessing probbility for clssiclly correlted plyers is equl to 1/ S. We cll these functions mximlly difficult. In non-distributed problems, the only mximlly difficult function is the trivil one in which the function mps ll the vlues into one, it erses ll the informtion. In distributed versions, there exist other non-trivil mximlly difficult functions. Correltions violting LO turn mximlly difficult functions for clssicl plyers into non-mximlly difficult.
16 LO nd quntum correltions Quntum correltions stisfy LO. Proof: Event Input Output e 1 x 1 x i x N 1 i N e 2 x 1 x i x N 1 i N mx p e 1 + p e 2 = mx ψ Π x 1, 1 Π x i, i Π x N, N + Π x 1, 1 Π x i, i Π x N, N ψ ψ I ψ =1 Locl orthogonlity is stisfied both by clssicl nd quntum theory. Indeed, while quntum physics breks the orthogonlity of preprtions, it keeps the orthogonlity of mesurement outcomes. Intuition: mesurement outcomes re lwys of clssicl nture.
17 LO nd the no-signlling principle For two prties: comptibility with LO non-signlling correltions. Cbello, Severini nd Winter For more prties: LO is strictly more restrictive thn no-signlling. Exmple: GYNI. p p p p All events in GYNI re pirwise orthogonl.
18 LO nd grph theory How to get LO inequlities in generl scenrio consisting of N prties mking M mesurements of R possible outcomes? There re M N possible combintion of inputs. For ech of them, there re R N possible results. This mkes MR N different events. e 1 e j = 1 i N x 1 x i x N e 2 e k = 1 i N x 1 x i x N e MR N Cbello, Severini nd Winter We construct grph of events: Nodes: events. Edges: orthogonlity condition.
19 LO nd grph theory e 1 e j = 1 i N x 1 x i x N e 2 e k = 1 i N x 1 x i x N e MR N Clique: fully connected subgrph set of pirwise orthogonl events. Mximum clique optiml LO inequlity. There exist lgorithm to find cliques of grph. Recll tht finding the mximum clique of n rbitrry grph is n NP-hrd problem. These grphs re not rbitrry.
20 LO nd extreml triprtite correltions All extreml non-signlling correltions for 3 observers performing 2 mesurements of 2 outcomes were listed in S. Pironio et l, JPA 11. They cn be clssified into 46 clsses (one of them corresponding to locl points). All but one of the 45 clsses of non-locl correltions cn be ruled out by informtion cuslity (Tzyh Hur et l, NJP 12). The remining point, box 4, is n exmple of point tht cnnot be flsified by biprtite principles. All the triprtite boxes contrdict LO nd, thus, do not hve quntum reliztion. In prticulr, it rules out box 4 becuse of its intrinsiclly multiprtite formultion.
21 LO nd biprtite correltions Popescu-Rohrlich (PR)-box NS Q L x = 0,1 = 0,1 y = 0,1 b = 0,1 p b xy = 1 2, 0,0, 1 2 ; 1 2, 0,0, 1 2 ; 1 2, 0,0, 1 2 ; 0, 1 2, 1 2, 0
22 LO nd biprtite correltions Despite the equivlence with NS for two prties, LO cn be used to rule out suprquntum biprtite correltions. How? Use networks. x 1 = 0,1 A 1 A 2 1 = 0,1 x 1 = 0,1 PR-box x 2 = 0,1 2 = 0,1 A 3 A 4 1 = 0,1 PR-box x 2 = 0,1 2 = 0,1 Check now for violtion of LO inequlities for 4 prties.
23 LO nd biprtite correltions Two PR-boxes distributed mong 4 observers violte the LO inequlity: p p p p p NS All supr-quntum correltions in this region violte LO. Q L
24 Conjecture Conjecture: Locl orthogonlity defines the quntum set. Principle: there is lwys someone smrter thn you! Nvscués: there re supr-quntum correltions comptible with LO! In fct, the set of LO correltions is not even convex!
25 LO nd contextulity Our pproch esily extends to non-contextulity scenrio. This hs been studied for instnce in: T. Frizt, A. Leverrier nd A.B. Sinz, rxiv: A. Cbello, Phys. Rev. Lett. 110 (2013) B. Yn, rxiv:
26 Conclusions Multiprtite principle re needed for our understnding of quntum correltions. Locl orthogonlity is n intrinsiclly multiprtite principle. It cptures the clssicl nture of mesurement outcomes: outcomes of the sme mesurement define incomptible events. It is powerful method when combined with grph-theory concepts nd network geometries. It rules out supr-quntum correltions, both in the biprtite nd multiprtite cse. The principle lone does not give quntum correltions. Wht else is needed to define quntum correltions?
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