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Bell inequality for qubits based on the Cauchy-Schwarz inequality Jing-Ling Chen1,∗ and Dong-Ling Deng1 1Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin 300071, People’s Republic of China (Dated: January 14, 2009) We develop a systematic approach to establish Bell inequalities for qubits based on the Cauchy- Schwarz inequality. We also use the concept of distinct “roots” of Bell function to classify some well-known Bell inequalities for qubits. As applications of the approach, we present three new and tight Bell inequalities for four and threequbits,respectively. PACSnumbers: 03.65.Ud,03.65.Ta,03.67.Mn 9 0 0 Bell inequality has been regarded as “the most pro- N = odd, the WWZB inequality cannot yet be violated 2 found discovery in science” [1]. It is at the heart of the for the whole region ξ [0,π/2]. For the three-qubit ∈ n study of nonlocality and is the most famous legacy of case, such a difficulty has been overcome in Ref. [11], a the late physicist John S. Bell [2]. The inequality shows where a probabilistic Bell inequality was proposed and J that the predictions of quantum mechanics are not in- consequently Gisin’s theorem for three qubits naturally 4 tuitive, and touches upon fundamental philosophical is- returned. Recent development also indicates that Bell 1 suesthatrelatetomodernphysics. Bell-testexperiments inequality is not unique when one studies Gisin’s theo- serve to investigate the validity of the entanglement ef- rem for three qubits, in Ref. [12] three of such inequali- ] h fect in quantum mechanics by using some kinds of Bell ties have been listed and compared. All the inequalities p inequalities, however they to date overwhelmingly show mentioned above belong to the two-setting Bell inequal- - that Bell inequalities are violated. These experimental ities for N qubits, i.e., they are based on the standard t n results provide empirical evidence against local realism Bell experiment, in which each local observer is given a a [3] and in favor of quantum mechanics. choice between two dichotomic observables. Some sig- u nificantgeneralizationshavebeenmadeformulti-setting q Bell’s applaudable progress has stirred a great furor. Bell inequalities for N qubits [13] as well as two-setting [ Many people have been attracted in this problem and Bell inequalities for high-dimensional systems [14, 15]. 2 extensive work on Bell inequalities has been done, in- Notably, in Ref. [16] a multi-setting Bell inequality that v cluding both theoretical analysis and experimental test. violate the generalized GHZ state for the whole region 7 ThefamousClauser-Horne-Shimony-Holt(CHSH)[4]in- ξ [0,π/2] was found. 28 equality is a kind of improved Bell inequality that is ∈In this paper, we shall focus on Bell inequality for 2 more convenient for experiments. Now, it is well-known qubits. It is natural to ask two questions: (i) As an in- . thatallpureentangledstatesoftwotwo-dimensionalsys- equality, does Bell inequality have any deep connections 8 tems (i.e., qubits) violate the CHSH inequality and the with some ancient mathematic inequalities, such as the 0 8 maximum quantum violation is the so-called Tsirelson’s Cauchy-Schwarzinequalityormoregenerally,theHo¨lder 0 bound 2√2. Mermin, Ardehali, Belinskii, and Klyshko inequality? (ii)Sofar,manykindsofBellinequalitiesfor : have separately generalized the CHSH inequality to the qubitshaveemerged,evenforthreequbits. Canthesein- v N-qubit case, which now known as the MABK inequalities be classified in an efficient way? The purpose i X equality, and proved that quantum violation of this of this paper is to (i) develop a systematic approach to r inequality increases with the number of particles [5]. establishBellinequalitiesforqubitsbasedontheCauchy- a In 2001, Scarani and Gisin [6] noticed that the gen- Schwarzinequality,and(ii)classifysomewell-knownBell eralized Greenberger-Horne-Zeilinger (GHZ) states [7]: inequalities based on their distinct “roots”. Let us start ψ GHZ = cosξ 0 0 +sinξ 1 1 do not violate the from the weighed Ho¨lder inequality since it is a more | i | ··· i | ··· i MABKinequalityforsin2ξ 1/√2N−1 (TheGHZ state generalinequalitywhichcontainstheCauchy-Schwarzin- ≤ is for ξ = π/4). In Refs. [8, 9] a general correlation- equality as a special case. function N-qubit Bell inequality has been derived, here- The Cauchy-Schwarz inequality and Bell inequalities. after we call it as the Werner-Wolf-Z˙ukowski-Brukner Let f(λ) and g(λ) be any two real functions for which (WWZB)inequality. TheWWZBinequalityincludesthe f(λ)p and g(λ)q are integrable in Γ with p > 1 and MABK inequality as a special case. Ref. [10] shows that |1+1|=1,th|enth|eweighedHo¨lderinequalityreads[17]: p q (a) For N = even, although the generalized GHZ state 1 1 doesnotviolatetheMABKinequality,itdoesviolatethe p q WWZB inequality, and (b) For sin2ξ 1/√2N−1 and fgρ(λ)dλ  f pρ(λ)dλ  g qρ(λ)dλ ,(1) ≤ Z ≤ Z | | Z | | Γ  Γ   Γ  where Γ is the total λ space and ρ(λ) is a statistical dis- ∗Electronicaddress: [email protected] tribution of λ, which satisfies ρ(λ) 0 and dλρ(λ) = ≥ Γ R 2 1. When p = q = 2, the above inequality reduces each inequality. In this step, we finally achieve some to the Cauchy-Schwarz inequality: fgρ(λ)dλ 2 nontrivialBellinequalities,suchasthetightinequalities. f 2ρ(λ)dλ g 2ρ(λ)dλ . It may(cid:2)RbΓe very inte(cid:3)res≤t- Herewepresentanexampletoilluminatethismethod. (cid:2)inRgΓ|to|conside(cid:3)r(cid:2)tRhΓe|ca|ses whe(cid:3)n p, q take various values. Example 1: Derivation of theCHSH inequality. Letus However,in this paper, we restrict our study to the case look at the simplest case, i.e., N =2,M =2. Let with p=q =2, which is easier for the calculations. f(λ) = C +C [X +X ]+C [X +X ] Consider N spatially separatedparties and allow each 0 1 1,1 2,1 2 1,2 2,2 of them to choose independently among M observables, +C3X1,1X2,1+C4[X1,1X2,2+X1,2X2,1] determined by some local parameters denoted by λ . +C X X , 5 1,2 2,2 Let X (n ,λ), or X for simplicity, denote observ- j kj j,kj g(λ) = D0+D1[X1,1+X2,1]+D2[X1,2+X2,2] ables on the j-th qubit, each of which has two possible +D X X +D [X X +X X ] outcomes 1 and 1. From the viewpoint of local real- 3 1,1 2,1 4 1,1 2,2 1,2 2,1 − ism, the values of Xj’s are predetermined by the local +D5X1,2X2,2, (3) hidden variable (LHV) λ before measurement, and in- dependent of any measurements, orientations or actions then we get a series of equations of Cj’s and Dj’s (here performed on the other parties at spacelike separation. we omit them for sententiousness). After solving these The correlation function, in the case of a local real- equations,wefinallychoosethesolutions: C0 andD0 are flikRasoΓjttliliΠ=coowNjnt=1ih,nf1ue2Xgon,,rj·cwy(·tn,i·eok,inpjsM,rQλet.hs)(eρenFnn(kotλ1r)d,adcneλosfikyn,2nsv,etwe·den·mh·iaeea,srnnteciQkcejj,(a)nwpa=kpes1r,dQon1eakn,kc122ohk,,t2··et···o··t·k·hje,,.senNtIkacnjob)arltirnhs=edhe- aiQDhsfr21b(n1λi=ot−)trihaD2Qirhny22gg2=r()beλ2a0u)≤l,it2Dnt4wuh3,mee=ohbfraaDem(vrQ4seo,1=ua1CsB+−1CeD=lHQl5iS1nC,2Het2+hqiu=enQnaelq20fi1rtu,yoCa−mla3iQsthy=:f2!(2(QC)λ/41)21g=+(≤λQ)−1i1.2C2+≤I5t, Bellinequalitiesforqubits basedonthe Cauchy-Schwarz Similarly,the MABK andthe WWZB inequalities can inequality. It contains three main steps: also be derived with the same approach, although the Step 1: Connecting functions f(λ) and g(λ) with the computationbecomesabitmorecomplicatedwhenN in- observables. f(λ) andg(λ) in inequality (1) canbe func- creases. However, the above approach can be improved tions of the observables of the parties. To express the further with the aid of the distinct “roots” of the Bell functions more concisely, let X =1, then one has function. Whatdowemeanthe“roots”ofthe Bellfunc- j,0 tion? For instance, let the left-hand side of the CHSH inequality (λ) = (X X + X X + X X N N 1,1 2,1 1,1 2,2 1,2 2,1 B − f(λ)=Xχ CχjY=1Xj,kj, g(λ)=Xχ DχjY=1Xj,kj. (2) Xue1s,2X[su2,c2h)/a2sb{eXt1h,1e =Bel1l,fXun1,c2tio=n,1f,oXr2e,1ach=s1et,Xo2f,2va=l- 1 ], the Bell function corresponds to a number [such } Hereweassociatetoeachobservables,X ,X X ,or as (λ) = 1]. This number is called a “root” of the generally Nj=1Xj,kj,asinglesymbolχ,w1,h1ich1s,t1an2d,1sfor sBeetlsBlofufnvactluioens BX(λ).,XObvi,oXusly,,Xthere,asroe t(oλta)lhlyas2t4o=tal1ly6 N pairsofQindices(onepairforeachobserver). Obviously, { 1,1 1,2 2,1 2,2} B 16 roots. However, 8 roots equal to 1, the other 8 there are Nχ =(1+M)N distinct values of χ. The con- roots equal to 1, therefore (λ) has o−nly two distinct stant numbers C and D are coefficients of N X . B χ χ j=1 j,kj “roots”: Λ1 = 1 and Λ2 = 1. Then for any set of val- Step 2: Establishing Bell inequalities by dQetermining ues X ,X ,−X ,X ,one alwayshas the algebraic 1,1 1,2 2,1 2,2 coefficients Cχ and Dχ. Note that for the qubit case, equa{tion: [ (λ) Λ1][ (λ}) Λ2]=0, or 2(λ)=1. We one always has Xj2,kj = 1, which is useful for simplify- have the foBllowin−g TheBorem−. B ing f(λ)g(λ), f2(λ) and g2(λ) in the inequality (1). One Theorem 1. LetS = (λ) 2(λ)=1 ,for (λ) 2 will find that some terms in f(λ)g(λ), f2(λ) and g2(λ), S , we have Bell inequal{itBy =|B (λ) } 1∀.B ∈ 2 LHV I |hB i |≤ such as X1,1X1,2, X1,1X1,2X2,1, or X1,1X1,2X2,1X2,2, Proof. Letf(λ)=1+ (λ),g(λ)=1 (λ),thenfrom B −B etc., are impossible to calculate in quantum mechan- the Cauchy-Schwarzinequality we have [ [1+ (λ)][1 ics. Consequently, we shall set all the coefficients of (λ))ρ(λ)dλ] [1+ (λ)]2ρ(λ)dλ [1 R (λ)B]2ρ(λ)dλ−, sfourchCχteramnds bDeχ.zerBoy, asonldvitnhgenthgeseet eaquseartiieosnso,fweqeuoabtitoanins BwS2h,icwhhyiciehldims hp≤Bli(eλRs)i2L2H(λVB)≤=h1B,2t(hλu)isLwHReVa.r−BrievBceaautsethBe(λB)e∈ll B the solutions of Cχ and Dχ. Substituting these solu- inequality (λ) LHV 1. This ends the proof. |hB i |≤ tions into Eqs. (1) (2), one gets a set of Bell inequal- Based on Theorem 1, there is a simpler way to derive ities: f(λ)g(λ) 2 f(λ) 2 g(λ) 2 . Here Bell inequality as follows: Let f(λ)g(hλ) =iLHV f≤(λ)gh(λ)ρi(LλH)dVλh, andiLHfV(λ) 2 , h iLHV Γ h iLHV g(λ) 2 are definRed similarly. 1 h iLHV (λ)= X(λ)+Y(λ)+Z(λ) X(λ)Y(λ)Z(λ) , (4) Step 3: Ruling out the trivial Bell inequalities. Some B 2(cid:20) − (cid:21) inequalities obtained in Step 2 are trivial, i.e., they cannot be violated in quantum mechanics. Thus we should one easily proves that 2(λ) = 1, provided X2(λ) = rule them out by calculating the quantum violation of Y2(λ) = Z2(λ) = 1, Bthen from Theorem 1 one has 3 a Bell inequality = (λ) 1. For in- ture belong to S . The first example is the three-setting LHV 3 I |hB i | ≤ stance, let X(λ) = A (λ)B (λ), Y(λ) = A (λ)B (λ), Bell inequality for N qubits proposed in Ref. [20] [see 1 1 1 2 Z(λ) = A (λ)B (λ), which yields X(λ)Y(λ)Z(λ) = inequality (9) in [20] or inequality (2) in [21]]. Although 2 1 A (λ)B (λ) [using A 2(λ) = 1, B 2(λ) = 1], then one this inequality is not tight, it has a particular property 2 2 j j has the CHSH inequality: = (λ) = that for the GHZ state it is more resistant to noise than CHSH LHV I |hB i | A B + A B + A B A B /2 1; Let X(λ) = the MABK inequality when N 4. The second exam- 1 1 1 2 2 1 2 2 |h − i | ≤ ≥ A (λ)B (λ)C (λ), Y(λ) = A (λ)B (λ)C (λ), Z(λ) = ple is the two two-setting Bell inequalities listed in Ref. 1 1 2 1 2 1 A (λ)B (λ)C (λ), one has the MABK inequality for [12] [see inequalities (3) and (4) in [12]]; These two in- 2 1 1 three qubits as = (λ) = A B C + equalitiesarenottightbuttheyareviolatedbyanypure MABK LHV 1 1 2 I |hB i | |h A B C +A B C A B C /2 1. entangled state of three qubits. Moreover, there indeed 1 2 1 2 1 1 2 2 2 − i |≤ The Bell functions of the MABK and the WWZB exist some Bell inequalities belonging to S . The first 4 inequalities for N qubits have two distinct “roots” 1 example is a three-setting Bell inequality for two qubits and 1, so they belong to S . The Bell function proposed in Ref. [22] [see inequality (19) in [22]]. It is 2 − (λ) in Eq. (4) naturally connects with the MABK a relevant two-qubit Bell inequality inequivalent to the B inequality or the WWZB inequality in the following CHSH inequality. The most interesting feature of this way: Let X(λ) = N−1(λ)XN,1, Y(λ) = N−1(λ)XN,2, tight inequality is that there exist states that violate it, Z(λ) = BN′ −1(λ)XBN,1, where BN−1(λ) isBthe Bell func- but do not violate the CHSH inequality. The second ex- tion of MABK inequality or the WWZB inequality for ampleisthetwo-settingBellinequalitypresentedinRef. (N 1) qubits, and ′ (λ) is obtained through the [12] [see inequality (8) in [12]]. It is a tight Bell inequal- − BN−1 interchanges X X . After substituting them into ity and is violated by any pure entangled state of three j,1 j,2 Eq. (4) and using↔X(λ)Y(λ)Z(λ) = ′ (λ)X , one qubits. To our knowledge, Bell inequalities belonging to has B(λ)= 21{BN−1(λ)[XN,1+XN,2]+BNB−N′1−1(λ)N[X,2N,1− Sn (n≥5) have seldom appeared in the literature. X ] , which is just the Bell function of MABK and As applications of the approach,we present three new N,2 fbWaymWiidlZye}Bnotfiitntyewq1ou-Nasle−itt1tiein=s.gBQByeNjl=lt−h1i1neXewqjau,0ya,,liittfhyBefnN′or−on1m(eλan)reyicsoqrvueebprilstasctheades aqunEbdixttsaigmhhIp4tl,e2Bi2e≤l:l iA1n.enLqeuewtalBittw(iλeos)-st=ehtat1i7tn6gb+eB1l9o6enIll4g,i2nt(oeλqS)u,3awlaihtsyefrofeollIro4wf,2osu.=r showninRef. [18](seeinequality(4)in[18]). Also,both (1/9)( 5Q1111 2Q2222 [Q1120+Q1210+Q2110+Q1102] − − − − thethree-settingBellinequality(28)andthefour-setting [Q2200+Q2020+Q2002+Q0220+Q0202+Q0022]+[Q2000+ Bell inequality (35) presented in Ref. [19] belong to S2. Q0200+Q0020+Q0002]+2[Q1000+Q0100+Q0010+Q0001]+ For Bellfunctions with three or moredistinct “roots”, [Q1110+Q1101+Q1011+Q0111]+[Q1201+Q2101+Q1012+ we have the following Theorem. Q1102+Q2011+Q0112+Q0121+Q0211]+2[Q1112+Q1121+ S2T,hie.oe.r,em(λ2).mLuesttSh3av=et{hBr(eλe)d|isBti3n(cλt)“=rooBt(sλ”)Λ,1B=(λ)1∈/, QQ12221011++QQ21101212]+−Q[Q20122120++QQ20211220++QQ02122120++QQ01222012++QQ02211022]++ Λ2}= 0, aBnd Λ3 = 1, then for (λ) S3, one has−the [Q1122+Q1212+Q1221+Q2211+Q2121+Q2112]),Qmnkl = Bell inequality: (λ) LHV ∀ B1. In∈general, if Sn = ΓX1,mX2,nX3,kX4,lρ(λ)dλ, (m,n,k,l =0,1,2), are the (λ) j=n−1(|hB Λi) =|0,≤ (λ) / k=n−1S , n cRorrelationfunctions,andherewehavesetXj,0 =1(j = {inBteger|s,Qnj=0 3 ,Bw−hichj meansBthat∈nS“rko=o2ts” okf (λ∈) 1,2,3,4). It is easy to check B(λ) ∈ S3, therefore from 2ujn/i(f(oλnr)m−l1y),dif≥sotrri1∀b}.uBt(eλb)e∈twSene,no−ne1haansdth1ewBitehllΛinje=qu−aBl1ity+: soTyrhmehoIm4r,ee2mtiri≤c2uw1n.edehTrahvtiehsetinhpeeeqrBumeaulllittiaynteihoqInu4s,a2lioitf≤yX|h1jB,ki(’ssλ.t)iigLhHtVa|n≤d1is, |hBProoifL.HVF|ir≤st, we prove that for n = 3 the theorem For four qubits, the Werner states read ρW = V ψ ψ +(1 V)ρ , where ψ = (1/√2)(0000 + is valid. Since (λ) 2 2(λ) , what we need to | ih | − noise | i | i gd(oλ)is=to1prove2(hλhBB)2,(λfr)oiim≤≤th1eh.BCLaeutcihfy(-λS)ch=war1z+inBeq2u(λa)l-, i|1b1il1it1yi,)ρinsotihsee=fouIr1-6q/u1b6i,taGndHZI16stiastteh,eV1i6s×th1e6siod-ecnatlilteydmvias-- ity one obt−ainBs 2(λ) 2 4(λ) . By using (λ)3 = trix. Inquantummechanics,theobservablesreadXj,k = (λ), we have hB 2(λ)i2≤ hB 4(λi) = 2(λB) , i.e., ~σ·~nkj, (j = 1,2,3,4;k = 1,2), ~σ is the vector of Pauli B 2(λ) [ 2(λ) hB1] i1.≤BehcBausei 2(λh)B 0i, then matrices, ~nkj = (sinθkjcosφkj,sinθkj sinφkj,cosθkj), twhiBoenhamvieehtBhhBod2(λtio)i−LpHroVv≤e≤th1i.s tSheecoornedm,.hwBeSuuspeipot≥shee fionrduacl-l Xlatj,i0on=funI2ctiiosntshefor2 t×he2Wideernnteirtystmataetsrixre.adTQhemnckolrr=e- Tr(ρ X X X X ). For the GHZ state, none≤hkas, Bthke+1th=eor(eBmk2+i1s−va1li)d, jjt==hk1e−n1(foBrk+B1k+−1Λ∈j)S=k+01,. wthheictWhhriseslh1a,ormglde⊗rvtihsi2ab,nnili⊗1t/y2i√s3,2kV,G⊗4tH−hqeZu4rb,eiltsu≈lt9g/iv1e5n.5b6y≈th0e.5M78e4r-, If ( 2 1) = 0, Theorem Q1 yields 1. If jj==Bk1k−+11(B−k+1 − Λj) = 0, from the pr|ihnBckip+l1ei|of≤induc- rmeisnistianneqtutaolintyo.iseN.aHmoewlye,vtehr,eninuemqeuraicliatlyrheIs4u,2ltis≤ho1wisstlheasst Qtion method, one obtains |hBk+1i|=|hk−k2Bk−1i|≤ k−k2. the generalized GHZ states violate hI4,2i ≤ 1 for the Thus, one has (λ) 1, which completes the whole region, which cannot be done for the MABK in- LHV |hB i | ≤ proof. equality. By setting X = X = 1, I 1 reduces 4,1 4,2 4,2 h i ≤ Let us see what Bell inequalities appeared in litera- to an equivalent form of the three-qubit Bell inequality 4 (4) in Ref. [12]. Remarkably, numerical calculation in- proach to establish Bell inequalities for qubits based on dicates that Bell inequality I 1 is violated by all the Cauchy-Schwarz inequality. We have also used the 4,2 h i ≤ pure entangled states of four qubits, thus it is a good concept of distinct “roots” of Bell function to classify candidate for proving Gisin’s theorem for four qubits. some well-known Bell inequalities for qubits. As appli- Example 3: Another new two-setting Bell inequality cations of the approach, we have presented three new for four qubits I′ 1. Let (λ) = 6 + 10I′ (λ), andtight Bellinequalities. In addition,there is analter- where I′ = (1/h140,)2(i ≤[Q +QB +Q16 +16Q4,2 + native way to derive the CHSH inequality: From The- 4,2 − 1200 2100 1020 2010 Q +Q +Q +Q +Q +Q +Q + orem 1 and Eq. (4), one may have = 1+A B 1002 2001 0120 0210 0102 0201 0012 1 1 1 I h − Q ] Q +[Q +Q +Q +Q ]+3[Q + A B +A A B B /2 1, = 1+A B +A B 0021 2222 1112 1121 1211 2111 1000 2 2 1 2 1 2 2 1 2 2 1 − i ≤ I h − Q +Q +Q ]+[Q +Q +Q +Q ] A A B B /2 1. By adding up these two inequalities, 0100 0010 0001 2000 0200 0020 0002 1 2 1 2 − i ≤ [Q +Q +Q +Q +Q +Q +Q + one arrives at the famous CHSH inequality: = 1220 2120 2210 1202 2102 2201 1022 CHSH I Q +Q +Q +Q +Q ]+[Q +Q + + = A B +A B +A B A B /2 1. Usu- 2021 2012 0122 0221 0212 1122 1212 1 2 1 1 1 2 2 1 2 2 I I h − i ≤ Q +Q +Q +Q ] [Q +Q +Q + ally, combining two inequalities directly will lead to an 1221 2211 2121 2112 2220 2202 2022 − Q ] 3Q ). One may verify that (λ) S , thus inequality with looser constraint than before. In such a 0222 1111 3 − ′ B ∈ we have the Bell inequality I 1. This inequality is sense,inequalities and arestrongerthantheCHSH h 4,2i≤ I1 I2 tight and is violated by the generalized GHZ states for inequality, but the difficult point is that it is impossible the whole region. It is also a good candidate for proving to compute the term A A B B for quantum mechan- 1 2 1 2 h i the Gisin’s theorem for four qubits. ics. Some time ago, Gisin posed a question to find Bell Example 4: A new three-setting Bell inequality for inequalities which are more efficient than the CHSH one three qubits I 1. The Bell inequality reads for Werner states [23] (see also [24]). Recently, Vértesi 3,3 h i≤ hasgivenapositiveanswertoGisin’squestionbyprovid- I3,3 = [(Q223+Q232+Q322) 2(Q211+Q121+Q112) ing a new family of multi-setting (at least M =465 set- h i − +(Q +Q +Q ) (Q +Q +Q ) tings for each party) Bell inequalities [25], which proves 221 122 212 331 313 133 − the two-qubit Werner states to be nonlocal for a wider +(Q +Q +Q +Q +Q +Q ) 321 312 213 231 123 132 parameter range 0.7056<V 1. How to use Theorems +2Q +4Q Q ]/8 1, (5) ≤ 111 222 333 1 and2 toconstructevenmoreefficientBellinequalities − ≤ for Werner states is a significant topic, which we shall and (λ) = I (λ) S . This inequality is also tight, B 3,3 ∈ 3 investigate subsequently. which reduces to the CHSH inequality when X = 1, 3,1 X =X = 1. 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