Multi-Component Bell Inequality and its Violation for Continuous Variable Systems

0.14 MB·English

by  Jing-Ling Chen

Multi-Component Bell Inequality and its Violation for Continuous Variable Systems
Jing-Ling Chen,1, Chunfeng Wu,1 L. C. Kwek,1,2 D. Kaszlikowski,1 M. Z˙ukowski,3 and C. H. Oh1
∗
1Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542
2Nanyang Technological University, National Institute of Education, 1, Nanyang Walk, Singapore 637616
3Instytut Fizyki Teoretycznej i Astrofizyki, Uniwersytet Gdan´ski, PL-80-952, Gdan´sk, Poland
Multi-component correlation functions are developed by utilizing d-outcome measurements.
Based on the multi-component correlation functions, we propose a Bell inequality for bipartite
d-dimensional systems. Violation of the Bell inequality for continuous variable (CV) systems is
investigated. The violation of the original Einstein-Podolsky-Rosen state can exceed the Cirel’son
bound,themaximalviolationis2.96981. Forfinitevalueofsqueezingparameter,violationstrength
of CV states increases with dimension d. Numerical results show that theviolation strength of CV
states with finitesqueezing parameter is stronger than that of original EPR state.
5
0 PACSnumbers: 03.65.Ud,03.65.Ta,03.67.-a
0
2
n I. INTRODUCTION ables for testing nonlocality for a given state. Bell has
a presentedalocalrealisticmodelforpositionandmomen-
J In their famous paper of 1935 [1], Einstein, Podolsky tum measurements based on spin-1/2 particles. He has

2 and Rosen (EPR) questioned the completeness of quan- alsoarguedthattheoriginalEPRstatewouldnotexhibit
2 tum mechanics, based on a gedankenexperiment involv- nonlocal effects since the Wigner function representa-
ing the position and momentum of two entangled par- tionofthe originalEPRstateis positiveeverywhereand
2
ticles. Einstein believed that there must be elements of thereforeadmitsalocalhiddenvariablemodel. Recently,
v
2 realitythatquantummechanicsignores. Itisarguedthat Banaszek and W´odkiewicz [4] invokedthe notion of par-
0 the incompletedescriptioncouldbe avoidedbypostulat- ity as the measurement operator and interpreted the
1 ing the presence of hidden variables that permit deter- Wigner function as a correlation function for these par-
0 ministicpredictionsformicroscopicevents. Furthermore, itymeasurements. TheythenshowedthattheEPRstate
1 hiddenvariablescouldeliminateconcernsfornonlocality. and the two-mode squeezed vacuum state do not have a
3 For a long time, EPR argument remained a philosophi- local realistic description in the sense that they violate
0
cal debacle on the foundation of quantum mechanics. In Bell inequalities such as the Clauser and Horne inequal-
/
h 1964, John Bell made an important step forward in this ity [5] and the Clauser-Horne-Shimony-Holt (CHSH)[6]
p direction by considering a version based on the entan- inequality. Thestartingpointofthedemonstrationin[4]
t- glement of spin-1/2 particles introduced by Bohm. Bell isthatthe two-modeWignerfunctioncanbe interpreted
n [2] showed that the assumption of local realism had ex- asacorrelationfunctionforthejointmeasurementofthe
a
u perimental consequences,andwas not simply anappeal- parity operator. In the limit r → ∞, when the original
q ing world view. In particular, local realism implies con- EPR state is recovered,a significant violation of Bell in-
: straints on the statistics of two or more physically sepa- equality takes place, however, the violation is not very
v ratedsystems. Theseconstraints,calledBellinequalities, strong. To avoid the unsatisfactory feature, Chen et al.
i
X canbe violatedby the statisticalpredictionsofquantum [7] introduced “pseudospin” operators based on parity,
r mechanics. due to the fact that the degree of quantum nonlocality
a Although most of the concepts in quantum informa- that we can uncover crucially depends not only on the
tiontheorywereinitiallydevelopedforquantumsystems given quantum state but also on the “Bell operator” [8].
with discrete variables, many quantum information pro- From reference [7], the violation of CHSH inequality for
tocols of continuous variables have also been proposed the original EPR state can reach the Cirel’son bound
[3]. In recent years, quantum nonlocality for position- 2√2.
momentumvariablesassociatedwithoriginalEPRstates
has attracted much attention. The originalEPRstate is
a commoneigenstateofthe relativepositionxˆ xˆ and In this paper, we propose a Bell inequality, which is
1 2
−
the totallinear momentum pˆ +pˆ andcan be expressed based on multi-component correlation functions, for bi-
1 2
as a δ-function: partite systems by utilizing d-outcome measurements.
Wetheninvestigateviolationoftheinequalityforcontin-
Ψ(x1,x2)= ∞ e(2πi/h¯)(x1−x2+x0)pdp. (1) uous variable systems. Due to the considered d-outcome
Z−∞ measurements, violation of original EPR state can ex-
It is importantto choosethe appropriatetype ofobserv- ceedCirel’son’sbound,themaximalvioaltionis2.96981.
The CV case with finite value of squeezing parameter is
alsostudied. TheviolationstrengthofCVstateswithfi-
nitesqueezingparameterisstrongerthanthatoforiginal
∗Electronicaddress: [email protected] EPR state.
2
II. BELL INEQUALITY FOR different local observables of d outcomes, labelled by
MULTI-COMPONENT CORRELATION 0,1,...,N(= d 1). We denote X the observable mea-
i
FUNCTIONS suredbyparty−X andx theoutcomewithX =A,B(x=
i
a,b). If the observers decide to measure A ,B , the re-
1 2
We consider a Bell-type scenario: two space-separated sultis(0,4)withprobabilityP(a1 =0,b2 =4). Thenlet
observers, denoted by Alice and Bob, measure two us introduce d N-dimensional unit vectors
v =(1,0,0,0, ,0,0)
0
···
1 √N2 1
v = , − ,0,0, ,0,0
1
− N N ···
(cid:18) (cid:19)
1 1 (N +1)N N 2 (N +1)N
v = , , − ,0, ,0,0
2
− N −NsN(N 1) N s(N 1)(N 2) ···
(cid:18) − − − (cid:19)
.
.
.
1 1 (N +1)N 1 (N +1)N 1 (N +1)N 1 (N +1)N
v = , , , , ,
N 1
− (cid:18)− N −NsN(N −1) −Ns(N −1)(N −2) ··· −Nr 3·2 Nr 2·1 (cid:19)
1 1 (N +1)N 1 (N +1)N 1 (N +1)N 1 (N +1)N
v = , , , , , (2)
N
(cid:18)− N −NsN(N −1) −Ns(N −1)(N −2) ··· −Nr 3·2 −Nr 2·1 (cid:19)
These d vectors satisfy following properties: WenowdefineaBellquantityforthemulti-component
correlationfunctions,
N N(N 1) (N 1)(N 2)
(i) vj =0 Bd = B(0)+s(N +−1)NB(1)+s (−N +1)N− B(2)
j=0
X
(ii) vj ·vk ≡−N1 (j 6=k) (3) +···+s(N2+·11)NB(N−1)
N 1
− (N +1 k)(N k)
= − − (k) (5)
Ford=2,itisjusttwovaluedvariables(i.e.,v0 =1,v1 = s (N +1)N B
k=0
1)obtainedfromameasurement. Ifthe measureresult X
−
of Alice is m, and Bob’s result is n (where m and n are where
less than N), we then associate a vector v for the
correlationbetween Alice and Bob [vm+n unmd+ernstood as B(0) = Q(101)+Q(102)−Q(201)+Q(202)
vt, where t=(m+n),modd]. Based on which, we now B(k) = Q(1k1)−Q(1k2)−Q(2k1)+Q(2k2), (k 6=0) (6)
construct multi-component correlation functions:
Any local realistic description of the previous Gedanken
experiment imposes the following inequality:
Q~ij = vm+nP(ai =m,bj =n) d+1
2 δ +(1 δ ) 2 (7)
Xm,n − 2d − 2d d 1 ≤Bd ≤
(cid:18) − (cid:19)
d
= v P(m+n=t) (4) Obviously,this inequalityreducesto the usualCHSHin-
t
equality for d=2.
t=0
X
Thequantumpredictionforthejointprobabilityreads
PQM(a =m,b =n)= ψ Pˆ(a =m) Pˆ(b =n)ψ (8)
where P(ai = m,bj = n) is the joint probability of ai i j h | i ⊗ j | i
obtain outcome m and bj obtain outcome n , and Q~ij = where i,j = 1,2; m,n = 0,...,N, Pˆ(ai = m) =
(Q(ij0),Q(ij1),Q(ij2),··· ,Q(ijN−1)), Q(ijk) represents the k-th UA†|mihm|UA is the projector of Alice for the i-th mea-
component of the vector correlation function Q~ . surement and similar definition for Pˆ(b =n).
ij j
3
It is well known that the two-mode squeezed vacuum Local realistic description imposes 4 2. Nu-
− ≤ B ≤
statecanbegeneratedinthenondegenerateopticalpara- merical results show that (r = 1.4068) 2.90638;
d=3
B ≃
metric amplifier (NOPA)[9] is (r ) = 4/(6√3 9) 2.87293. For (r
d=3 d=3
B → ∞ − ≃ B →
), the four optimal two-component quantum correla-
∞
tions read:
|NOPAi=er(a†1a†2−a1a2)|00i=nX∞=0(tcaonshhrr)n|nni, (9) Q~11 =Q~22 =Q~∗12 =(2√36+1,−2−6√3),
wherer >0isthesqueezingparameterand|nni≡|ni1⊗ Q~21 =( 1, 2),
a|nlsio2 =ben1w!(rait†1t)enn(aa†2s)n[4|]0:0i. The NOPA states |NOPAi can Q~ =−3(Q−x3)2+(Qy )2 = √5. (14)
| ij| ij ij 3
q
NOPA = 1 tanh2r dq dq′g(q,q′;tanhr)qq′ , One thing worth to note that one can treat those three
| i − | i
p Z Z two-dimensional vectors in terms of complex numbers,
where g(q,q′;x) ≡ √π(11−x2)exp −q2+2q(1′2−−x22q)q′x and nωa=meelxyp,[vi02π=/31],.vN1o=wωthaenBdevl2l i=neωq2uafolirtysibmepcloimciteys,[w11h]ere
qq q q , with q beihng the usual ieigen-
|sta′tie o≡f th|ei1po⊗sit|io′ni2operator|.iSince limx 1g(q,q′;x)= −4≤ Re[Q11+Q12−Q21+Q22]
δ(q q ), one has lim dq dq g(q,q→;tanhr)qq = + 1 Im[Q Q Q +Q ] 2. (15)
dq−qq′ = EPR , whr→ich∞is just the′ origi′nal EPR| st′aite. √3 11− 12− 21 22 ≤
| i | i R R
Thus, in the infinite squeezing limit, NOPA be- In this sense, we can check the generalized “parity” op-
r
Rcomes the original, normalized EPR|state. i|Fo→ll∞owing erator (ω)nˆ other than usual parity operator ( 1)nˆ, the
Brukner et al. [10], we can map the two-mode squeezed two-component correlation function also reads−
state onto a d-dimensional pure state:
Qij =hNOPA|UA† ⊗UB† (ω)nˆa+nˆb UA⊗UB|NOPAi(16)
d 1
sechr −
|ψdi= 1 tanh2dr (tanhr)n|nni. (10) wcahnerseuffiUAci,eBntalyretagkeneetrhaellmy aUs(3th)etrparnosdfuorcmtsaotfiotnhsreaensdpiwne-
n=0
− X
coherent operators:
p
If the measurement result of Alice is m photons, and
Bvob’s froersutlhteicsornrelpahtoiotnonbse,twweeenthAenliceasacnridbeBoab.veActnodr UA =eξ3Uˆ+−ξ3∗Uˆ− eξ2Vˆ+−ξ2∗Vˆ− eξ1Iˆ+−ξ1∗Iˆ−
m+n
P(ai = m,bj = n) is the joint probability of ai obtain where ξj = θ2je−iϕj [actually, the phases ϕj can be set
m photons and b obtain n photons. More precisely, for to be zero, since they do not affect the maximal viola-
j
the two-mode squeezed state one obtains following joint tion. Hence is a SO(3) rotation]. Iˆ , Uˆ and Vˆ
A,B
probability are pseudo-suU(3)-spinwhich can be realize±d by±the Foc±k
states as
PQM(a =m,b =n)= ψ Pˆ(a =m) Pˆ(b =n)ψ(11)
i j d i j d
h | ⊗ | i
Iˆ = ∞ 3n 3n+1, Iˆ = ∞ 3n+1 3n,
+
| ih | − | ih |
n=0 n=0
III. SOME EXAMPLES X X
1
Iˆ = (3n 3n 3n+1 3n+1),
z
2 | ih |−| ih |
For d = 3, we have three outcomes v = (1,0),
0
vth1e=NO(−P1A/2s,ta√t3e/i2s)d,ivvi2de=d(in−t1o/t2h,r−ee√g3r/o2u).psA, ncacomredliyn,gly Uˆ+ = ∞ |3n+1ih3n+2|, Uˆ− = ∞ |3n+2ih3n+1|,
n=0 n=0
X X
1
1 ∞ Uˆz = (3n+1 3n+1 3n+2 3n+2)
NOPA = tanh3nr3n 3n 2 | ih |−| ih |
| i coshr | i| i
nX=0(cid:18) Vˆ = ∞ 3n 3n+2, Vˆ = ∞ 3n+2 3n,
+tanh3n+1r3n+1 3n+1 + | ih | − | ih |
| i| i n=0 n=0
X X
+tanh3n+2r3n+2 3n+2 (12) Vˆ = 1(3n 3n 3n+2 3n+2).
| i| i z 2 | ih |−| ih |
(cid:19)
The two-component correlation functions depend on Operators Iˆ ,Iˆ , Uˆ ,Uˆ and Vˆ ,Vˆ form three
z z z
quantum version of joint probabilities SU(2) grou{ps±, re}spe{cti±vely,}and I{ˆ ,±Uˆ ,}Vˆ ,Iˆ,(Uˆ +
z z
Vˆ )/√3 forms a SU(3) group. { ± ± ±
PQM(a =m,b =n) z }
i j We can similiarly get (r =finite value) and (r
d d
= NOPAPˆ(a =m) Pˆ(b =n)NOPA (13) ) with different d. WBe list them partly in TBable→I.
i j
h | ⊗ | i ∞
4
Obviously, the degree of the violation increases with di- ity,i.e.,originalEPRstategotten,wecanfindthebound
mension d, and the violation strength of CV states with of the violation of multi-component correlation-function
finite squeezing parameter is stronger than that of origi- Bell inequality [12].
nalEPRstate. When squeezingparametergoesto infin-
[d/2] 1
− 2k 1 1
lim (r ) = lim 4d (1 )( )
d Bd →∞ d − d 1 2d3sin2[π(k+ 1)/d] − 2d3sin2[π( k 1+ 1)/d]
→∞ →∞ Xk=0 − 4 − − 4
2 ∞ 1 1
= [ ] 2.96981 (17)
π2 (k+1/4)2 − (k+3/4)2 ≃
k=0
X
hBdi d=5 d=10 d=15 d=20 d=25 equalityforbipartited-dimensionalsystemsisdeveloped
EPR 2.91055 2.9398 2.94973 2.95473 2.9577 accordingly. We then investigate violation of such Bell
NOPA 2.9886 3.03842 3.06836 3.08273 3.08932
with r (1.44614) (1.72082) (1.8366) (1.96562) (2.07377)
3.15
TABLE I: Violation of multi-component Bell inequality for
3.10
|NOPAi and |EPRi states with different d.
n
o
olati3.05
vi
It is interesting to note that for (r ), the four m
optimalmulti-componentquantumBcdorre→lati∞onssharethe antu3.00
u
q
same module: Q~ = 2d 1. When d tends to infinity,
| ij| 3−d 2.95
Q~ = 2/3. Howevqer, we do not have an analytical
ij
| |
way to find a bound for violation with finite squeezing 2.90
p
parameter. For this case, what we do is draw a graph to
0 50 100 150 200 250 300 350
see the variation of (r = finite value) with increasing
Bd Dimension d
dimension,seeFig.1. Wecalculatethemaximalquantum
violation for CV states with different d. The more the
value of dimension, the more difficult to find a maximal
FIG. 1: Violation of multi-component Bell inequality for CV
violation. Hence the violationstrengthpoints we getare
states with finitesqueezing parameter for different d.
for d 330. With these values, it is easy to see that the
≤
violation increases from slowly to slowly with increasing
of d. Which means that there exists a limit for quantum
violation when d goes to infinity. Until now, we do not inequality for continuous variables case. The degree of
haveanexactvalueofthelimit. Weuseasoftware,which the violation increases with dimension d, and the limit
can give experimence expressiongiven enough points, to of the violation for the original EPR state is found to
find a expression that describes the curve in Fig.1, be 2.96981,whichexceeds the Cirel’sonbound. The rea-
son for this is due to the fact that we consider d(> 2)-
B =3.12885 1.06535/d+2.13122/d2 2.19262e−d(18) outcomemeasurements. Numericalresultsshowthatthe
− −
violation strength of CV states with finite squeezing pa-
When d , quantum violation (B) goes to 3.12885.
→ ∞ rameter is stronger than that of original EPR state.
Hence,suchvaluecanbethoughtasanapproximatevio-
lationlimitforCVstateswithfinitesqueezingparameter. This work is supported by NUS academic research
grant WBS: R-144-000-089-112. J.L.C acknowledges fi-
nancial support from Singapore Millennium Foundation
IV. CONCLUSION and (in part) by NSF of China (No. 10201015).
Note added: While completing this work we learn a
Insummary,weconstructmulti-componentcorrelation similar result obtained in Ref. [13], which based on the
functions based on d-outcome measurements. A Bell in- CGLMP inequality [12].
5
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