Experimental fully contextual correlations

Expe imen al Fully Con ex ual Co ela ions Elias Amselem, 1 La s Ei ik Danielsen, 2 An onio J. Lo ´pez-Ta ida, 3 Jose ´R. Po illo, 4 Mohamed Bou ennane, 1 and Ada ´n Cabello 3,1 1 Depa men o Physics, S ockholm Uni e si y, S-10691 S ockholm, Sweden 2 Depa men o In o ma ics, Uni e si y o Be gen, P.O. Box 7803, Be gen N-5020, No way 3 4 Depa amen o de Fı ´sica Aplicada II, Uni e sidad de Se illa, E-41012 Se illa, Spain Depa amen o de Ma ema ´ ica Aplicada I, Uni e sidad de Se illa, E-41012 Se illa, Spain Quan um co ela ions a e con ex ual ye , in gene al, no hing p e en s he exis ence o e en mo e con ex ual co ela ions. We iden i y and es a noncon ex uali y inequali y in which he quan um iola ion canno be imp o ed by any hypo he ical pos quan um heo y, and use i o expe imen ally ob ain co ela ions in which he ac ion o noncon ex ual co ela ions is less han 0.06. Ou co ela ions a e expe imen ally gene a ed om he esul s o sequen ial compa ible es s on a ou -s a e quan um sys em encoded in he pola iza ion and pa h o a single pho on. In oduc ion.—Quan um con ex uali y [1–3] e e s o he ac ha he p edic ions o quan um mechanics (QM) canno be ep oduced assuming noncon ex uali y o esul s (i.e., ha he esul s a e p ede ined and independen o o he compa ible es s) o , equi alen ly, noncon ex ual hidden a iable heo ies. By compa ible es s we mean hose sa is ying he ollowing heo y-independen de ini- ion: ‘‘I a physical sys em is p epa ed in such a way ha he esul o es xi is p edic able and epea able, and i a compa ible es xj is hen pe o med (ins ead o es xi)a subsequen execu ion o es xi shall yield he same esul as i es xj had no been pe o med’’ [4] (see [5] o o he de ini ions o compa ibili y). In QM, wo es s ep esen ed by sel -adjoin ope a o s A and B a e compa ible when A and B commu e. This gua an ees ha he quan um p edic- ions o compa ible es s a e gi en by a single p obabili y measu e on a single p obabili y space. Compa ibili y im- plies ha he p obabili y PðaijxiÞ o ob aining he esul ai o he es xi is independen o o he compa ible es s x1; ... ;xi1, xiþ1; ... ;xn, i.e., PðaijxiÞ¼ X a1;...;ai1;aiþ1;...;an Pða1;...;a njx1;...;x nÞ;(1) o all se s x1;...;x no compa ible es s, and whe e Pða1;...;a njx1;...;x nÞis he join p obabili y o ob ain- ing he esul s a1;...;a n o he compa ible es s x1;...;x n, espec i ely. Assump ion (1) is o mally equi a- len o he no-signaling p inciple, bu in ol es compa ible es s ins ead o spacelike sepa a ed es s. The assump ion o he noncon ex uali y o esul s s a es ha he esul aio es xiis he same ega dless o o he compa ible es s being pe o med; i only depends on xi and some hidden a iables . This implies ha he co e- la ion among he esul s o compa ible es s can be ex- p essed as Pða1;...;a njx1;...;x nÞ¼X  PðÞY n i¼1 Pðaijxi;Þ;(2) o some common dis ibu ion PðÞ. Noncon ex uali y inequali ies a e exp essions o he o m SXTa1;...;an;x1;...;xnPða1;...;anjx1;...;xnÞNC NC;(3) whe e Ta1;...;an;x1;...;xna e eal numbe s and NC NC de- no es ha he maximum alue o S o any noncon ex ual co ela ions [ he e o e sa is ying (2)] is NC. Quan um con ex uali y is expe imen ally obse ed h ough he io- la ion o noncon ex uali y inequali ies [6–9]. Quan um nonlocali y [10] is a pa icula o m o quan- um con ex uali y which occu s when he es s a e no only compa ible bu also spacelike sepa a ed. In his case, non- con ex uali y inequali ies a e called Bell inequali ies [10]. In addi ion o applica ions such as de ice-independen quan um key dis ibu ion [11,12] and andom numbe gene a ion [13], which equi e spacelike sepa a ion, quan- um con ex uali y also o e s ad an ages in scena ios wi h- ou spacelike sepa a ion. Examples a e communica ion complexi y [14], pa i y-obli ious mul iplexing [15], ze o- e o classical communica ion [16], and quan um c yp og- aphy secu e agains speci ic a acks [17,18]. The goal o his wo k is o iden i y and pe o m an expe imen wi h sequen ial quan um compa ible es s, which p oduces co ela ions wi h he la ges con ex uali y allowed unde he assump ion (1), which is assumed o be alid also o pos quan um heo ies. Fo his pu pose, we i s in oduce a measu e o con ex uali y o he co ela- ions, he noncon ex ual con en WNC, so ha WNC ¼0 co esponds o he maximum con ex uali y. Then, we show how o expe imen ally ob ain es able uppe bounds o WNC. Nex , we show how g aph heo y allows us o iden i y expe imen s in which he uppe bound o WNC p edic ed by QM is ze o, and apply his me hod o single ou an expe imen o which WNC ¼0. Finally, we pe o m his expe imen and ob ain co ela ions in which WNC <0:06. Noncon ex ual con en .—E e y co ela ion among com- pa ible es s [ he e o e sa is ying (1)] can be exp essed as Pða1;...;anjx1;...;xnÞ¼wNCPNCða1;...;anjx1;...;xnÞ þð1wNCÞPCða1;...;anjx1;...;xnÞ; (4) whe e 0wNC 1,PNCða1;...;a njx1;...;x nÞcan be ex- p essed as (2), and PCða1;...;a njx1;...;x nÞsa is ies (1) bu canno be exp essed as (2). We de ine he noncon ex- ual con en WNC o he co ela ions as he maximum alue o wNC o e all possible decomposi ions as (4), i.e., WNC max PNC;PCgwNC:(5) This de ini ion is pa allel o he de ini ion o local con en in oduced in [19]. In ac , o co ela ions gene a ed h ough spacelike sepa a ed es s, he noncon ex ual con- en equals he local con en . NC,Q, and Cwill deno e, espec i ely, he maxi- mum alue o S o noncon ex ual co ela ions [i.e., which can be exp essed as (2)], quan um co ela ions, and co e- la ions sa is ying (1). Now conside co ela ions sa is ying (1) and sa u a ing Q. Then, gi en a decomposi ion o such co ela ions as (4), wi h wNC ¼WNC,Qcan be exp essed as Q¼PTa1;...;an;x1;...;xn½WNCPNCða1;...;anjx1;...;xnÞþ ð1WNCÞPCða1;...;anjx1;...;xnÞ¼WNC PTa1;...;an;x1;...;xn PNCða1;...;anjx1;...;xnÞþð1WNCÞPTa1;...;an;x1;...;xn PCða1;...;anjx1;...;xnÞ. The i s sum can be exp essed in a noncon ex ual o m, so i is uppe bounded by NC. The second sum canno be exp essed in a noncon ex ual o m, so i can only be uppe bounded by C. Hence, Q WNCNC þð1WNCÞC, and, aking in o accoun ha NC QC, hen WNC CQ CNC :(6) Any expe imen al iola ion Sexp o a noncon ex uali y in- equali y indica es ha C>NC and, he e o e, p o ides an uppe bound on WNC, namely WNC ðC SexpÞ=ðCNCÞ. Assuming ha he maximum Sexp in an ideal expe imen is gi en by Q, o obse e co ela ions wi h ze o noncon ex ual con en , he e called ully con ex- ual co ela ions, one has o es a noncon ex uali y inequal- i y such ha i s maximum quan um iola ion equals i s maximum possible iola ion unde he assump ion (1), i.e., an inequali y o which NC <Q¼C. Howe e , e en i Q¼C, inhe en impe ec ions o ac ual expe imen s will p e en he obse a ion o WNC ¼0. In gene al, he mo e complex he expe imen o p oduce he equi ed quan um co ela ions is, he highe he p obabili y ha expe imen al impe ec ions lead o a highe uppe bound o he noncon ex ual con en . The e o e, he ask is o iden i y he simples noncon ex- uali y inequali y iola ed by QM and such ha Q¼C. G aph app oach.—We add essed his p oblem by using a connec ion be ween g aph heo y and noncon ex uali y inequali ies no iced in [20]: Fo any g aph he e is a non- con ex uali y inequali y o which NC,Q, and Ca e gi en, espec i ely, by he independence numbe , he Lo a ´sz numbe , and he ac ional packing numbe o he g aph [21]. We calcula ed hese h ee numbe s o all nonisomo phic g aphs wi h less han 11 e ices, and ound ha he e a e no g aphs wi h less han 10 e ices wi h NC <Q¼C, and he e a e only ou 10- e ex g aphs wi h hese p ope ies [21]. The maximum quan um iola ion o noncon ex uali y inequali ies associa ed wi h h ee o hem equi es quan um sys ems o dimension highe han ou , while dimension ou is enough o he g aph in Fig. 1. The inequali y associa ed wi h he g aph is cons uc ed by looking o p oposi ions in ol ing compa - ible es s, such ha each e ex ep esen s one p oposi ion in he inequali y and he edges only link p oposi ions ha canno be simul aneously ue. Then, he inequali y is simply gi en by he sum o all he p obabili ies o he p oposi ions ep esen ed in he g aph. Fo he g aph in Fig. 1, i can be easily seen ha he ollowing noncon ex uali y inequali y is in one- o-one co - espondence wi h he g aph: SPð010j012ÞþPð111j012ÞþPð01j02ÞþPð00j03Þ þPð11j03ÞþPð00j14ÞþPð01j25ÞþPð010j345Þ þPð111j345ÞþPð10j35Þ NC 3;(7) whe e Pð10j35Þis he p obabili y o ob aining esul 1 when es 3 is pe o med and esul 0 when es 5 is pe o med. In his case, he coe icien s Ta1;...;an;x1;...;xnin (3) a e all 1. The noncon ex ual bound, NC ¼3, can be ob ained om he independence numbe o he g aph in 01|02 10|35 010|012 111|345 11|03 00|03 111|012010|345 01|25 00|14 11 ... | ... nn 1... : es s n 1... : esul s n FIG. 1. G aph co esponding o inequali y (7). Ve ices ep e- sen p oposi ions. Fo example, 01j25 means ‘‘ esul 0 is ob- ained when es 2 is pe o med, and esul 1 is ob ained when es 5 is pe o med.’’ Edges link p oposi ions ha canno be simul aneously ue. Fo example, 01j25 and 01j02 a e linked, since in he i s p oposi ion he esul o es 2 is 0, while in he second p oposi ion he esul o es 2 is 1. Fig. 1. The maximum quan um iola ion o inequali y (7) and i s maximum possible iola ion unde he assump ion (1) can be ob ained om he Lo a ´sz and he ac ional packing numbe s o he g aph in Fig. 1, espec i ely [21]. This gi es Q¼C¼3:5:(8) The maximum quan um iola ion can be achie ed by p epa ing a ou -s a e quan um sys em in he s a e j c i¼ 1 ffiffiffi 2 pðj0iþj3iÞ;(9) whe e h0j¼ð1;0;0;0Þ,h1j¼ð0;1;0;0Þ,h2j¼ð0;0;1;0Þ, and h3j¼ð0;0;0;1Þ, and wi h he es s ep esen ed by he ollowing enso p oduc s o Pauli ma ices iand he 22iden i y ma ix 1: 0¼x1;1¼1z;2¼xz; 3¼1x;4¼z1;5¼zx:(10) The esul s 0 and 1 co espond o he eigen alues 1and þ1, espec i ely, o he ope a o s in (10). No ice ha e e y p obabili y in (7) includes only pai s o ios o mu ually compa ible es s. Expe imen .—The expe imen equi ed wo- es sequen- ces [ o ins ance, o ob ain Pð00j14Þ], and h ee- es se- quences [ o ins ance, o ob ain Pð010j012Þ]. We buil six de ices o he six dicho omic es s de ined in (10). The sequen ial es s we e pe o med using cascade se ups [9] like he one shown in Fig. 2. We es ed inequali y (7) using he spa ial pa h and pola iza ion o a single pho on ca ying a ou -s a e quan um sys em wi h he ollowing encoding: j0i¼j ;Hi;j1i¼j ;Vi;j2i¼j ;Hi;j3i¼j ;Vi;(11) whe e , ,H, and Vdeno e he ansmi ed pa h, e lec ed pa h, ho izon al, and e ical pola iza ion o he pho on, espec i ely. The cascade se up used o implemen wo sequen ial es s on a single pho on consis s o h ee pa s: s a e p epa a ion, es ing de ices, and de ec o s. The p epa a ion o he pola iza ion-spa ial pa h-encoded single-pho on s a e j c iis achie ed using a sou ce o H-pola ized single pho ons. This single-pho on sou ce consis s on an a enu- a ed s abilized na ow bandwid h diode lase emi ing a he wa eleng h o 780 nm. This lase o e s a long cohe - ence leng h. The wo-pho on coincidences we e se o a negligible le el by a enua ing he lase o a mean pho on numbe o 0.06 pe ime coincidence window. This sou ce is ollowed by a hal -wa e pla e (HWP) se a 22.5and a pola izing beam spli e (PBS), allowing he pho on o be dis ibu ed wi h equal p obabili y be ween he wo pa hs and wi h he igh pola iza ion Hand V, espec i ely [see Fig. 2]. Then, he pho on in he wo pa hs en e s he de ice o es ing x1 h ough he de ice’s inpu and ollows one o he wo possible ou pu s, which co espond o he alues þ1 and 1. A e each o he wo ou pu s, we placed a de ice o es ing x2. We used wo iden ical de ices o es ing x2. Finally, we placed a single-pho on de ec o (D) a he ou pu o he wo de ices x2. The same idea is used o sequences o h ee es s x1,x2, and x3, by adding ou de ices o measu ing x3and using eigh single-pho on de ec o s. De ices o measu ing he six es s de ined in (10) a e gi en in Fig. 3. Measu emen s 1 and 3 a e s anda d pola - iza ion measu emen s using a PBS and a HWP which map he pola iza ion eigens a e o he ope a o o j ; Hiand j ; Vi. The mapping o he eigens a es o es 0, namely ðj ij iÞ=ffiffiffi 2 p, was accomplished by in e e ing he wo pa hs in a 50=50 beam spli e (BS). A wedge (W) is placed in one o he pa hs o se he phase be ween bo h pa hs [see Fig. 3]. Tes s 2 and 5 a e ep esen ed by he enso p oduc o a spa ial pa h and a pola iza ion ope a o so hey ha e a ou -dimensional eigenspace. Howe e , since he es s need o be owwise and columnwise compa ible, only hei common eigens a es can be used o dis inguishing he eigen alues. Measu emen 4 equi es us only o dis inguish be ween pa hs and . We needed o ec ea e he eigen- s a es o he pe o med es s a e each mapping and be o e en e ing he nex es , since ou single- es de ices map eigens a es o a ixed spa ial pa h and pola iza ion. All in e e ome e s in he expe imen al se up we e based on a displaced Sagnac con igu a ion. The s abili y o hese in e e ome e s is e y high. We ob ained isibili ies o e 99% o phase insensi i e in e e ome e s, and anging be ween 90% and 95% o phase sensi i e in e e ome e s. We used silicon a alanche pho odiodes calib a ed o ha e FIG. 2 (colo online). (a) Scheme o sequen ial es s o x1and x2. The wo possible esul s o each es a e assigned he alues þ1and 1, and a e ep esen ed by whiche e lamp is lashing. (b) Cascade se up used o implemen wo sequen ial es s on a single pho on. I consis s o h ee pa s: s a e p epa a ion, es ing de ices, and de ec o s. The p epa a ion pa p oduces he pola iza ion-spa ial pa h-encoded single-pho on s a e j c i. The wo ou pu s o he de ice o es ing x1co espond o he wo possible esul s. A e each o hese wo ou pu s, we placed a de ice o es ing x2. Single-pho on de ec o s a e placed a each o he ou ou pu s o he wo de ices x2(see he main ex o de ails). he same de ec ion e iciency o single-pho on de ec ion. All single coun s we e egis e ed using an eigh -channel coincidence logic wi h a ime window o 1.7 ns. The aw de ec ion e en s we e ga he ed in a 10-second ime pe iod o each o he six expe imen al con igu a ions. The expe imen al esul s a e p esen ed in Table I. The e o s in he esul s we e deduced om he s anda d de- ia ion o 50 samples in he 10-second ime pe iod. The main sou ces o sys ema ic e o s we e he small impe - ec ions in he in e e ome e s and in he o e lapping o he ligh modes and he pola iza ion componen s. These a e he causes o he de ia ion o he expe imen al esul s om he ideal case obse ed in Table I. The ac ha some o he expe imen al esul s exceeded he co esponding ideal p e- dic ions was due o he lack o pe ec compa ibili y be- ween he sequen ial es s caused by he nonpe ec isibili ies o he in e e ome e s. Re e ence [5] explains how o deal wi h his loophole. F om he esul s in Table I, we can es ablish he ollow- ing expe imen al uppe bound o he noncon ex ual con en o he co ela ions: WNC 0:0658 0:0019:(12) This is he lowes expe imen al bound on he noncon ex- ual con en e e epo ed in any Bell o noncon ex uali y inequali ies expe imen . The p e ious lowes expe imen al uppe bound on he noncon ex ual (local) con en was 0:218 0:014 [22]. As in mos expe imen s o Bell and noncon ex uali y inequali ies wi h pho ons, we assumed ha he de ec ed pho ons we e an unbiased sample o he p epa ed pho ons. This assump ion is necessa y, since he de ec ion e i- ciency, wi hou aking in o accoun he losses in he se up, was 0.50 (a alue ob ained conside ing ha he de ec ion e iciency o he single-pho on de ec o s was 55% and he e iciency o he ibe coupling was 90%). Fu u e expe i- men s using he alded sou ces and single-pho on de ec o s o e y high e iciency [23,24] may close his loophole. Ou expe imen was in ended o be a p oo -o -p inciple expe imen o illus a e he powe o he g aph app oach [20] o single ou expe imen s wi h p ope ies on demand (in ou case, NC <Q¼C), and o expe imen ally obse e ully con ex ual co ela ions. Conclusions.—By using a new echnique based on g aph heo y [20], we ha e iden i ied and pe o med an expe i- men in which no hypo he ical pos quan um co ela ions sa is ying (1) can ou pe o m he con ex uali y o quan um co ela ions. Assuming ha he de ec ed pho ons a e a ai sample o hose emi ed by he sou ce and assuming ha he compa ibili y o he sequen ial es s is pe ec , he co ela ions obse ed in ou expe imen exhibi he la ges con ex uali y e e epo ed in any expe imen o Bell o noncon ex uali y inequali ies, and p o ide compelling e i- dence o he exis ence o ully con ex ual co ela ions (i.e., hose wi hou noncon ex ual con en ) in na u e. Mo eo e , we ha e demons a ed he use ulness o he app oach o quan um co ela ions based on g aph heo y [20] in iden i ying expe imen s wi h p ope ies on demand. We expec ha u he de elopmen s along hese lines will p o ide be e ools o iden i y and obse e phenomena o physical in e es . The au ho s hank M. Ra ˚dma k o his help du ing he expe imen , and A. Acı ´n, L. Aoli a, C. Bud oni, R. Gallego, P. Ma aloni, S. Se e ini, G. Vallone, and A. Win e , o s imula ing discussions. This wo k was suppo ed by he Swedish Resea ch Council (VR), he Resea ch Council o No way, he Spanish P ojec s No. FIS2008-05596, MTM2008-05866, and FIS2011- 29400, and he Wenne -G en Founda ion. [1] E. P. Specke , Dialec ica 14, 239 (1960). [2] J. S. Bell, Re . Mod. Phys. 38, 447 (1966). [3] S. Kochen and E. P. Specke , J. Ma h. Mech. 17,59 (1967). [4] A. Pe es, Quan um Theo y: Concep s and Me hods (Kluwe , Do d ech , 1995), p. 203. FIG. 3 (colo online). De ices o measu ing he six es s de ined in (10). 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