Noncontextual wirings

Noncon ex ual Wi ings Ba ba a Ama al,1,2,3 Adán Cabello,4Ma celo Te a Cunha,5and Leand o Aoli a6,3,7 1Depa amen o de Ma emá ica, Uni e sidade Fede al de Ou o P e o, 35400-000 Ou o P e o, Minas Ge ais, B azil 2Depa amen o de Física e Ma emá ica, CAP—Uni e sidade Fede al de São João del-Rei, 36.420-000 Ou o B anco, Minas Ge ais, B azil 3In e na ional Ins i u e o Physics, Fede al Uni e si y o Rio G ande do No e, 59070-405 Na al, B azil 4Depa amen o de Física Aplicada II, Uni e sidad de Se illa, E-41012 Se illa, Spain 5Depa amen o de Ma emá ica Aplicada, IMECC-Unicamp, 13084-970 Campinas, São Paulo, B azil 6Ins i u o de Física, Uni e sidade Fede al do Rio de Janei o, Caixa Pos al 68528, Rio de Janei o, Rio de Janei o 21941-972, B azil 7ICTP Sou h Ame ican Ins i u e o Fundamen al Resea ch, Ins i u o de Física Teó ica, UNESP-Uni e sidade Es adual Paulis a R. D . Ben o T. Fe az 271, Bl. II, São Paulo 01140-070, São Paulo, B azil (Recei ed 6 June 2017; e ised manusc ip ecei ed 5 Decembe 2017; published 29 Ma ch 2018) Con ex uali y is a undamen al ea u e o quan um heo y necessa y o ce ain models o quan um compu a ion and communica ion. Se ious s eps ha e he e o e been aken owa ds a o mal amewo k o con ex uali y as an ope a ional esou ce. Howe e , he main ing edien o a esou ce heo y—a conc e e, explici o m o ee ope a ions o con ex uali y—was s ill missing. He e we p o ide such a componen by in oducing noncon ex ual wi ings: a class o con ex uali y- ee ope a ions wi h a clea ope a ional in e p e a ion and a iendly pa ame iza ion. We cha ac e ize hem comple ely o gene al black-box measu emen de ices wi h a bi a ily many inpu s and ou pu s. As applica ions, we show ha he ela i e en opy o con ex uali y is a con ex uali y mono one and ha maximally con ex ual boxes ha se e as con ex uali y bi s exis o a b oad class o scena ios. Ou esul s comple e a uni ied esou ce- heo e ic amewo k o con ex uali y and Bell nonlocali y. DOI: 10.1103/PhysRe Le .120.130403 In oduc ion.—Quan um con ex uali y e e s o he impossibili y o explaining he s a is ical p edic ions o quan um heo y in e ms o models whe e he measu emen ou comes e eal p eexis en sys em p ope ies ha a e independen o he con ex , i.e., on which o he compa ible measu emen s a e join ly pe o med [1,2]. Con ex uali y can be seen as a gene aliza ion o Bell nonlocali y [3] o he case whe e he spacelike sepa a ion es ic ion is emo ed, so ha single sys ems a e included. I hus ep esen s an exo ic, in insically quan um phenomenon wi h bo h un- damen al and p ac ical implica ions. Con ex uali y has ecei ed lo s o a en ion o e he las decade. On one hand, i has been expe imen ally s udied in a a ie y o physical se ups [4–8]. On he o he one, i is known o be a esou ce in magic-s a e [9–12] and measu emen -based [13] quan um compu ing, o andom numbe ce i ica ion [14], and o se e al o he in o ma ion-p ocessing asks in he Bell scena io o spacelike sepa a ed measu emen s [15]. This has mo i a ed conside able in e es in esou ce heo ies o bo h con ex uali y [16–18] and Bell nonlocali y [19–21]. Resou ce heo ies gi e powe ul amewo ks o he o mal ea men o a physical p ope y as an ope a- ional esou ce, adequa e o i s cha ac e iza ion, quan i i- ca ion, and manipula ion [22,23]. Thei cen al componen is a special class o ans o ma ions, called he ee ope a ions, ha ul ill he essen ial equi emen o mapping e e y ee (i.e., esou celess) objec o he heo y in o a ee objec . Whe eas esou ce- heo e ic app oaches o quan um nonlocali y a e highly de eloped [19–21,24–28], he ope a- ional amewo k o con ex uali y as a esou ce is s ill less de eloped. In Re s. [16,17], an abs ac cha ac e iza ion o he axioma ic s uc u e o a esou ce heo y o con ex uali y was done. Howe e , a conc e e speci ica ion o he ee ope a ions o con ex uali y was no gi en. Wi hou an explici pa ame iza ion o a physically mo i a ed class o ee ope a ions, a esou ce heo y signi ican ly loses applicabili y. Fo ins ance, in Re s. [16,17], an in e es ing measu e o con ex uali y, called he ela i e en opy o con ex uali y, was p oposed, bu only pa ial mono onici y unde a a he es ic ed subse o con ex uali y ee ope a ions was shown. Mono onici y (noninc ease unde he co esponding ee ope a ions) is he undamen al equi emen o a unc ion o be a alid quan i ie o a esou ce. He e, we ill his gap by in oducing he class o noncon ex ual wi ings. These a e he na u al noncon ex- uali y p ese ing physical ope a ions a hand in he de ice- independen scena io o black-box measu emen de ices, whe e one does no assume any a p io i knowledge o he s a e o he obse ables in ques ion. We de i e a iendly analy ical exp ession o gene ic noncon ex ual wi ings applicable o all nondis u bing boxes, so ha bo h quan um PHYSICAL REVIEW LETTERS 120, 130403 (2018) 0031-9007=18=120(13)=130403(6) 130403-1 © 2018 Ame ican Physical Socie y and pos quan um boxes a e co e ed. In addi ion, he amewo k is e sa ile in ha i allows o ans o ma ions be ween sys ems wi h di e en numbe s o inpu s and ou pu s as well as di e en compa ibili y cons ain s. Fu he mo e, we show ha , o he case o Bell es s, he wi ings educe o he canonical ee ope a ions o Bell nonlocali y [19–21]. Hence, he amewo k cons i u es a uni ied esou ce heo y o bo h con ex uali y and Bell nonlocali y in hei mos gene al o ms. As applica ions, i s we show ha an impo an quan i ie called ela i e en opy o con ex uali y is mono onic unde all noncon- ex ual wi ings, a p oblem le open in Re s. [16,17]. Then, o he b oad class o so-called cycle boxes, we show ha con ex ali y bi s exis s in he s onges possible sense: single boxes om which he en i e nondis u bing se can be eely ob ained wi h noncon ex ual wi ings. Nondis u bing boxes.—We conside a measu emen de ice wi h Nbu ons (inpu s) and Mligh s (ou pu s), wi h N,M∈N. No all bu ons a e compa ible, i.e., can be p essed join ly. Each subse o compa ible bu ons de ines a con ex [29–31]. Le X¼ 1;2;…;Ng ep esen he se o bu ons. The con ex s can be encoded in an inpu compa - ibili y hype g aph IX≔ χj⊆Xgj¼1;…;jIXj, whe e each hype edge χjcon ains he bu ons ha can be join ly p essed in con ex j, wi h jIXj he numbe o con ex s [30,31]. We say ha jis a maximal con ex i , o all 1≤j0≤jIXj,χj⊆χj0implies χj0¼χj. Simila ly, no all ligh s can u n on join ly. Le A¼ 1;2;…;Mgbe he se o ligh s. Then, each k h bu on has a se AðkÞ⊆Ao ligh s associa ed, one—and only one—o which u ns on upon p essing ha bu on. The numbe o ligh s on is hus always equal o he numbe o bu ons p essed. Hence, o he ligh s i is mo e con enien o wo k wi h mu ual exclusi i y cons ain s. These can be encoded in an ou pu exclusi i y hype g aph OA≔ AðkÞgk¼1;…;N, whe e AðkÞencodes he exclusi i y hype edge o bu on k∈X. We deno e by AðχÞ≔⋃ k∈χAðkÞ he subse o ligh s associa ed wi h all he bu ons in χ∈IX. In u n, no e ha di e en bu ons may sha e associa ed ligh s. We e e o XðlÞ≔ k∈X∶l∈AðkÞgas he subse o bu ons associ- a ed wi h ligh l∈A. We es ic h oughou o he case whe e only incompa ible bu ons can ha e common asso- cia ed ligh s. Tha is, o e e y l∈A, k; k0g⊆XðlÞis allowed only i k; k0g∩χ⊂ k; k0g o all χ∈IX. Fo any inpu hype g aph IXand ou pu hype g aph OA, we conside condi ional p obabili y dis ibu ions PAjX≔ pAjXða;χÞga∈ 0;1gM;χ∈IX:ð1Þ The M-bi s ing a≔ða1;aMÞ∈ 0;1gM ep esen s he s a e o all Mligh s: al¼0s ands o “l h ligh o ”and al¼1 o “l h ligh on”. Hence, pAjXða;χÞis he p ob- abili y o he ligh s being in s a e aupon p essing he bu ons in he subse χ, which is nonze o only i aassigns he s a e “on” o one, and only one, o he ligh s associa ed wi h each bu on in χ. Tha is, o each χ∈Iχ, pAjXða;χÞ≠0only i kaðkÞkh¼1, wi h aðkÞ≔ðalÞl∈AðkÞ he subs ing o ao ligh s associa ed wi h bu on kand kaðkÞkh he Hamming no m o (numbe o ones in) aðkÞ, o all k∈χ. We e e o any such PAjXas a box beha io ela i e o IXand OA. A specially ele an class is ha o nondis u bing beha io s: PAjXis nondis u bing i , o all χ, χ0∈IXwi h χ0⊂χ, X al∶l∉Aðχ0Þ pAjXða;χÞ¼pAðχ0ÞjX0ðaðχ0Þ;χ0Þ;ð2Þ wi h aðχ0Þ≔ðalÞl∈Aðχ0Þ he subs ing o ao ligh s associa ed wi h he bu ons in χ0(ins ead o he en i e con ex χ). The nondis u bance condi ion demands ha whene e wo con ex s ha e bu ons in common he ma ginal dis ibu ion o e he common bu ons is independen o he con ex . I is hus he analogue o he no-signaling condi ion in Bell scena ios [15]. Wi h his, we can a las p o ide a p ecise o mal de ini ion o he gene al ma hema ical objec s o he esou ce heo y. Namely, we call e e y se o inpu and ou pu hype g aphs IXand OA, espec i ely, oge he wi h a nondis u bing beha io PAjX ela i e o hem, a box, B≔ IX;OA;PAjXg:ð3Þ We call he se o all such nondis u bing boxes ND. In u n, he ee objec s o he heo y, i.e., he esou ce- less ones, a e gi en by he class NC ⊂ND o noncon ex- ual (NC) boxes, de ined by NC box beha io s. A beha io PAjXis NC i i admi s a NC hidden- a iable model, i.e., i , o all χ∈IXand a∈ 0;1gMwe ha e pAjXða;χÞ¼X λ pΛðλÞY l∈A DlðaljχðlÞ;λÞ;ð4Þ whe e Λis he hidden a iable, aking he alue λwi h p obabili y pΛðλÞ,χðlÞ≔χ∩XðlÞis he single-elemen subse [32] o χassocia ed wi h ligh l, and DlðaljχðlÞ;λÞ≔δ(al; lðχðlÞ;λÞ), whe e δ(al; lðχðlÞ;λÞ), wi h δ he K onecke del a, is he λ h NC de e minis ic esponse unc ion o he l h ligh gi en he inpu χðlÞ. The unc ion lencodes he de e minis ic assignmen o χðlÞ in o al o he λ h global de e minis ic s a egy inco po- a ing he cons ain s o OA. Tha is, i is such ha , o all λ, lð∅;λÞ¼0(l h ligh is o i no associa ed bu on is p essed, i.e., i χðlÞ¼∅) and lðχðlÞ;λÞ× l0ðχðl0Þ;λÞ¼0, whene e l; l0g⊆AðkÞ o any k∈X(no mu ually exclu- si e ligh s simul aneously on). No e ha , since ldepends only on χðlÞ(ins ead o he en i e con ex χ), Dlcan only gene a e NC beha io s in Eq. (4). In ac , we show in PHYSICAL REVIEW LETTERS 120, 130403 (2018) 130403-2 Sec. IV o he Supplemen al Ma e ial [33] ha , when he con ex s a e de ined by spacelike sepa a ed bu ons, exp ession (4) educes o he usual local hidden- a iable models o Bell nonlocali y [15]. Any box ou side NC is called con ex ual. I is a well-known ac ha measu emen s on quan um s a es can yield con ex ual boxes. Con ex uali y- ee ope a ions.—We conside composi- ions o he ini ial box Bwi h a p ep ocessing box BPRE ≔ IY;OB;PBjYg∈NC;ð5Þ and a ðb;ψÞ-dependen pos p ocessing box BPOSTðb;ψÞ≔ IZ;OC;PCjZ;ψ;bg∈NC;ð6Þ o all b∈ 0;1gjBjand ψ∈IY, as shown in Fig. 1.Yand Ba e, espec i ely, he se s o bu ons and ligh s o BPRE, and Zand C hose o BPOSTðb;ψÞ. Fo he composi ion o be possible, we demand ha he se o allowed ou pu s o BPRE is a subse o he allowed inpu s o B, and he same o Bwi h BPOST. To his end, we need o in oduce he ou pu compa ibili y hype g aph ¯ OAassocia ed o OA, gi en by all subse s α⊂Ao ou pu ligh s wi h a mos one ligh pe exclusi i y hype edge in OA:¯ OA≔ α⊂A∶jα∩AðkÞj≤1; k¼1;…;Ng, and simila ly o ¯ OB. Tha is, ¯ OAand ¯ OB gi e he compa ible combina ions o ligh s on, hose no iola ing any o he cons ain s in OAand OB, espec i ely. Then, we demand ha ¯ OB⊆IXand ¯ OA⊆IZ. Mo eo e , we allow PCjZ;b;ψ o ha e only a es ic ed dependence on ðb;ψÞ, in such a way ha each ou pu ligh o he pos p ocessing box is causally in luenced only by he inpu s and ou pu s o he p ep ocessing box ha a e associa ed wi h i . Tha is, we demand ha , o all b∈ 0;1gjBj,c∈ 0;1gjCj,ψ∈IY, and ζ∈IZ, pCjZ;b;ψðc;ζÞ¼X ϕ pΦðϕÞY n∈C DnðcnjζðnÞ;χðbÞ ½n;ψ½n;ϕÞ;ð7Þ wi h DnðcnjζðnÞ;χðbÞ ½n;ψ½n;ϕÞde ined analogously o DlðaljχðlÞ;λÞin Eq. (4). Simila ly o χðlÞ he e, ζðnÞis he single-elemen subse o ζassocia ed wi h ligh n∈C. In u n, we now in oduce he sho -hand no a ions χðbÞ ½nand ψ½n≔ψðχðbÞ ½nÞ[34]. The subse χðbÞ ½nis composed o he single bu on in χðζðnÞÞdi ec ly wi ed o some ligh on in b, whe eas ψ½nis he single-bu on subse o ψassocia ed o he ligh di ec ly wi ed o he bu on o χðbÞ ½n. These subse s a e all well de ined h ough he hype g aphs IXand OA, independen ly o he speci ic beha io PAjXin ques ion, as shown in Sec. I o he Supplemen al Ma e ial [33]. This is c ucial o he composi ion no o c ea e con ex uali y. Wi h his, we a e now in a good posi ion o in oduce he ee ope a ions o con ex uali y. De ini ion 1: Noncon ex ual wi ings.—We de ine he noncon ex ual wi ing wi h espec o he p e- and pos - p ocessing boxes desc ibed abo e, as he linea map WNC ha akes any ini ial box B∈ND, gi en by Eq. (3), in o a inal box B ≔WNCðBÞwi h N ≔jYjbu ons and M ≔jCjligh s, wi h WNCðBÞ≔ IY;OC;PCjYg;ð8Þ whe e PCjYis he inal beha io , gi en by pCjYðc;ψÞ ¼X a∈ 0;1gjAj b∈ 0;1gjBj pCjZ;b;ψðc;ζðaÞÞpAjXða;χðbÞÞpBjYðb;ψÞ;ð9Þ o all c∈ 0;1gjCjand ψ∈IY. We deno e he class o all such wi ings by NCW. Sel -consis ency o he heo y equi es ha NCW sa - is ies he ollowing p ope y, p o en in Sec. II o he Supplemen al Ma e ial [33]. Lemma 1: Nondis u bance p ese a ion.—The class o boxes ND is closed unde all wi ings in NCW. In addi ion, o gi e alid ee ope a ions, NCW mus ul ill he ollowing equi emen , p o en in Sec. III o he Supplemen al Ma e ial [33]. Theo em 1: Noncon ex uali y p ese a ion.—The class o boxes NC is closed unde all wi ings in NCW. In ui i ely, his is connec ed o he ac ha he compo- si ion o any h ee independen noncon ex ual boxes yields a inal box ha is also noncon ex ual (wi h h ee indepen- den noncon ex ual hidden a iables). NCW is, howe e , FIG. 1. A noncon ex ual wi ing WNC wi h espec o p e- and pos p ocessing boxes BPRE and BPOST, espec i ely, mapping an ini ial box Bin o a inal box WNCðBÞ. The bu ons and ligh s o WNCðBÞa e gi en by he bu ons o BPRE and he ligh s o BPOST, espec i ely. Only he ligh s (bu ons) o Bo he same colo can be on (p essed) a he same ime. The beha io o BPOST is causally in luenced by BPRE, bu in a es ic ed way such ha he s a is ics o each ou pu ligh o BPOST depends only on he bu ons and ligh s o BPRE ha a e associa ed wi h i (see ex ). As a esul , i Bis noncon ex ual so is WNCðBÞ. PHYSICAL REVIEW LETTERS 120, 130403 (2018) 130403-3 mo e powe ul han such composi ions because he p e- and pos p ocessing boxes he e a e no independen . S ill, he es ic ion o Eq. (7) enables noncon ex uali y p ese a ion (see Sec. III o he Supplemen al Ma e ial). Finally, in Sec. IV o he Supplemen al Ma e ial [33], we show ha , o spacelike sepa a ed measu emen s, NCW educes o local ope a ions assis ed by sha ed andomness, he canonical ee ope a ions o Bell nonlocali y [19–21]. Con ex uali y mono ones.—In Re . [16], a measu e o con ex uali y called he ela i e en opy o con ex uali y, RC, was in oduced. Fo an a bi a y box B∈ND, RCðBÞ≔min B∈NCSðBkBÞ:ð10Þ SðBkBÞis he ela i e en opy o Bwi h espec o B(see Sec. IVo he Supplemen al Ma e ial [33]), which measu es he dis inguishabili y o B om Bin a b oad class o scena ios [21]. Hence, RCðBÞquan i ies he dis inguish- abili y o B om i s closes (wi h espec o S) non- con ex ual box B, p o iding a di ec gene aliza ion o con ex uali y o he s a is ical s eng h o Bell nonlocali y p oo s [35]. The essen ial equi emen o a unc ion o be a alid measu e o a esou ce is ha i is mono onic (i.e., non- inc easing) unde he co esponding ee ope a ions. In Re . [16], he au ho s show, o quan um boxes, mono o- nici y o RCunde p obabilis ic mix u es o independen channels on each quan um obse able (each con ex ). This co esponds o a es ic ed subse o NCW [36]. He e, we show mono onici y o RCunde he whole class NCW and o all boxes B∈ND. Lemma 2: Mono onici y o RC.—Le B∈ND. Then, RC½WNCðBÞ ≤RCðBÞ o all WNC ∈NCW. The p oo (gi en in Sec. Vo he Supplemen al Ma e ial [33]) elies explici ly on he pa ame iza ion o NCW in Eq. (9). In e es ingly, also, ano he measu e o con ex uali y, he con ex ual ac ion CðBÞ[29,37], was ecen ly shown o be mono onic unde some speci ic classes o con ex uali y- ee ope a ions [18]. A s aigh o wa d calcula ion (see Sec. VI o he Supplemen al Ma e ial [33]) shows ha CðBÞ is also mono onic unde he NCW class. Lemma 3: Mono onici y o C.—Le B∈ND. Then, C½WNCðBÞ ≤CðBÞ o all WNC ∈NCW. Con ex uali y bi s.—The ope a ional amewo k de el- oped allows us o s udy con ex uali y in e con e sions. A na u al ques ion is whe he he e exis s a box om which all boxes, o ixed inpu and ou pu hype g aphs, can be ob ained o ee (i.e., h ough noncon ex ual wi ings). This is in ima ely connec ed o quan i ica ion: such a supe io box can be aken as a uni o con ex uali y, o con ex uali y bi , yielding a na u al and unambiguous (measu e-inde- penden ) de ini ion o maximally con ex ual boxes. He e we answe ha ques ion a i ma i ely o a b oad class gi en by he so-called N-cycle boxes (see Fig. 2). A N-cycle box has as many maximal con ex s as bu ons (N), each k h maximal con ex consis s o wo bu ons (kand kþ1), each k h bu on belongs o wo maximal con ex s (χkand χk−1) and has wo associa ed ou pu ligh s, he (2k−1) h and he ð2kÞ h ligh s, so ha M¼2N. Modulo Nis implici ly assumed o he labels o bu ons, con ex s, and ligh s. These boxes admi 2N−1con ex uali y bi s: Lemma 4: Exis ence o con ex uali y bi s.—Fo any N≥3, all N-cycle boxes in ND can be eely ob ained om an N-cycle box wi h beha io PðγÞ AjXo componen s pðγÞ AjXða;χÞ ≔1 2;i χ¼ k; k þ1gand a2k−s¼a2ðkþ1Þ−sþγk; 0;o he wise; ð11Þ o all s∈ 0;1gand k∈X, wi h γ≔ðγ1;…;γNÞ, such ha γk¼0o 1 and kγkhis an odd in ege . Equa ion (11) desc ibes any o he 2N−1con ex ual N-cycle beha io s ex emal in ND, de i ed (in a di e en no a ion) and shown o be equi alen unde noncon ex ual elabelings o ou pu s in Re . [41]. The p oo o he lemma, gi en in Sec. VII o he Supplemen al Ma e ial [33], consis s hen o showing ha any con ex mixing o such elabelings is in NCW. Fo he pa icula case N¼4( he CHSH scena io), he beha io s in Eq. (11) become equi - alen o he no-signaling ex emal box o [42], known o gene a e all no-signaling boxes unde local wi ings assis ed by sha ed andomness [19–21]. Lemma 5 hus gene alizes his ac o a bi a y N≥3and noncon ex ual wi ings. Finally, i is impo an o men ion ha , o e en N, he bu ons can be spli in o wo disjoin subse s o N=2 incompa ible bu ons each, and he ligh s can be educed om 2N o only 4 (one mu ually exclusi e pai pe subse o bu ons), as in he chained inequali ies [38]. This is an al e na i e ep esen a ion o he same physical box. Ou o malism is o ally e sa ile in his sense, as i can di ec ly deal wi h any chosen ep esen a ion o a box. Final discussion.—Recen in es iga ions sugges ha con ex uali y may be a key esou ce o quan um FIG. 2. N-cycle g aphs CN o N¼3, 4, 5, and 6 bu ons. A N- cycle box is such ha he union o all hype edges in IXequals CNand each inpu bu on has i s own pai o ou pu ligh s. Fo e en N, he class is also in ima ely connec ed o he well-known chained inequali ies o Bell nonlocali y [38]. I includes he Clause -Ho ne-Shimony-Hol (CHSH) scena io [39], whe e he inpu s de ine he squa e C4, and he Klyachko-Can-Binicioˇ glu- Shumo sky one [40], whe e he inpu s o m he pen agon C5. Fo any N≥3, he e exis con ex uali y bi s, i.e., maximally con- ex ual N-cycle boxes om which all o he N-cycle boxes can be ob ained o ee (see ex ). PHYSICAL REVIEW LETTERS 120, 130403 (2018) 130403-4 ad an ages in a ious in o ma ion-p ocessing asks [9–14]. He e we ake a s ep o wa d owa ds con ex uali y as an ope a ional esou ce by in oducing and cha ac e izing noncon ex ual wi ings. In con as o mo e abs ac app oaches [16,17], noncon ex ual wi ings ha e a clea ope a ional in e p e a ion and admi a iendly analy ical pa ame iza ion. This is use ul o classi y, quan i y, and manipula e con ex uali y as a o mal esou ce. Fo ins ance, he ques ion o mono onici y o con ex uali y was un il ecen ly unclea . While in Re s. [16–18] mono onoci y o he ela i e en opy o con ex ually and o he con ex ual ac ion is p o en unde some speci ic ope a ions, he e we ha e se led he p oblem o mono onici y unde all non- con ex ual wi ings o bo h con ex ually measu es. Fu he mo e, we ha e also shown ha maximally con- ex ual single boxes ha se e as con ex uali y bi s exis o all cycle boxes, which encompass impo an Bell scena ios [39,40] and play a c ucial ole in con ex uali y heo y [43–48]. This esul can also be ex ended o boxes wi h mo e ou pu s [24]. In e es ing ques ions a e, e.g., wha he simples box admi ing inequi alen (no eely in e con- e ible) classes o con ex uali y is and wha he simples one allowing o con ex uali y dis illa ion. Finally, we ha e shown ha , o Bell scena ios, noncon ex ual wi ings educe o he usual ee ope a ions o Bell nonlocali y [19–21], which is in e es ing in i sel . Hence, ou indings yield a main missing ing edien o a comple e, uni ied esou ce heo y o con ex uali y and Bell nonlocali y. The p esen wo k was ini ia ed du ing he wo kshop “Quan um Co ela ions, Con ex uali y, and All Tha … Again”a he In e na ional Ins i u e o Physics (IIP), Na al, B azil. The pa icipan s o he wo kshop as well as he hospi ali y o IIP a e g a e ully acknowledged. We hank D. Ca alcan i, C. Dua e, M. Pusey, R. Soa es Ba bosa, S. Mans ield and an anonymous Re e ee o ui ul discus- sions. B. A. hanks he Ins i u o de Ma emá ica Pu a e Aplicada (IMPA) o he hospi ali y a Rio de Janei o, B azil. B. A., M. T. C., and L. A. acknowledge inancial suppo om he B azilian minis ies MEC and MCTIC and agencies CNPq, CAPES, FAEPEX, FAPERJ, FAPESP, and INCT-IQ. A. 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