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). The echnique used consis s o mapping he
eigens a es o he ope a o o he wo s a es j ; iand j ; i,
whe e is a pola iza ion s a e (see he main ex o de ails).
TABLE I. Expe imen al esul s o inequali y (7). The column
‘‘Ideal’’ e e s o he p edic ions o QM o an ideal expe imen .
P obabili y Expe imen al esul Ideal
Pð010j012Þ0:240 91 0:000 21 0.25
Pð111j012Þ0:301 87 0:000 20 0.25
Pð01j02Þ0:280 57 0:000 20 0.25
Pð00j03Þ0:503 75 0:000 14 0.5
Pð11j03Þ0:479 76 0:000 14 0.5
Pð00j14Þ0:475 11 0:000 34 0.5
Pð01j25Þ0:437 65 0:000 15 0.5
Pð010j345Þ0:242 96 0:000 51 0.25
Pð111j345Þ0:257 04 0:000 52 0.25
Pð10j35Þ0:247 51 0:000 35 0.25
3:4671 0:0010 3.5
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