All-Ve sus-No hing P oo o
Eins ein-Podolsky-Rosen S ee ing
Jing-Ling Chen
1,2
, Xiang-Jun Ye
1,2
, Chun eng Wu
2,6
, Hong-Yi Su
1,2
, Ada
´n Cabello
3
, L. C. Kwek
2,4
&C.H.Oh
2,5
1
Theo e ical Physics Di ision, Che n Ins i u e o Ma hema ics, Nankai Uni e si y, Tianjin 300071, People’s Republic o China,
2
Cen e o Quan um Technologies, Na ional Uni e si y o Singapo e, 3 Science D i e 2, Singapo e 117543,
3
Depa amen o de
Fı
´sica Aplicada II, Uni e sidad de Se illa, E-41012 Se illa, Spain,
4
Na ional Ins i u e o Educa ion and Ins i u e o Ad anced S udies,
Nanyang Technological Uni e si y, 1 Nanyang Walk, Singapo e 637616,
5
Depa men o Physics, Na ional Uni e si y o
Singapo e, 2 Science D i e 3, Singapo e 117542,
6
Pilla o Enginee ing P oduc De elopmen , Singapo e Uni e si y o Technology
and Design, 20 Do e D i e, Singapo e 138682.
Eins ein-Podolsky-Rosen s ee ing is a o m o quan um nonlocali y in e media e be ween en anglemen
and Bell nonlocali y. Al hough Sch o
¨dinge al eady moo ed he idea in 1935, s ee ing s ill de ies a comple e
unde s anding. In analogy o ‘‘all- e sus-no hing’’ p oo s o Bell nonlocali y, he e we p esen a p oo o
s ee ing wi hou inequali ies ende ing he de ec ion o co ela ions leading o a iola ion o s ee ing
inequali ies unnecessa y. We show ha , gi en any wo-qubi en angled s a e, he exis ence o ce ain
p ojec i e measu emen by Alice so ha Bob’s no malized condi ional s a es can be ega ded as wo
di e en pu e s a es p o ides a c i e ion o Alice- o-Bob s ee abili y. A s ee ing inequali y equi alen o he
all- e sus-no hing p oo is also ob ained. Ou esul clea ly demons a es ha he e exis many quan um
s a es which do no iola e any p e iously known s ee ing inequali y bu a e indeed s ee able. Ou me hod
o e s ad an ages o e he exis ing me hods o expe imen ally es ing s ee abili y, and sheds new ligh on
he asymme ic s ee ing p oblem.
Quan um nonlocali y is an in aluable esou ce in nume ous quan um in o ma ion p o ocols. I is pa o a
hie a chical s uc u e
1
: quan um s a es ha ha e Bell nonlocali y
2
o m a subse o Eins ein-Podolsky-
Rosen s ee able s a es which, in u n, o m a subse o en angled s a es. The concep o s ee ing can
his o ically be aced back o Sch o¨dinge ’s eply
3
o he Eins ein-Podolsky-Rosen a gumen
4
, and i has since
been igo ously o mula ed by Wiseman, Jones, and Dohe y
1
.
Wi hin he s ee ing scena io, Alice p epa es a bipa i e sys em, keeps one pa icle and sends he o he one o
Bob. She announces ha he Bob’s pa icle is en angled wi h he s, and hus ha she has he abili y o ‘‘s ee ’’ he
s a e o Bob’s pa icle a a dis ance. This means ha she could p epa e Bob’s pa icle in di e en s a es by
measu ing he pa icle using di e en se ings. Howe e , Bob does no us Alice; Bob wo ies ha she may
send him some unen angled pa icles and ab ica e he esul s using he knowledge abou he local hidden s a e
(LHS) o his pa icles. Bob’s ask is o p o e ha no such hidden s a es exis .
The s udy o Bell nonlocali y ha e wi nessed phenomenal de elopmen s o da e wi h impo an widesp ead
applica ions
5–7
. I s exis ence can be demons a ed h ough wo di e en app oaches: he i s conce ns he
iola ions o Bell inequali ies, and he second elies on an all- e sus-no hing (AVN) p oo wi hou inequal-
i ies
8–11
. The AVN p oo shows a logical con adic ion be ween he local-hidden- a iable models and quan um
mechanics, and hus o e s an elegan a gumen o he nonexis ence o local-hidden- a iable models. Wha is
possible wi h Bell nonlocali y and local hidden a iables should also be possible wi h s ee ing and local hidden
s a es. In s a k con as o Bell nonlocali y, he s udy o s ee ing is s ill a i s in ancy. Recen wo ks like Re s. 1,12
pu s ee ing on i me g ounds. Like Bell nonlocali y, his opic is gene ally o b oad in e es , as i hinges on
ques ions pe aining o he ounda ions o quan um physics
13
, and a he same ime e eals new possibili ies o
quan um in o ma ion
14
. Eins ein-Podolsky-Rosen s ee ing can be de ec ed h ough he iola ion o a s ee ing
inequali y, which ules ou he LHS model in he same spi i in which he iola ion o a Bell inequali y ules ou he
local-hidden- a iable model. Recen ly, se e al s ee ing inequali ies ha e been p oposed and expe imen ally
es ed
15–18
. Ne e heless, s ee ing is a om being comple ely unde s ood and he subjec dese es u he
in es iga ion.
The AVN p oo o Bell nonlocali y
8–11
has been de eloped o ule ou any local-hidden- a iable models. Likewise,
i is in e es ing o ind ou i he e an analogous AVN p oo which can ule ou any LHS models o s ee ing. The
OPEN
SUBJECT AREAS:
QUANTUM
INFORMATION
THEORETICAL PHYSICS
QUBITS
QUANTUM MECHANICS
Recei ed
17 Janua y 2013
Accep ed
11 June 2013
Published
5 July 2013
Co espondence and
eques s o ma e ials
should be add essed o
J.-L.C. (cq chenj@nus.
edu.sg)
SCIENTIFIC REPORTS | 3 : 2143 | DOI: 10.1038/s ep02143 1
pu pose o his wo k is o p esen an a i ma i e answe o his ques ion
by showing ha Eins ein-Podolsky-Rosen s ee ing wi hou inequal-
i ies exis s in a wo-qubi sys em. This p oo is an analogy o AVN
a gumen o Bell nonlocali y wi hou inequali ies, and o e s ad an-
ages o e he exis ing me hods o expe imen ally es ing s ee abili y
as well as shedding new ligh on he asymme ic s ee ing p oblem. In
addi ion, a s ee ing inequali y based on he AVN p oo is also
ob ained.
Resul s
S ee ing wi hou inequali ies o wo qubi s.The wo-se ing s ee -
ing scena io can be desc ibed as ollows: a he beginning, Alice
p epa es a wo-qubi s a e
AB
. She keeps one qubi and sends he
o he o Bob. She hen announces ha i is en angled wi h he one she
holds (see Fig. 1), and ha she could emo ely ‘‘s ee ’’ his s a e by
p ojec i e measu emen s P^
n
a~z{1ðÞ
a^
n:~
s½
=2, wi h ^
n he mea-
su emen di ec ion, a(wi h a50, 1) he Alice’s measu emen
esul , he 2 32 iden i y ma ix, and ~
s~sx,sy,sz

he ec o
o he Pauli ma ices. Bob hen asks Alice o pe o m wo p ojec i e
measu emen s P^
n1
aand P^
n2
a(wi h ^
n1=^
n2) on he qubi and o ell
him he measu emen esul s o a. A e Alice’s measu emen has
been done, Bob ob ains he ou condi ional s a es ~
^
nj
a. Alice could
chea Bob i he e exis s an ensemble 2j j
no
(see he g ay box wi h
colo ed pa icles in Fig. 1) and a s ochas ic map 2aj^
A,j

om j o
a, such ha he ollowing equa ions hold,
~
^
nj
a~X
j
2a^
nj,j


2j j,a~0,1;j~1,2ðÞ:ð1Þ
In o de o Bob o be con inced ha Alice can s ee his s a e, Bob
needs o be su e ha no such hidden s a es a e indeed possible. I we
demand ha Bob’s s a es possess an LHS desc ip ion, hen his
densi y ma ices should sa is y Eq. (1). A con adic ion among he
ou equa ions, meaning ha hey canno ha e a common solu ion o
2j j
no
and 2a^
n,jjðÞÞ, con inces Bob ha an LHS model does no
exis and ha Alice can s ee he s a e o his qubi .
I is wo h men ioning ha he se o equa ions (1) plays an ana-
logous ole o he one in he s anda d G eenbe ge -Ho ne-Zeilinge
(GHZ) a gumen
8
. The p incipal di e ence be ween he a gumen s
is ha he se o equa ions in (1) deal wi h densi y ma ices whe eas
in he GHZ a gumen , each equa ion pe ains o he ou comes o
measu emen s and he e o e co esponds o eal numbe s. The
cons ain s imposed by LHS model on densi y ma ices a e much
s ic e han cons ain s imposed by eal numbe s. This p o ides an
in ui i e explana ion as o why AVN p oo would wo k o he
Eins ein-Podolsky-Rosen s ee ing o wo-qubi s a es.
Suppose ha Alice ini ially p epa es a p oduc s a e
AB
5
jy
A
æÆy
A
jfljy
B
æÆy
B
j. I can be e i ied ha , o any p ojec i e mea-
su emen P^
n
a(wi h P^
n
a=yA
ji
yA
hj
and y
A
y
A
) pe o med by
Alice, Bob always ob ains wo iden ical pu e no malized condi ional
s a es as ^
n
a~~
^
n
a ~
^
n
a~yB
ji
yB
hj
,(a50, 1), which means ha Alice
canno s ee Bob’s s a e. Mo eo e , Bob can ob ain wo iden ical pu e
no malized condi ional s a es i and only i
AB
is a di ec -p oduc
s a e. Hence, he ea e we assume ha ^
n
0and ^
n
1a e wo di e en
pu e s a es, i.e., ^
n
0= ^
n
1.
Fo a gene al
AB
, ^
n
aa e no pu e. I hey a e pu e, hen
AB
possesses he ollowing uni o m o m:
AB~P^
n
06~
^
n
0zP^
n
16~
^
n
1zz^
n
ji
{^
n
hj
6M
z{^
n
ji
z^
n
hj
6M{
,
whe e +^
njia e eigens a es o ^
n:~
s,Mis a 2 32 complex ma ix
unde he posi i i y condi ion o
AB
, and M{is he He mi ian
conjuga ion o M.
Fo
AB
, i is no di icul o ind ha M~0 i and only i
AB
is
sepa able, and he s a e
AB
admi s a LHS (which means ha i is no
s ee able) i and only i M~0 (see he Me hods sec ion). In a wo-
se ing s ee ing p o ocol o ^
n1,^
n2
g
, i Bob can ob ain wo di e en
pu e no malized condi ional s a es along Alice’s p ojec i e di ec ion
^
n1(o ^
n2), he ollowing h ee p oposi ions a e equi alen : (i) M=0.
(ii)
AB
is en angled. (iii) No LHS model exis s o Bob’s s a es, so
AB
is s ee able (in he sense o Alice s ee ing Bob’s s a e). We hus ha e
ou s ee ing a gumen concluded, and ha is gi en any wo-qubi
en angled s a e, he exis ence o ce ain p ojec i e measu emen by
Alice so ha Bob’s no malized condi ional s a es a e wo di e en
pu e s a es p o ides a c i e ion o Alice- o-Bob s ee abili y.
Al hough he s anda d GHZ a gumen is elegan o p o iding
a ull con adic ion be ween local-hidden- a iable model and quan-
um mechanics (wi h 100% success p obabili y) , i s alidi y is only
limi ed o some pu e s a es wi h high symme y, such as N-qubi
GHZ s a es and clus e s a es wi h N$3
19
. Ha dy a emp ed o
ex end he GHZ a gumen o an a bi a y wo-qubi sys em
9
.
Howe e , Ha dy’s a gumen wo ks o only 9% o he uns o a
specially cons uc ed expe imen . Mo eo e , Ha dy’s p oo is no
Figu e 1
|
The s ee ing scena io illus a ion. Alice i s p epa es a wo-qubi s a e and keeps one qubi . She hen sends he o he qubi o Bob and
announces ha i is en angled wi h he one she possesses (see he pai o ed balls and g een a ows). Thus she could emo ely ‘‘s ee ’’ Bob’s s a e by
p ojec i e measu emen s. Howe e , Bob does no us Alice and he wo ies ha she may ab ica e he esul s using he knowledge abou LHS. In he wo-
se ing s ee ing scena io, Bob asks Alice o pe o m wo speci ic p ojec i e measu emen s on he qubi (see he ed dashed a ow) and o le him know he
measu emen esul s (see he blue dashed a ow). A e Alice’s measu emen (see he measu emen de ice), Bob ob ains ou condi ional s a es (see he
dashed ci cle). Alice could chea Bob i he e exis s an ensemble (see he g ay box wi h colo ed pa icles) and a s ochas ic map, such ha he se o
equa ions (1) holds. To be con inced ha Alice can s ee his s a e, Bob needs o con i m ha no such hidden s a es a e possible.
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SCIENTIFIC REPORTS | 3 : 2143 | DOI: 10.1038/s ep02143 2
alid o wo-qubi maximally en angled s a e. To o e come his,
Cabello p oposed an AVN p oo o wo obse e s, each possessing
a wo-qubi maximally en angled s a e
10,11
. Nowadays, he e is no
AVN p oo o Bell nonlocali y o a genuine wo-qubi s a e p e-
sen ed. Howe e , we show ha o any wo-qubi en angled s a e
AB
, i he e exis s a p ojec i e di ec ion ^
nsuch ha Bob’s no malized
condi ional s a es ^
n
abecome wo di e en pu e s a es, hen Alice can
s ee Bob’s s a e. Ou s ee ing a gumen is no only alid o wo-
qubi pu e s a es, bu i is also applicable o a wide class o s a es
including mixed s a es.
The AVN p oo e sus he known s ee ing inequali ies.Le us
compa e ou esul wi h he known s ee ing inequali ies. Fi s , hey
play di e en oles in demons a ing s ee ing: s ee ing inequali y
ollows a simila app oach o he Bell inequali y o Bell nonlo-
cali y, while s ee ing wi hou inequali y se es as an analogous coun-
e pa o he GHZ es o Bell nonlocali y wi hou Bell inequali ies.
Secondly, ou a gumen shows ha he e a e many quan um s ee -
able s a es ha do no iola e any known s ee ing inequali ies. Fo an
example, conside he s a e
h
V~VYhðÞ
ji
YhðÞ
hj
z1{VðÞWhðÞ
ji
WhðÞ
hj
,ð2Þ
whe e jY(h)æ5cos hj00æ1sin hj11æ,jW(h)æ5cos hj10æ1sin hj01æ.
I is en angled when Vg[0, 1/2) <(1/2, 1] and hg(0, p/2). I can
be easily e i ied ha , o s a e (2), a e Alice pe o ms an ^
x-di -
ec ion measu emen on he qubi , Bob’s no malized condi ional
s a es a e jus wo di e en pu e s a es, cos hj0æ1sin hj1æand cos
hj0æ2sin hj1æ. Thus, based on ou AVN p oo o s ee ing, Alice can
always s ee Bob’s s a e using jus a wo-se ing p o ocol ^
x,^
z
g
.On
he o he hand, a class o N-se ing s ee ing inequali y SN~
1
NXN
k~1Ak~
sB
k

{CNƒ0 has been in oduced in Re . 15 o show
he abili y o Alice s ee ing Bob’s s a e. By unning a nume ical check
o a 10-se ing s ee ing inequali y o he abo e o m, we obse e ha ,
o some egions o Vand h, he s ee ing inequali y canno de ec he
s ee ing o s a e (2)(as shown in Fig. 2 a). The colo s deno e di e en
iola ion alues, as shown in he legend. The blank egion indica es
ha he s ee abili y o s a e (2) canno be de ec ed by eso ing o his
inequali y.
Finally, unlike quan um en anglemen and Bell nonlocali y, he
de ini ion o s ee ing is asymme ic
1,20
. Ou AVN p oo can shed ligh
on his p oblem. The s a e (2) is no symme ic unde a pe mu a ion
o Alice and Bob (e en wi h local uni a y ans o ma ions ac ing on
he s a e). The known s ee ing inequali ies in Re . 15 do no e eal
asymme ic s ee ing (see Fig. 2 a). Howe e , ou a gumen p esen s a
p omising way o e eal asymme ic s ee ing. Acco ding o ou AVN
p oo , he s a e (2) exhibi s wo-se ing asymme ic s ee ing. On one
hand, Alice can always s ee Bob’s s a e using jus he wo-se ing
p o ocol ^
x,^
z
g
. On he o he hand, a e Bob has pe o med a p o-
jec i e measu emen along an a bi a y ^
n-di ec ion on his qubi ,
Alice’s no malized condi ional s a es can ne e be cas in o wo
di e en pu e s a es, allowing o he exis ence o LHS models.
Take he s a e wi h pa ame e s V53/5 and h5p/8 as an example
(whose co esponding poin is ou side o he colo ed egion in Fig. 2
a): Nume ical esul s show ha , o any wo-se ing p o ocol
^
n1,^
n2
g
, he e is always a solu ion o LHS o Alice’s condi ional
s a es. In sho , his example illus a es a s a e in which he s ee ing
scena io is no in e changeable. This esul can be o p ac ical
impo ance, since asymme ic s ee ing has applica ions in one-way
quan um c yp og aphy
21
and may ha e po en ial applica ions in
o he ields o quan um in o ma ion p ocessing.
A s ee ing inequali y.I is known ha a Bell inequali y can be
de i ed om he GHZ a gumen
22
. This is also he case o he
s ee ing wi hou inequali ies a gumen . The s ee ing inequali y
equi alen o he AVN p oo eads
W3
hi
{CLHSƒ0,ð3Þ
subjec o he cons ain W1
hi
~W2
hi
~0. He e Wja e p ojec o s as
W1~P^
n
06 ^
n
0,W2~P^
n
16 ^
n
1,W3~z
ji
z
hj
6^
nB
ji
^
nB
hj
, wi h ^
n
a
o hogonal o ^
n
a,zji~z
^
njiz{
^
njiðÞ
ffiffiffi
2
p,^
nB
ji~cos hB
20jiz
sin hB
2eiQB1
ji
,Wj

~ Wj%AB

, and CLHS~max^
nB ^
nB
ji
^
nB
hj
~
^
n
0z

~
^
n
1Þ2Þ is he uppe bound o he LHS model. I s physical
implica ion can be desc ibed as ollows: Suppose Alice pe o ms a
p ojec i e measu emen in he ^
n-di ec ion and inds ha Bob can
ob ain wo di e en pu e no malized condi ional s a es, hen
W1
hi
~W2
hi
~0. They hen pe o m a join -measu emen W3(in
which Alice’s measu emen di ec ion is pe pendicula o ^
n-
di ec ion). Acco ding o Lemma 2 (see he Me hods sec ion), he
LHS model equi es M~0, hus he p obabili y W3
hi
is bounded
by C
LHS
. Howe e , wi h quan um mechanics, his bound is always
exceeded due o a non- anishing M.
Figu e 2
|
(a) De ec ing s ee abili y o he s a e (2) using he en-se ing s ee ing inequali ies. We explo e he s ee ing o s a e (2) ia iola ion o
he en-se ing inequali y p esen ed in Re . 15. The colo s deno e di e en alues o quan um iola ion, as scaled in he legend. The blank egion indica es
ha s ee abili y o (2) canno be de ec ed by his inequali y. Wi h he eplacemen Ak?
~
sA
kand ~
sB
k?Bkin he abo e inequali y, one ob ains a simila
s ee ing inequali y S’Nƒ0 o show Bob’s abili y o s ee ing Alice’s s a e. The inequali y S’N~10ƒ0yields he same iola ion egion. This indica es ha
s ee ing inequali ies in Re . 15 canno e eal asymme ic s ee ing. (b) De ec ing s ee abili y using he s ee ing inequali y (3). We show he s ee ing o he
s a e
col
h ough iola ion o inequali y (3). Quan um p edic ion o he le -hand-side o he inequali y always succeeds 0 unless V50o h50, p/2.
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SCIENTIFIC REPORTS | 3 : 2143 | DOI: 10.1038/s ep02143 3
As an ins ance, we in es iga e he s ee ing o s a e col~
VYhðÞ
ji
YhðÞ
hj
z1{VðÞ
col, wi h colo noise col~00
ji
00
hj
zð
11ji11hjÞ=2 by using ou inequali y (3). We ind ha Bob’s
condi ional s a es on Alice’s p ojec i e measu emen in he z-di -
ec ion a e wo di e en pu e s a es j0æÆ0jand j1æÆ1j, and he uppe
bound is C
LHS
5(1 1Vjcos 2hj)/4. The quan um p edic ion o he
le -hand-side o inequali y (3) eads 1
2Vsin2h o hg[0, p/4], and
1
2Vcos2h o hg[p/4, p/2], which do no anish unless V50o h
50, p/2 (see Fig. 2 b). The iola ion o he inequali y clea ly demon-
s a es ha he s a e
col
possesses s ee abili y excep V50o h50,
p/2.
Discussion
We ha e p esen ed an AVN p oo o Eins ein-Podolsky-Rosen s ee -
ing o wo qubi s wi hou inequali ies based on a wo-se ing s ee -
ing p o ocol. The a gumen is alid o any wo-qubi en angled s a e,
bo h pu e and mixed. We show ha many quan um s a es ha do no
iola e any known s ee ing inequali ies a e indeed s ee able s a es.
This p o ides a new pe spec i e o unde s anding s ee abili y and
o e s an elegan a gumen o he nonexis ence o LHS models wi h-
ou eso ing o s ee ing inequali ies. The esul also sheds new ligh
on he asymme ic s ee abili y – a phenomenon wi h no coun e pa
in quan um en anglemen and Bell nonlocali y. The esul is es able
h ough measu emen s o Bob’s condi ional s a es and p o ides a
simple al e na i e o he exis ing expe imen al me hod o de ec ing
s ee abili y
15–18
. Theo e ically, a wo-se ing s ee ing p o ocol can be
used o show ha no LHS models exis o
AB
i he s a e sa is ies he
condi ion gi en in ou AVN a gumen . Expe imen ally, he de e -
mina ion o he s ee abili y o a quan um s a e can be done by pe -
o ming quan um s a e omog aphy
23
on Bob’s qubi . Mo eo e , a
s ee ing inequali y is ob ained om ou AVN a gumen , and his
inequali y o e s ano he way o es s ee abili y o s a es. Like Bell
nonlocali y whose impo ance has only been ealized wi h he apid
de elopmen o quan um in o ma ion science, we an icipa e u he
de elopmen s in his exci ing a ea.
Me hods
We p o e wo Lemmas in he sec ion. The s ee abili y o
AB
is equi alen o ha o
he s a e %AB~UA6ðÞ AB U{
A6

. I is always possible o Alice o choose an
app op ia e uni a y ma ix U ha o a es he di ec ion ^
n o he di ec ion ^
z. The e o e,
we can ini ially se ^
n~^
zby s udying he s a e %AB ins ead o
AB
. A e Alice pe o ms
a p ojec i e measu emen in he ^
z-di ec ion, Bob’s unno malized condi ional s a es
a e
~
^
z
0~ A0
ji0
hj
6ðÞ%AB
½~m1Q1
ji
Q1
hj
,ð4aÞ
~
^
z
1~ A1ji1hj6
ðÞ
%AB
½
~m2Q2
jiQ2
hj,ð4bÞ
wi h m1~ ~
^
z
0

,m2~ ~
^
z
1

, ^
z
0~Q1
jiQ1
hj, and ^
z
1~Q2
jiQ2
hj. Then one has
%AB~m10ji0hj6Q1
jiQ1
hjzm21ji1hj6Q2
jiQ2
hj
z0ji1hj6Mz1ji0hj6M{:
Lemma 1.M~0 i and only i %AB is sepa able.
P oo .Look a he o m o %AB, ob iously M~0 implies %AB is sepa able. To p o e
he con e se, one needs he de ini ion o sepa abili y: %AB~Pipi Ai6 Bi, whe e
Ai
and
Bi
a e, espec i ely, Alice and Bob’s local densi y ma ices, and p
i
.0 sa is y
Pipi~1. Fo con enience, le mn
Ai m,n~1,2ðÞdeno e he elemen o Alice’s densi y
ma ix
Ai
. By calcula ing A0
ji0
hj
6ðÞ%AB
½and A1
ji1
hj
6ðÞ%AB
½, one has
Pipi 11
Ai Bi~m1Q1
ji
Q1
hj
,Pipi 22
Ai Bi~m2Q2
ji
Q2
hj
. Le Q
1
and Q
2
be wo pu e
s a es ha a e o hogonal o jQ
1
æand jQ
2
æ, espec i ely. No ice ha
Pipi mm
Ai Bi|Q
m
Q
m


~0,m~1,2ðÞ, hus, o any index i, we ha e
mm
Ai Bi Q
m
Q
m


~0, which esul s in
11
Ai 22
Ai Bi Q
1
Q
1


z Bi Q
2
Q
2


~0:ð5Þ
Since Q
1
=Q
2
, hey canno be simul aneously pe pendicula o he s a e
Bi
, hus
11
Ai 22
Ai~0, which yields 12
Ai~ 21
Ai~0 due o posi i i y condi ion o
Ai
.So
M~Pipi 12
Ai Bi~0. Lemma 1 is hence o h p o ed.
Lemma 2.The s a e %AB admi s a local-hidden-s a e (LHS) model (which means ha
i is no s ee able) i and only i M~0.
P oo .M~0 implies %AB is sepa able, hus %AB admi s a LHS model. Now we ocus
on he p oo o necessi y. I Alice’s measu emen se ing is ^
z,^
x
g
, hen one has
~
^
x
0~1
2m1Q1
jiQ1
hjzm2Q2
jiQ2
hjzMzM{

,ð6aÞ
~
^
x
1~1
2m1Q1
jiQ1
hjzm2Q2
jiQ2
hj{M{M{

:ð6bÞ
Subs i u e Eqs. (4a)(4b)(6a)(6b) in o Eq. (1) and due o Q
1
~
^
z
0Q
1
~0 and
Q
2
~
^
z
1Q
2
~0, one immedia ely has
j
g{jQ
1
æÆQ
1
j,jQ
2
æÆQ
2
j} o any j. Based on
which, Eqs. (6a) (6b) a e alid only i MzM{~axQ1
ji
Q1
hj
zbxQ2
ji
Q2
hj
ðÞ
=2, wi h
a
x
,bx[R. Simila ly, i Alice’s measu emen se ing is ^
z,^
y g, hen one has
M{M{~iayQ1
jiQ1
hjzbyQ2
jiQ2
hj
.
2, wi h a
y
,by[R. I he e exis s a LHS model
o Bob’s s a es, hen M~aQ
1
ji
Q1
hj
zbQ
2
ji
Q2
hj
, wi h a5a
x
1ia
y
,b5b
x
1ib
y
.
Subs i u e Min o Eq. (5), we ha e
%AB~m1Ta6Q1
ji
Q1
hj
zm2Tb6Q2
ji
Q2
hj
,
wi h Ta~1a
a0

and Tb~0b
b1

. Now we cons uc he ollowing wo
p ojec o s: Q1~x1
ji
x1
hj
6Q
2
Q
2
,Q2~x2
ji
x2
hj
6Q
1
Q
1
, whe e jx
1
æis he
eigen ec o o T
a
wi h eigen alue 1~1{ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1z4a
jj
2
q

2ƒ0, and jx
2
æis he
eigen ec o o T
b
wi h eigen alue 2~1{ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1z4b
jj
2
q

2ƒ0. Because %AB is a
densi y ma ix, one has
%ABQ1
ðÞ~ 1m1Q
2Q1


2§0,
%ABQ2
ðÞ~ 2m2Q
1Q2


2§0:
This leads o M~0. Lemma 2 is hence o h p o ed.
Th ee measu emen se ings we e men ioned in he p oo o Lemma 2. This does
no mean ha we need a h ee-se ing p o ocol o show s ee ing. Fo a gi en en angled
s a e %AB, a wo-se ing p o ocol is enough o demons a e s ee ing. Lemma 2 shows
ha MzM{and M{M{canno be linea ly expanded o jQ
1
æÆQ
1
jand jQ
2
æÆQ
2
j
simul aneously (because ha means M~0 and
AB
is sepa able). Fo a gi en %AB,i
MzM{=axQ1
ji
Q1
hj
zbxQ2
ji
Q2
hj
ðÞ
=2, hen using ^
z,^
x
g
o demons a e s ee ing,
o he wise using ^
z,^
y
g
.
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Acknowledgemen s
J.L.C. is suppo ed by he Na ional Basic Resea ch P og am (973 P og am) o China unde
G an No. 2012CB921900 and he NSF o China (G an Nos. 10975075 and 11175089). A.C.
is suppo ed by he Spanish P ojec No. FIS2011-29400. This wo k is also pa ly suppo ed
by he Na ional Resea ch Founda ion and he Minis y o Educa ion, Singapo e (G an No.
WBS: R-710-000-008-271).
Au ho con ibu ions
J.L.C. ini ia ed he idea. J.L.C., X.J.Y., H.Y.S. and C.W. es ablished he p oo . J.L.C., C.W.,
A.C., L.C.K. and C.H.O. w o e he main manusc ip ex . H.Y.S. and X.J.Y. p epa ed igu es
1 and 2. All au ho s e iewed he manusc ip .
Addi ional in o ma ion
Rep in s and pe mission in o ma ion is a ailable a www.na u e.com/ ep in s.
Compe ing inancial in e es s: The au ho s decla e no compe ing inancial in e es s.
How o ci e his a icle: Chen, J.-L. e al. All-Ve sus-No hing P oo o
Eins ein-Podolsky-Rosen S ee ing. Sci. Rep. 3, 2143; DOI:10.1038/s ep02143 (2013).
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