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. C. acknowledges suppo om P ojec
No. FIS2014-60843-P, “Ad anced Quan um In o ma ion”
(MINECO, Spain), wi h FEDER unds, he FQXi La ge
G an “The Obse e Obse ed: A Bayesian Rou e o he
Recons uc ion o Quan um Theo y,”and he p ojec
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Wallenbe g Founda ion, Sweden).
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