Table Of ContentCompleteproofofGisin’s theorem forthree qubits
Sujit K. Choudhary,1,∗ Sibasish Ghosh,1,† GuruprasadKar,2,‡ and Ramij Rahaman2,§
1TheInstituteof Mathematical Sciences, C. I.T. Campus, Taramani, Chennai 600113, India
2PhysicsandAppliedMathematicsUnit,IndianStatisticalInstitute,203,B.T.Road,Kolkata700108,India
Gisin’stheoremassuresthatforanypurebipartiteentangledstate,thereisviolationofBell-CHSHinequality
revealingitscontradictionwithlocalrealisticmodel.Whether,similarresultholdsforthree-qubitpureentangled
states,remainedunresolved. Weshowanalyticallythatallthree-qubitpureentangledstatesviolateaBell-type
inequality,derivedonthebasisoflocalrealism,byexploitingtheHardy’snon-localityargument.
PACSnumbers:03.67.-a,03.65.Ud,42.50.-p
9
0 Not all measurementcorrelations in some state of a com- latesaBell-typeinequality.Thissituationchangesdrastically
0
positequantumsystemcanbedescribedbylocalhiddenvari- whenweconsiderHardy-typeNLWIforthreetwo-levelsys-
2
abletheory(LHVT)[1], a factwhichissaid to be“the most tems [14, 15], where all but one of the joint probabilities in
n
profounddiscoveryofscience”[2]. Experimentalverification theabove-mentionedsetarezero.Everymaximallyentangled
a
ofthisfact(i.e.,whethermeasurementcorrelationsinNature stateofthreequbits[16]satisfiesHardy-typeNLWIforsuit-
J
obey quantum rules or LHVT) goes also in favour of Quna- ablychosenpairsofnon-commutingdichotomicobservables
5
1 tum Theory, modulo some loopholes [3]. Every LHVT de- perqubit[17].Moreover,anattemptwasmadeinref.[17]to-
scription[4] ofthe measurementcorrelationsof a composite wardsachievingtheresultthateverypurestateofthreequbits,
] system(assumedtobefinitedimensionalinthepresentpaper) havinggenuinetri-partite entanglement, satisfies Hardy-type
h
givesrisetoone(or,morethanone)linearinequality(or,in- NLWI.Butinabsenceofthediscoveryofcanonicalformfor
p
- equalities)involvingthesecorrelations[5].Therearestatesof three-qubitpurestates (whichwasdonelater in ref. [18]), it
t
n acompositequantumsystem,whichviolatesomeorallthese didnotyieldacompleteproof.
a inequalitiesforsuitablechoicesofthesubsystemobservables. Now, from the set of joint probabilities in any NLWI ar-
u
gument, one can, in principle, construct a linear inequality
q Gisin’s theorem assures that for any pure entangled state
involvingthese joint probabilitiesby usinglocal realistic as-
[ of two-qudits, the above-mentioned violation is generic for
sumption. Thisinequalityis automaticallyviolatedby every
two settings per site (i.e., for the choice of one between two
2 quantumstate whichsatisfiesthecorrespondingNLWIargu-
v non-commutingobservablesperqudit)[6].Inotherwords,all
ment.InthecaseofHardy-typeNLWIargumentfortwotwo-
9 pureentangledstatesoftwod-dimensionalquantumsystems
level systems, this inequality (given in equation (11) of ref.
6 violateasingleBell-typeinequalitywithtwosettingspersite,
6 [19],equation(11)ofref.[20],andequation(26)ofref.[21])
where the choice of the observables depends on that of the
1 is nothing butthe correspondingCH inequality[22]. So, by
state. However,validityofthistheoremformulti-partitesys-
1. temsisstillnotguaranteed. Forexample,foroddN,thereis Gisin’s theorem, every two-qubit pure entangled state (irre-
0 spective of its amount of entanglement) will violate the for-
a familyofentangledpurestatesofN qubits, eachofwhich
9 merinequality. Inthisletter,weshowanalyticallythatevery
satisfies allBell-typeinequalitiesinvolvingcorrelationfunc-
0
three-qubitpureentangledstateviolatesalinearinequalityof
: tions, arising out of measurement of one between two non-
v theabove-mentionedtype(seeeqn.(3)below)involvingjoint
commutingdichotomicobservablesperqubit[7].Later,Chen
i probabilitiesassociatedwiththeHardy-typeNLWI,irrespec-
X etal. [8]providedaBell-typeinequalityinvolvingjointprob-
tive of whether the state has genuine tripartite entanglement
r abilities,associatedtomeasurementofonebetweentwonon-
a orpurebi-partiteentanglement.
commuting dichotomic observables per qubit, which is vio-
Hardy-typeNLWIargumentstartsfromthefollowingsetof
latedbyallthestatesoftheabove-mentionedfamily[9]. But
fivejointprobabilityconditionsforthreetwo-levelsystems:
a single Bell-type inequalityis notguaranteedto be violated
byallpureentangledstatesofthree-qubits,althoughthereare
P(D1 =+1,U2 =+1,U3 =+1) = 0,
claimsofhavingnumericalevidencesinfavourofthisviola-
P(U1 =+1,D2 =+1,U3 =+1) = 0,
tion[8,10,11].
P(U1 =+1,U2 =+1,D3 =+1) = 0, (1)
QuantumTheoryalsoshowscontradictionwithLHVTvia P(D1 = 1,D2 = 1,D3 = 1) = 0,
− − −
‘non-localitywithoutinequalities’(NLWI)[12]. Inthiscase, P(U1 =+1,U2 =+1,U3 =+1) > 0,
a set of values of joint probabilities of outcomes of mea-
surements of one between two non-commuting observables whereeachU (aswellasD )isa +1, 1 -valuedrandom
j j
{ − }
per site, has contradictionwith LHVT butcanbe realized in variable. Thissetofconditionscannotbesatisfiedbyalocal
QuantumTheory. Unfortunately,NLWIisweakerthanBell- realistictheory,andhenceithascontradictionwithLHVT[14,
type inequalities, as no maximally entangled state of two- 17].
quditsseemstoshowNLWI(inthecaseofHardy-typeNLWI, Toshowthatinquantumtheorytherearestates whichex-
this has been shown in [13]) even though each of them vio- hibit this kind of non-locality, we replace U and D by the
j j
2
+1, 1 -valued observables Uˆ and Dˆ respectively with
j j
{ − } TABLEI:Classificationof ψ .
[Uˆj,Dˆj] = 0. The probabilities appearing in (1) are expec- | i
6 Condition case
tationvaluesofonedimensionalprojectorscorrespondingto
λ0λ1 =0, λ2 =λ3 =λ4=0 (A.1)
thefollowingfiveproductvectors: 6
λ0=0,λ1 =λ2 =λ3 =λ4=0 (A.2)
6
|Dˆ1 =+1i|Uˆ2 =+1i|Uˆ3 =+1i, λλ00λ=1λ02,λ=1λ0,4eλi3φ==λλ42λ=30 ((AB..13))
Uˆ1 =+1 Dˆ2 =+1 Uˆ3 =+1 , λ0λ1λ3 6=0,λ2 =λ4 =0 (B.2)
| i| i| i 6
Uˆ1 =+1 Uˆ2 =+1 Dˆ3 =+1 , 0<λ0λ2 <1/2,λ1 =λ3 =λ4=0 (B.3)
| i| i| i 0<λ0λ3 <1/2,λ1 =λ2 =λ4=0 (B.4)
|DˆUˆ11==−+11i|DˆUˆ22==−+11i|DˆUˆ33==−+11i,. λ0=0and√2(cid:18)λ1λe3iφ λλ42 (cid:19) (B.5)
| i| i| i
isneitherasingularmatrixnora
unitarymatrix
λ0λ2 =1/2,λ1 =λ3 =λ4 =0 (C.1)
One can easily check that these five vectors are linearly in-
λ0λ3 =1/2,λ1 =λ2 =λ4 =0 (C.2)
dependent and hence span five dimensional subspace of the
eight dimensional Hilbert space associated to the total sys- λ0=0and√2 λ1eiφ λ2 (C.3)
(cid:18) λ3 λ4 (cid:19)
tem. Henceonecanchooseanyone(amonginfinitelymany) isaunitarymatrix
vector which is orthogonalto the first four vectors and non- λ0λ1λ2λ3λ4 =0,φ>0 (D.1)
6
orthogonal to the last one. Actually this result shows that λ0λ1λ2λ3λ4 =0,φ=0,λ2λ3 =λ1λ4 (D.2)
6 6
for any choice of observables in the above-mentioned non- λ0λ1λ2λ3λ4 =0,φ=0,λ2λ3 =λ1λ4 (D.3)
6
commuting fashion, one can always find a quantum state λ0λ1λ2λ3 =0,λ4 =0 (D.4)
6
whichexhibitscontradictionwithlocalrealism[23]. λ0λ1λ2λ4 6=0,λ3 =0 (D.5)
λ0λ1λ3λ4 =0,λ2 =0, λ0 =λ4 (D.6)
But, in this letter, our purposeis to find the converse. We 6 6
λ0λ1λ3λ4 =0,λ2 =0, λ0 =λ4 (D.7)
would like to see whether every three-qubit pure entangled λ0λ1λ4 =60,λ2 =λ3 =0 (D.8)
stateexhibitscontradictionwithlocalrealism. Inthiscontext, λ0λ3λ4 6=0,λ1 =λ2 =0 (D.9)
6
it has to be mentioned that Gisin’s theorem could provide a λ0λ2λ3λ4 =0,λ1 =0,λ2 =λ4 (D.10)
6 6
single prescriptionfor finding the observablesfor any bipar- λ0λ2λ3λ4 =0,λ1 =0,λ2 =λ4 (D.11)
6
tite pure state to show violation of the Bell-CHSH inequal- λ0λ2λ3 =0,λ1 =λ4 =0 (D.12)
6
ity, duetothe existenceofSchmidtdecomposition. Schmidt λ0λ2λ4 6=0,λ1 =λ3 =0 (D.13)
λ0λ4 =0,λ1=λ2 =λ3 =0 (D.14)
decomposition, in its strict sense [24], is absent for systems 6
comprisingofthreeandmoresubsystems. Thisgivesrise to
complicationsandoneneedstofindtheobservablesforeach
qubitsinsuchawaythattheconditionofnon-localitywithout
inequivalentcase (dependinguponthevaluesoftheparame-
inequalityholds. Nowinourthree-qubitcase,wefirstchoose
tersdescribingthestate)separately. Inthisdirection,westart
withanarbitrarythree-qubitpurestate ψ ,whichcanalways |Uˆ3 = +1i = (1/√2)(|χi+|χ⊥i)and|Dˆ3 = +1i = |χ⊥i,
| i where χ⊥ χ = 0. We can then choose two pairs of non-
betakeninthecanonicalform[18]: commuhting|diichotomicobservables(Uˆ1,Dˆ1)and(Uˆ2,Dˆ2)in
ψ =λ0 000 +λ1eiφ 100 +λ2 101 +λ3 110 +λ4 111 , such a way that the state |ηi satisfies Hardy’s NLWI condi-
| i | i | i | i | i | i tionsfortwotwo-levelsystemscorrespondingtotheseobserv-
(2)
where0 λ (forj = 0,1,2,3,4), 4 λ2 = 1, and0 ables. Thisimmediatelyshowsthatthestate ψ satisfiesthe
≤ j j=0 j ≤ Hardy-typeNLWIcondition(1). Ascondition|(1i)issymmet-
φ π. P
≤ ricwithrespecttothequbits,weseethatforeachofthecases
We now fully classify the above-mentionedthree-qubitstate
(B.1)-(B.5), thestate ψ willsatisfytheHardy-typeNLWI
ψ into four major classes: (A) ψ is a fully productstate,
| i
| i | i argument(1).
(B) ψ has pure two-qubit non-maximal entanglement, (C)
ψ |hais pure two-qubit maximal entanglement, and (D) ψ Again,let ψ = η χ ,where η isatwo-qubitmaxi-
|hasigenuinepurethree-qubitentanglement.Dependingon|thei mallyentangl|edistat|eoif⊗th|eifirstandt|heisecondqubits,while
valuesofλ′sandφ,intableI,wefurtherclassifyeachofthese χ is a state of the third qubit. If we now demand ψ to
i | i | i
four classes into several sub-classes: (A) consists of (A.1) - satisfy (1), it will immediately follow that η must satisfy
| i
(A.3); (B) consists of (B.1) - (B.5); (C) consists of (C.1) - Hardy’sNLWIconditionsfortwotwo-levelsystems–anim-
(C.3);(D)consistsof(D.1)-(D.14). possibility [12]. As above, we see that in none of the cases
If ψ has only bi-partite non-maximal entanglement we (C.1)- (C.3), state ψ willsatisfy the Hardy-typeNLWI ar-
thenfi|rsitconsiderthesituationwhere ψ = η χ ,with η gument. | i
| i | i⊗| i | i
being a two-qubitnon-maximallyentangled state of the first If ψ have genuine tri-partite entanglement then to show
| i
and the secondqubits, while χ is a state of the third qubit. that it satisfies the Hardy-type NLWI argument(1), one can
| i
Hardy [12] has shown that for all two-qubit non-maximally choosethethreepairsof +1, 1 -valuednon-commutating
entangledpurestates,onecanchooseobservablesforboththe observables(Uˆ ,Dˆ )(wh{erej−isa}ssociatedwithj-thsystem
j j
3
(j =1,2,3))asfollows: Condition (i) is equivalent to assigning definite values to
theobservables. InanyLHVT,theexperimentalprobabilities
Uˆj =+1 =kj(αj 0 +βj 1 ), Dˆj =+1 =lj(γj 0 +δj 1 ), wouldbereproducedinthefollowingway:
| i | i | i | i | i | i
P(A = i ) = ρ(ω)dωP (A = i ), P(A =
where 0 < k l (α γ∗ +β δ∗), k l (α δ β γ ) < 1, k k Ω ω k k k
k α 2 + k|βj j2 =j jl γ 2j+j l|δ| j2j= 1j,jan−dkj’sj, l|’sare ik,Al = il) = Ωρ(ω)RdωPω(Ak = ik,Al = il), P(Ak =
|thej njo|rmal|izjatijo|n con|sjtanjt|s (fo|rjj j=| 1,2,3). Thje valjues of ik,Al = il,AmR = im) = Ωρ(ω)dωPω(Ak = ik,Al =
α , β , γ , δ are given in table II for all the cases (D.1) - il,Am =im),where Ωρ(ω)dRω =1.
j j j j NowconsiderthefoRllowingquantity
(D.14).
We now derive a linear inequality (mentionedin equation
(7) of ref. [25] for n qubits) involving the joint probabili- B(ω) = Pω(D1 = 1)Pω(D2 = 1)Pω(D3 = 1)
− − −
ties in equation (1), starting from local realistic theory. For + Pω(D1 =+1)Pω(U2 =+1)Pω(U3 =+1)
this, we first assume that all the experimental probabilities + Pω(U1 =+1)Pω(D2 =+1)Pω(U3 =+1)
P(A = i ), P(A = i ,A = i ), P(A = i ,A =
k k k k l l k k l + Pω(U1 =+1)Pω(U2 =+1)Pω(D3 =+1)
i ,A = i ) (with A U ,D , and i +1, 1
l m m k k k k
for k,l,m = 1,2,3, and∈k ={ l = m}) can be d∈es{cribed−by} Pω(U1 =+1)Pω(U2 =+1)Pω(U3 =+1).
−
6 6
alocalhiddenvariableω,definedontheprobabilityspaceΩ
with probability density ρ(ω). For local realistic theory, the Onecaneasily checkthatB(ω) 0 forallω Ω. Then
≥ ∈
probabilitieswouldsatisfythefollowingconditions: obviously
i)P (A =i )(fori +1, 1 withk =1,2,3)canonly
ω k k k
∈{ − }
takevalues1or0.
ρ(ω)dωB(ω) 0,
ii)Pω(Ak = ik,Al = il) = Pω(Ak = ik)Pω(Al = il), ZΩ ≥
P (A = i ,A = i ,A = i ) = P (A = i )P (A =
ω k k l l m m ω k k ω l
i )P (A =i ). which,inturn,givesrisetothefollowingBell-typeinequality:
l ω m m
P(D1 = 1,D2 = 1,D3 = 1)+P(D1 =+1,U2 =+1,U3 =+1)+P(U1 =+1,D2 =+1,U3 =+1) (3)
− − −
+P(U1 =+1,U2 =+1,D3 =+1) P(U1 =+1,U2 =+1,U3 =+1) 0.
− ≥
ThusweseethateveryLHVTsatisfiestheinequality(3). here as the ‘threshold visibility’ of the state ψ . Lower the
| i
From our above-mentioned discussion on Hardy-type amountofthresholdvisibility,highertheamountofnoisethe
NLWI, it follows that every three-qubit pure state will vio- inequality can sustain. The maximum negative violation of
latetheinequality(3)unlessitisafullyproductstateorithas theinequality(3)bytheGHZstateisnumericallyfoundtobe
purebi-partitemaximalentanglement.Wenowshowthatthis −0.175459(approx.),andsothethresholdvisibilityvGthHrZ of
inequalityisevenviolatedwhen ψ haspurebi-partitemaxi- thisstateturnsouttobe0.68125(approx.),whichisapprox-
malentanglement,although,inth|isicase, ψ doesnotsatisfy imately same as that found in [11]. On the other hand, the
theHardy-typeNLWIcondition(1). With|ouitlossofgeneral- maximum negative violation of the inequality (3) by the W-
ity, we can take ψ in this case as: ψ = (1/√2)(00 + state(1/√3)(001 + 010 + 100 )isnumericallyfoundto
|D1ˆ11i)=⊗+|01i. C=ho|o0sie, |UUˆˆ12 == ++11i ==| ((i√0.20.096|+0i√+00.9|.26|11ii)),, boeft−hi0s.s1t9a2te6t0u8|rn(sapoipurtotxo|.)b,ea0in.d66s|0o6t6h7ei6th(arpesphrooxld.),vwishibicilhityisvaWtlhsor
|Dˆ2 = +1i =| 1i ,| Uˆ3 = +i1 = (1|/√i 2)(0 +|1i) approximatelysamewiththevalue0.660668ofvWthr,foundin
|Dˆ3 = +1i= 1| .i W|ith this choiice, ψ will v|ioilate|thie [11]. ItistobenotedthatsofarasthevaluesofvGthHrZ,vWthr
i|nequality(3i). | i | i are concerned, although the probabilistic Bell-type inequal-
ity (18) of ref. [11] and the above-mentionedinequality (3)
Thuswehaveestablishedthedesiredresultthateverypure
provideapproximatelythesamevalues,unlikeinequality(3),
entangledstateofthreequbitsviolatestheBell-typeinequality
neitherinequality(18)of[11]noranyotherinequality,men-
(3).
tionedintheliteraturetilldate(see,forexample,[8,10,11]),
If a three-qubit state ψ violates the inequality (3) max-
isanalyticallyguaranteedtobeviolatedbyallpureentangled
| i
imally corresponding to the set (ψ) of three pairs of non-
statesofthreequbits. Byconsideringamodifiedformofthe
commuting observables (Uˆ1,Dˆ1S), (Uˆ2,Dˆ2), and (Uˆ3,Dˆ3), inequality (3) (e.g., inequality (11) of [20]), one may get a
thentheminimumvalueofthecoefficientv [0,1]forwhich
lowervalueofthethresholdvisibilityforthestates.
∈
the state ρ(ψ,v) v ψ ψ + ((1 v)/8)I (I being the
≡ | ih | −
8 8identitymatrix)alsoviolatestheinequality(3),iscalled One may also try to find similar feature (i.e., violation of
×
4
TABLEII:Observablesforgenuinetri-partitepureentanglement.
Case Setofobservablesfordifferentcases
(D.1),(D.2),(D.4),(D.5) α1 = λ1, β1 = λ0eiφ, γ1 = 0, δ1 = 1; α2 = 1, β2 = 0, γ2 = λ2λ3eiφ λ1λ4, δ2 = λ1λ2
;α3 =λ2eiφ, β3 =− λ1, γ3=1, δ3 =0 −
(D.3) αλ11+=λ02,,ββ13==1λ,2γ1−=λ1λ,0γλ31,=δ1λ2=, δ(13−=λ2o)λ;1α,w2h=erλe,1ττ−=λλ3ǫ20,λ3β(2λ=1+λ3λτ2+),λǫ1=ǫ, λγ220λ=1(λλ31,+δ2λ=2)−+λ(11; α3λ20=)
(D.6) α1 = 0, β1 = 1, −γ1 = λ1e−iφ(λ24 λ20−), δ1 = λ0(1 λ20); α2 = λ3(1 λ20), β2 = λ1e−iφ−(1
λ24), γ2=λ3, δ2 = λ1e−iφ; α3 =−1, β3 =0, γ3−=λ4(−1 λ24), δ3 =λ3(λ−24 λ20) − −
− − −
(D.7) α1 = λ1e−iφ, β1 = λ0, γ1 = 0, δ1 = 1; α2 = λ3, β2 = λ1e−iφ, γ2 = 1, δ2 = 0; α3 = 1, β3 =
− −
0, γ3 =λ0, δ3 = λ3
−
(D.8) α1 =0, β1 =1, γ1 =λ1e−iφ(ǫ+λ4), δ1 = λoǫ;α2 =1, β2 =1, γ2 =λ4, δ2 = ǫ;α3 =ǫ, β3 =
λ1e−iφ, γ3 =λ4, δ3 = λ1e−iφ;withǫbeing−asolutionofz2(1 λ24)+zλ4(1 λ20)+−λ44 =0
(D.9),(D.10) α1 =λ2(λ22+λ24)+λ4(−1 λ20), β1 = λ0λ3λ4, γ1=1, δ1 =−0;α2 =1, β2 =−1, γ2=λ4, δ2 = λ2;
α3 =0, β3 =1, γ3=λ3λ−4, δ3 =λ22+−λ24 −
(D.11) α1 = 0, β1 = 1, γ1 = λ22λ3, δ1 = λ0(λ22 + λ23); α2 = λ22 +λ23, β2 = λ22, γ2 = 1, δ2 = 0;
−
α3 =1, β3 =0, γ3=λ3, δ3 =λ2
(D.12) α1 =0, β1 =1, γ1 =δλ0λ2λ3, δ1 =λ32δ+λ33;α2 =1, β2 =1, γ2 =λ3, δ2 = λ2δ;α3 =1, β3 =
δ, γ3 =λ2, δ3 = λ3;withδbeingasolutionofz2λ42+zλ2λ3+λ43 =0 −
−
(D.13) α1 = 1, β1 = 1, γ1 = λ2, δ1 = λ0ǫ;α2 = 1, β2 = 0, γ2 = λ4, δ2 = (λ0ǫ+λ2);α3 = ǫ, β3 =
1, γ3 =λ2, δ3 = λ0;withǫbeing−asolutionofz2λ40+zλ0λ2(λ20+λ22)+λ−22(λ22+λ24)=0
(D.14) α1 =1, β1 =1, γ−1=iλ0, δ1 = λ4;α2 =1, β2 =1, γ2 =iλ0, δ2 = λ4;α3 =λ24, β3 =iλ20, γ3=
− −
λ4, δ3 = λ0
−
Bell-type inequality, derived from Hardy-type NLWI argu- [8] J.-L.Chen,C.Wu,L.C.Kwek,andC.H.Oh,Phys.Rev.Lett.
ment, by all pure entangled states) in the case of n-partite 93(2004)140407.
quantumsystems. [9] Needlesstosaythatthechoiceoftheobservablesheredepends
onthatofthestate.
[10] C.Wu,J.-L.Chen,L.C.Kwek,andC.H.Oh,Phys.Rev.A77
(2008)062309.
ACKNOWLEDGMENTS
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(2008)032107.
R.R. acknowledges the support by CSIR, Government of [12] L.Hardy,Phys.Rev.Lett.71(1993)1665;L.Hardy,Phys.Rev.
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KaushikandS.Ghosh(inpreparation).
ofprobabilisticinequalities,liketheonegiveninequation(3),
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in the case of any NLWI, and also pointed out the equiva-
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two two-level systems and the correspondingCH inequality. [16] Any pure stateof three qubits, which islocallyunitarilycon-
Authors would also like to thank Samir Kunkri for valuable nectedtotheGHZstate(1/√2)(000 + 111 ),istakenhere
| i | i
discussiononGisin’stheorem. asamaximallyentangledstate.
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‡ Electronicaddress:[email protected] [22] S. M. Roy, D. Atkinson, G. Auberson, G. Mahoux, and V.
§ Electronicaddress:ramij˙[email protected] Singh,Mod.Phys.Lett.A22(2007)1717.
[23] S. K. Choudhary, S. Ghosh, G. Kar, S. Kunkri, R. Rahaman,
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ψ CId1 CId2 ... CIdN is given by ψ =
[4] Whichisalsoassumedtobe‘EPRlocal’,i.e.,onesubsystem’s | i ∈ ⊗ ⊗ ⊗ | i
probabilitydistributionformeasurementoutcomesisindepen- di=1√λi|φ1ii⊗|φ2ii⊗...⊗|φNii, where λi’s are non-
[5] dEe.nRt.oLfothuebecnheotisc,earoXfitvh:e0m80e4a.s4u0r4e6m[eqnutaonnt-opthh]e.rsubsystems. inP,eig′a=tiv1e,,2hφ,.j.i|.φ,jdi′i =mδinii′df1o,rda2ll,.j..=,d1N,2,....,N andforall
≡ { }
[6] N.Gisin,Phys.Lett.A154(1991)201;N.GisinandA.Peres, [25] J.L.Cereceda,Phys.Lett.A327(2004)433.
Phys.Lett.A162(1992)15.
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Phys.Rev.Lett.88(2002)210402.
