Detection of Genuine Multipartite Entanglement in Quantum Network Scenario
Biswajit Paul, Kaushiki Mukherjee, Sumana Karmakar, Debasis Sarkar, Amit Mukherjee, Arup Roy, Some Sankar Bhattacharya
DDetection of Genuine Multipartite Entanglement in Quantum Network Scenario
Biswajit Paul, ∗ Kaushiki Mukherjee, † Sumana Karmakar, ‡ DebasisSarkar, § Amit Mukherjee, ¶ Arup Roy, ∗∗ and Some Sankar Bhattacharya †† Department of Mathematics, South Malda College, Malda, West Bengal, India Department of Mathematics, Government Girls’ General Degree College, Ekbalpore, Kolkata, India. Department of Applied Mathematics, University of Calcutta, 92, A.P.C. Road, Kolkata-700009, India. Physics and Applied Mathematics Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata 700108 , India.
Experimental demonstration of entanglement needs to have a precise control of experimentalistover the system on which the measurements are performed as prescribed by an appropriate entan-glement witness. To avoid such trust problem, recently device-independent entanglement witnesses(
DIEW s) for genuine tripartite entanglement have been proposed where witnesses are capable oftesting genuine entanglement without precise description of Hilbert space dimension and measuredoperators i.e apparatus are treated as black boxes. Here we design a protocol for enhancing thepossibility of identifying genuine tripartite entanglement in a device independent manner. We con-sider three mixed tripartite quantum states none of whose genuine entanglement can be detected byapplying standard
DIEW s, but their genuine tripartite entanglement can be detected by applyingthe same when distributed in some suitable entanglement swapping network.
PACS numbers: 03.65.Ud, 03.67.Mn
I. I. INTRODUCTION
Entanglement is one of the most intriguing andmost fundamentally non-classical phenomena in quantumphysics. A bipartite quantum state without entangle-ment is called separable. A multipartite quantum statethat is not separable with respect to any bi-partition issaid to be genuinely multipartite entangled [1]. This typeof entanglement is important not only for research con-cerning the foundations of quantum theory but also inquantum information protocols and quantum tasks suchas extreme spin squeezing [2], high sensitivity in somegeneral metrology tasks [3], quantum computing usingcluster states [4], measurement-based quantum computa-tion [5] and multiparty quantum network [6–9]. Severalexperiments have been conducted so far for generationof genuine multipartite entanglement [10–12]. However,detection of this kind of resource in an experiment turnsout to be quite difficult. Experimental demonstration ofgenuine multipartite entanglement is generally performedwith one of the two following techniques: tomography ofthe full quantum state [13, 14], or evaluation of an en-tanglement witness [1]. But both of these techniques facesome common drawbacks viz. requirement of precise con-trol(by the experimentalist) over the system subjected tomeasurements and sensitivity of these techniques to sys-tematic errors [15].However, there exists a way to avoid this sort of draw- ∗ Electronic address: [email protected] † Electronic address: kaushiki mukherjee@rediffmail.com ‡ Electronic address: [email protected] § Electronic address: [email protected],[email protected] ¶ Electronic address: [email protected] ∗∗ Electronic address: [email protected] †† Electronic address: [email protected] backs. Such an alternative method is provided by usingsome specific Bell-type inequalities [16]. Bell inequalitywas first designed by John Bell to explain the incompati-bility of quantum predictions with local-realism[16]. Tilldate various types of Bell inequalities have been designedfor the purpose of detection of nonlocality of correlationswhere any precise control of the device by the experimen-talist is not needed. Now presence of entanglement is anecessary resource for generation of nonlocal correlations.In this context some specific type of Bell inequalities havebeen proposed to detect entanglement, more specificallygenuine multipartite entanglement(GME) certified fromstatistical data only. To be precise, if the value of aBell expression in multipartite scenario exceeds the valueobtained due to measurements on biseparable quantumstates, then the presence of genuine entanglement canbe guaranteed. This technique to detect genuine multi-partite entanglement in device-independent manner wasfirst introduced in [17–20] followed by an extensive for-malization by Bancal et al. [21]. In particular theyintroduced the term Device-Independent EntanglementWitness(DIEW) of genuine multipartite entanglement forsuch Bell expressions. Later, Pal [22] and Liang et al.[23] developed other DIEWs for detecting genuine mul-tipartite entanglement. Throughout the paper, we referto the procedure of detecting genuine entanglement asdevice-independent entanglement detection(DIED) andthe entanglement detected in device-independent way asdevice-independent entanglement(DIE).Entanglement swapping[24] as a resource has been usedin a number of quantum information processing taskssuch as entanglement concentration or distillation, pu-rification, speeding up the distribution of entanglement,correction of amplitude errors developed due to propaga-tion, activation of nonlocality etc. Entanglement swap-ping in tripartite scenario mainly involves a network ofthree parties say, Alice, Bob and Charlie. The proce- a r X i v : . [ qu a n t - ph ] J a n dure of entanglement swapping was first generalized formulti-party scenario in [25]. In the present paper, weaddress the following questions: consider some tripartitestates whose genuine entanglement cannot be detectedby applying some standard DIEWs [17, 19, 21, 23], nowis it possible to find some suitable entanglement swap-ping process, after which the genuine entanglement ofswapped state can be detected by those DIEWs? Weanswer this question affirmatively and have designed aprotocol based on entanglement swapping procedure bywhich genuine entanglement of the tripartite state result-ing from multiple swapping can be detected in a deviceindependent way , i.e. without any reference of the deviceinvolved. Precisely speaking, this new protocol enhancesthe regime of DIED for tripartite quantum states. Inthis context another important question is whether onecan enhance detection of genuine entanglement in a semi-device independent way(corresponding to phenomenon ofquantum steering). We also answer this question affirma-tively.The rest of this paper has been organized as follows:section II deals with some mathematical preliminariesand a brief overview of some standard DIEWs. In sec-tion III we design the protocol involving entanglementswapping procedure followed by detailed discussion onenhancement of entanglement detection by using somestandard DIEWs in section IV. In section.V we have usedthis protocol to enhance genuine entanglement in a semidevice independent way. Finally we conclude with a briefdiscussion regarding the importance of this work and pos-sible further extensions. II. BACKGROUNDA. Notion of DIEWs
Violation of Bell inequality by quantum mechanicalsystems always indicates presence of entanglement. Thusa Bell inequality can be considered as a suitable candidatefor detecting the presence of entanglement in a device in-dependent way unlike the standard procedures like statetomography or use of entanglement witnesses where ex-perimentalist needs to trust the experimental apparatus.This is because detection of entanglement using Bell in-equality solely depends on the statistical data. To charac-terize genuine entanglement in a device-independent wayfor tripartite scenario where each of the three subsys-tems, one of m possible measurements can be performed,yielding one of two possible outcomes. The measurementsettings are denoted by x , y , z ∈ { , , , ...m − } andtheir outputs by a , b , c ∈ { -1,1 } for Alice, Bob and Char-lie respectively. The experiment is thus characterized bythe joint probability distribution p ( abc | xyz ). The corre-lations P ( abc | xyz ) can be categorized as bi-separable ifthey can be reproduced through the measurements on a tripartite bi-separable state ρ bi where ρ bi = (cid:88) λ p λ ρ Aλ (cid:79) ρ BCλ + (cid:88) µ p µ ρ Bµ (cid:79) ρ ACµ + (cid:88) ν p ν ρ Cν (cid:79) ρ ABν . (1)Here 0 ≤ p λ , p µ , p ν ≤ (cid:80) λ p λ + (cid:80) µ p µ + (cid:80) ν p ν =1 . To be precise, if there exists a state of the formgiven by Eq.(1) in some Hilbert space H and somesuitable local measurement operators M a | x , M b | y and M c | z (without loss of generality these operators can beconsidered to be projection operators satisfying the re-striction M a | x M a (cid:48) | x = δ a,a (cid:48) M a | x and (cid:80) a M a | x = I ) suchthat: p ( abc | xyz ) = tr [ M a | x (cid:79) M b | y (cid:79) M c | z ρ bi ] (2)If the correlations are not biseparable, then the stateused is surely a genuine tripartite entangled state.Such a conclusion can be drawn independent of thecorresponding Hilbert space dimension. Equivalently,biseparable quantum correlations can also be decom-posed as, P ( abc | xyz ) = (cid:88) k P kQ ( ab | xy ) P kQ ( c | z )+ (cid:88) k P kQ ( ac | xz ) P kQ ( b | y ) + (cid:88) k P kQ ( bc | yz ) P kQ ( a | x ) (3)where P kQ ( ab | xy ) and P kQ ( c | z ) denote arbitrary two partyand one party quantum correlations respectively. So theyare of the form: P kQ ( ab | xy ) = tr [ M ka | x a | x (cid:78) M kb | y ρ kAB ]and P kQ ( c | z ) = tr [ M kc | z ρ kC ] for some unnormalizedquantum states ρ kAB , ρ kC and measurement operators M ka | b , M kb | y , M kc | z .Let Q denotes the set of tripartite quantum correlationsand Q | denotes the set of biseparable quantum corre-lations. Clearly, Q | ⊆ Q . The set Q | being convex,can be characterized by linear inequalities. DIEWs ofgenuine tripartite entanglement correspond to those in-equalities(Bell inequalities) that separate the sets of Q and Q | . Now as Q | has infinite number of extremalpoints so there exist many such DIEWs separating gen-uine entanglement from bi-separable entanglement. Inrecent times many such DIEWs are designed for detectinggenuine tripartite entanglement in a device independentway [17–23]. As already mentioned in the introduction,Bancal [21] was the first to formalize the concept of de-vice independent detection of entanglement introducingthe term DIEW for detecting genuine multipartite en-tanglement. In this context, one can consider the DIEWprovided by the Mermin polynomial [26] as the most sim-ple example for detecting genuine tripartite entanglement[17]. In [19] Uffink, designed another non linear Bell-typeinequality which has been extensively used for this pur-pose. In recent times, Bancal et al. gave more efficient3-settings Bell inequality which can be used as a DIEWto detect genuine tripartite entanglement [21]. More than3 setting DIEWs are also provided in [22]. However inour present topic of discussion, we restrict our search fornot more than 3-settings Bell inequalities due to obvi-ous computational complexity. More recently another 2settings DIEW was designed by Liang et al.[23](see Ap-pendix.A). As detection of genuine nonlocality by anyBell inequality implies genuine entanglement [27, 28] soit is a DIEW for detecting genuine entanglement. How-ever the converse is not necessarily true. After discussingabout DIEW and their advantages over usual proceduresof detecting entanglement experimentally, we are now ina position to use them for our purpose of detecting gen-uine entanglement in an entanglement swapping proto-col. This in turn helps to enhance the chance of genuinetripartite entanglement being detected in a device inde-pendent way. But before that we illustrate our multipleentanglement swapping scenario. III. MULTIPLE ENTANGLEMENT SWAPPINGPROCEDURE
Consider the multiple entanglement swapping networkgiven in Fig.1. It is a network of six space-like separatedobservers. Three tripartite quantum states ρ i ( i = 1 , , ρ is shared among theparties A i ( i = 1 , ,
3) such that j th particle( ρ j ) of ρ iswith party A j ( j = 1 , ,
3) respectively. State ρ is sharedamong A , A and A with the specification that j th qubit( ρ j ) is sent to party A j +1 ( j = 1 , , ρ is shared among A , A and A such thatparty A j +3 holds j th ( j = 1 , ,
3) particle of ρ ( ρ j ). Soeach of the three parties A , A and A holds two parti-cles: A holds ρ and ρ ; A holds ρ and ρ ; A holds ρ and ρ . Now in the preparation stage, each of the threeparties A i ( i = 1 , ,
3) performs Bell basis measurementson two of the three particles that each of them holds: A performs Bell basis measurement on 3 rd particle of ρ ( ρ ) and 1 st particle of ρ ( ρ ); A performs Bell basisFIG. 1: Swapping scheme measurement on 2 nd particle of ρ ( ρ ) and 1 st particle of ρ ( ρ ); A performs Bell basis measurement on 3 rd par-ticle of ρ ( ρ ) and 2 nd particle of ρ ( ρ ). After all thethree parties have performed Bell basis measurement ontheir respective particles, they communicate the resultsamong themselves, as a result of which ρ is generated atthe end of the preparation stage. Clearly ρ varies withthe output of the Bell measurements. The final state ρ isobtained from the initial states ρ i ( i = 1 , ,
3) by meansof post-selecting on particular results of local measure-ments, in particular Bell basis measurements performedon these states( ρ i ( i = 1 , , | ψ ± (cid:105) = | (cid:105)±| (cid:105)√ . If the output of all mea-surements correspond to | ψ + (cid:105) ( | ψ − (cid:105) ), the resultant state ρ +4 ( ρ − ) is given by: ρ ± = (cid:104) ψ ± | A ⊗ (cid:104) ψ ± | A ⊗ (cid:104) ψ ± | A ( ρ ⊗ ρ ⊗ ρ ) | ψ ± (cid:105) A ⊗ | ψ ± (cid:105) A ⊗ | ψ ± (cid:105) A (4)So preparation stage of this protocol can be consid-ered as a particular instance of Stochastic Local Oper-ation and Classical Communication (SLOCC). After ρ ± is generated and shared among the parties in the prepa-ration stage, each of the three parties A , A and A performs projective measurement on the state ρ ± in themeasurement stage. Now if the correlations generatedfrom ρ ± exhibit violation of any DIEW under the con-text that the initial states ρ i ( i = 1 , ,
3) fail to reveal thesame, then that guarantees enhancement of DIED in ourprotocol.
IV. ENHANCEMENT OFDEVICE-INDEPENDENT ENTANGLEMENTDETECTION POSSIBILITY
In this section we deal with the procedure of enhancingDIED of tripartite quantum states in terms of expand-ing the set of states by using the multiple entanglementswapping protocol described in Fig.1. For this we providean explicit example. Initially we consider three tripartitequantum states ρ i ( i = 1 , ,
3) with some restricted rangeof state parameters, for each of which none of the DIEWsproposed in the literature[21, 23, 26] can detect genuineentanglement. These states, after being used in the mul-tiple entanglement swapping network(Fig.1), generates astate ρ whose genuine entanglement can be detected ina device-independent manner.Let the three initial states be given by: ρ = p | ψ f (cid:105)(cid:104) ψ f | + (1 − p ) | (cid:105)(cid:104) | (5)with | ψ f (cid:105) = cos θ | (cid:105) + sin θ | (cid:105) , 0 ≤ θ ≤ π and 0 ≤ p ≤ ρ = p | ψ + m (cid:105)(cid:104) ψ + m | + (1 − p ) | (cid:105)(cid:104) | (6)with | ψ + m (cid:105) = | (cid:105) + | (cid:105)√ and 0 ≤ p ≤ ρ = p | ψ l (cid:105)(cid:104) ψ l | + (1 − p ) | (cid:105)(cid:104) | (7)with | ψ l (cid:105) = sin θ | (cid:105) + cos θ | (cid:105) . Now each of the threeparties A , A and A performs Bell basis measurementon their respective particle. As already stated before, theoutput state depends on the outputs of the Bell measure-ments performed. For instance when | ψ ± (cid:105) = | (cid:105)±| (cid:105)√ isobtained as the output odd number of times, a resultantstate ρ ± is obtained which after correcting phase term isgiven by: ρ ± = p f | ψ ± m (cid:105)(cid:104) ψ ± m | + (1 − p f ) | (cid:105)(cid:104) | (8)where | ψ ± m (cid:105) = | (cid:105)±| (cid:105)√ and p f = p cos θ p cos 2 θ . Clearly ρ ± is independent of p , but the probability of obtaining ρ ± directly depends on it. Here reader must note that ρ ± can also be generated for some other combination ofswapping networks together with some different arrange-ment of particles in between the parties A i (1 , ...,
6) andfor different outputs of the Bell measurement. To detectDIE of each of the states ρ i ( i = 1 , , ,
4) we obtain thecondition for which they violate each of the DIEWs(seeAppendix.A) given in [17, 19, 21, 23]. Among all, the3-settings Bell inequality √
32 ( (cid:104) A B C (cid:105) − (cid:104) A B C (cid:105) − (cid:104) A B C (cid:105) − (cid:104) A B C (cid:105)−(cid:104) A B C (cid:105) − (cid:104) A B C (cid:105) − (cid:104) A B C (cid:105) − (cid:104) A B C (cid:105)−(cid:104) A B C (cid:105) − (cid:104) A B C (cid:105) − (cid:104) A B C (cid:105) + (cid:104) A B C (cid:105) − (cid:104) A B C (cid:105) − (cid:104) A B C (cid:105) − (cid:104) A B C (cid:105) + (cid:104) A B C (cid:105) + (cid:104) A B C (cid:105) + (cid:104) A B C (cid:105) ) ≤ ρ i ( i = 1 , , , (cid:104) A α B β C γ (cid:105) designate the expected value of the product of three ± A α , B β , C γ .Now the initial states ρ i ( i = 1 , ,
3) do not violateEq.(9) if and only if(see Appendix.A) p ≤ p ≤
23 sin 2 θ (10)The condition of violation of Eq.(9) for the final states ρ ± is given by(see Appendix.A): p > θ + 1 (11) FIG. 2: (color online) The shaded region denotes therange of parameters of states ρ and ρ ( p ≤ ) forwhich the -settings Bell inequality (Eq.(9)) is violatedonly after distributing them in the multipleentanglement swapping network as described in Sec.III,i.e. neither of the three initial states violate Eq.(9)whereas the final state violates it. Hence this regiongives the range where DIED is enhanced. There exists a range of the state parameters where theinitial states ρ i ( i = 1 , ,
3) do not violate Bancal’s 3-settings Bell inequality, but after distributing them in themultiple entanglement swapping network, final states ρ ± violates it. The range of state parameters in which detec-tion of DIE is enhanced by this entanglement swappingprotocol(see Fig.2) is given by: p ≤ and1cos θ + 1 < p ≤
23 sin 2 θ . (12)The restrictions imposed on the state parameters(Fig.2)indicate that DIED is enhanced at the end of the swap-ping procedure. In this context, it is interesting to notethat the probability of success of this protocol p succ isgiven by p succ = 12 pp [1 + p cos 2 θ ] sin θ. In recent times there has been experimental implementa-tion of DIED [29] and also experimental demonstrationof entanglement swapping [30, 31]. So our procedure ofenhancing DIED method can also be demonstrated ex-perimentally within the scope of current technology. Thefact that this explicit example shows enhancement of thepossibility of entanglement detection in a device indepen-dent manner indicates that for any genuinely entangledtripartite state( ρ , say) it may be possible to design a suit-able swapping protocol via which entanglement of the fi-nal state resulting from the protocol using many copies ofthe initial state( ρ ), can be detected even when the samecannot be detected for ρ itself. DIEW Violation by ρ Violation by ρ Violation by ρ Enhanced rangeMermin[26] p> √ θ p > √ p> √ θ , √ −
2) cos θ +1
√ θ p > √ p> √ θ , √ −
2) cos θ +1
23 sin 2 θ p > p>
23 sin 2 θ , θ +1
23 sin 2 θ Liang et al.[23] p> √
25 sin 2 θ p > √ p> √
25 sin 2 θ , √
25 sin 2 θ
2) cos θ +1 TABLE I: The condition of violation of each of the DIEWs given in [19, 21, 23, 26] for each of thestates( ρ i ( i = 1 , , V. ENHANCEMENT OF SEMIDEVICE-INDEPENDENT ENTANGLEMENTDETECTION POSSIBILITY
Cavalcanti et. al. in [32] have provided an inequalitywhich detect genuine entanglement in semi device inde-pendent way. The inequality looks like:1 − . (cid:104) A B (cid:105) + (cid:104) A Z (cid:105) + (cid:104) B Z (cid:105) ) − . (cid:104) A B X (cid:105)− (cid:104) A B Y (cid:105) − (cid:104) A B Y (cid:105) − (cid:104) A B X (cid:105) ) ≥ ρ i ( i = 1 , ,
3) do not violate Cavalcanti et al. inequalityif and only if p ≥ . . . θ ]and p ≥ . ρ is given by: p > p cos [ θ ]1 + p cos[2 θ ] (15)Thus there exists a range of the state parameters ( p, p )where the initial states ρ i ( i = 1 , ,
3) do not violate theinequality Eq.(13), but after distributing them in the net-work and executing the protocol, final states ρ violatesit. The range of state parameters in which enhancementof detection of semi-device independent entanglement isobserved by our protocol is given by: p ≤ . p cos [ θ ]1 + p cos[2 θ ] < p < . . . θ ] (16)which indicates a clear advantage as shown in Fig.3. FIG. 3: Shaded region gives the restrictions imposed onthe state parameters for which enhancement of DIED isobserved via the multiple swapping procedure under therestriction of p ≥ . over the state parameter p of ρ . VI. CONCLUSION
In a nutshell, our present topic of discussion may beconsidered as a contribution in the field of device in-dependent entanglement detection which minimizes therequirement of precise control over measurement devicesby an experimentalist in an experimental detection ofentanglement. More precisely, in our work we haveshown that it is possible to enhance device independentdetection of genuine tripartite entanglement in somesuitable measurement context. For our purpose, we haveconsidered four DIEWs given by Mermin[26], Uffink[19],Bancal[21] and Liang et. al.[23], out if which theDIEW given by Bancal et.al.[21] emerges to be the mostefficient. We have designed a state preparation protocol(prior to receiving final measurements), particularly anentanglement swapping procedure involving six distantobservers via which genuine tripartite entanglement ofthe resultant(swapped) state, generated by using threeinitial tripartite states(whose entanglement cannot bedetected by the standard DIEWs), can be detected bythese standard DIEWs(used for testing entanglement ofthe initial states) after performing the state preparation.
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We are now going to enlist the DIEWs which are usedas tools for DIED in main text. To start with one can con-sider the device-independent-entanglement-witness pro-vided by the Mermin polynomial [26] as the simplest ex-ample for detecting genuine tripartite entanglement [17]: M = |(cid:104) A B C (cid:105) + (cid:104) A B C (cid:105) + (cid:104) A B C (cid:105)−(cid:104) A B C (cid:105)| ≤ √ (cid:104) A B C + A B C + A B C − A B C (cid:105) + (cid:104) A B C + A B C + A B C − A B C (cid:105) ≤ (cid:104) A B C (cid:105) + (cid:104) A B C (cid:105) + (cid:104) A B C (cid:105) + (cid:104) A B C (cid:105) + (cid:104) A B C (cid:105) + (cid:104) A B C (cid:105) + (cid:104) A B C (cid:105) − (cid:104) A B C (cid:105) ) ≤ √ . (A3)Now we present the detailed proofs of the results statedin the main text. To obtain the condition of violation ofeach of the DIEWs (Eqs.(A1, A2, 9, A3)) in terms of stateparameters for each of the initial states ρ i ( i = 1 , , ρ , we apply the same method as used in[33]. First we find the condition of violation(in terms ofstate parameters) of the DIEW given in Eq.(A1) for theinitial state ρ . We consider the following measurements: A = (cid:126)x. (cid:126)σ or A = (cid:126) ´ x. (cid:126)σ on 1 st qubit, B = (cid:126)y. (cid:126)σ or B = (cid:126) ´ y. (cid:126)σ on 2 nd qubit, and C = (cid:126)z. (cid:126)σ or C = (cid:126) ´ z. (cid:126)σ on 3 rd qubit, where (cid:126)x, (cid:126) ´ x, (cid:126)y, (cid:126) ´ y and (cid:126)z, (cid:126) ´ z are unit vectorsand σ i are the spin projection operators that can bewritten in terms of the Pauli matrices. Representingthe unit vectors in spherical coordinates, we have, (cid:126)x = (sin θa cos φa , sin θa sin φa , cos θa ) , (cid:126)y =(sin αb cos βb , sin αb sin βb , cos αb ) and (cid:126)z =(sin ζc cos ηc , sin ζc sin ηc , cos ζc ) and similarly,we define, (cid:126) ´ x, (cid:126) ´ y and (cid:126) ´ z by replacing 0 in the indices by1. Then the value of the operator M (Eq.(A1)) withrespect to the state ρ (Eq.(5)) gives: M ( ρ ) = − cos αb ( − p + p cos 2 θ )(cos ζc cos θa + cos ζc cos θa ) − sin αb ( p sin 2 θ )(cos( βb + ηc + φa ) sin ζc sin θa + cos( βb + ηc + φa ) sin ζc sin θa ) + cos αb ( − p + p cos 2 θ )(cos ζc cos θa − cos ζc cos θa )+sin αb ( p sin 2 θ )(cos( βb + ηc + φa ) sin ζc sin θa − cos( βb + ηc + φa ) sin ζc sin θa ) . (A4)Hence in order to get maximum value of S ( ρ ), we have toperform maximization over 12 measurement angles. Now if we maximize the last equation with respect to αb and αb , we have M ( ρ ) ≤ (cid:112) (( X )(cos ζc cos θa + cos ζc cos θa )) + ( Y ) ( A sin ζc sin θa + A sin ζc sin θa ) + (cid:112) (( X )(cos ζc cos θa − cos ζc cos θa )) + ( Y ) ( A sin ζc sin θa − A sin ζc sin θa ) (A5)Where X = − p + p cos 2 θ , Y = p sin 2 θ , and A ijk =cos( βb i + ηc j + φa k )( i, j, k ∈ { , } ). The last inequal-ity is obtained by using the inequality x cos θ + y sin θ ≤ (cid:112) x + y . It is clear from the symmetry of the measure- ment angles θa , ζc and θa , ζc that the right handside of Eq.(A5) gives maximum value when θa = ζc and θa = ζc . Hence Eq.(A5) takes the form: M ( ρ ) ≤ (cid:112) (( X )(2 cos θa cos θa )) + ( Y sin θa sin θa ) ( A + A ) + (cid:113) (( X )(cos θa − cos θa )) + ( Y ) ( A sin θa − A sin θa ) (A6)Again we maximize it with respect to θa . Critical point0 or π gives the maximum value depending on values ofthe state parameters. For the critical point 0, Eq.(A6)becomes M ( ρ ) ≤ (cid:112) (2 X cos θa ) + (cid:113) sin θa ( X + Y ) (A7)where we have chosen A = 1. Maximizing over θa ,we get M ( ρ ) ≤ X + Y √ X + Y (A8)the maximum being obtained for cos θa = | X |√ X + Y . Forthe other critical point π , Eq.(A6) takes the form: M ( ρ ) ≤ (cid:112) ( Y sin θa ) ( A + A ) + (cid:113) X cos θa + Y ( A sin θa − A ) ≤ (cid:112) Y sin θa ) + (cid:113) X cos θa + Y (sin θa + 1) ≤ | Y | (A9)The second inequality in Eq.(A9) is obtained from thefirst by setting A = 1, A = 1, A = 1 and A = − . The final inequality is achieved when θa = π . Twosets of measurement angles which realize the two values X + Y √ X + Y (Eq.(A8)) and 4 | Y | (Eq.(A9)), are θa = αb = ζc = cos − ( | X |√ X + Y ), θa = αb = ζc = 0, βb i = ηc i = φa i = 0 (i = 0, 1) and θa i = αb i = ζc i = π (i =0, 1) , βb = ηc = φa = 0, βb = − ηc = − φa = π respectively. Hence from Eq.(A8) and Eq.(A9), we have M ( ρ ) ≤ max[ 2 X + Y √ X + Y , | Y | ] . (A10)Clearly, X + Y √ X + Y ≤ < √ p ∈ [0 , ≤ θ ≤ π . So the initial state ρ violates the DIEWbased on Mermin expression (Eq.(A1)) if4 | Y | = 4 | p | sin 2 θ > √ . (A11)The last inequality is considered as the condition of vio-lation of the DIEW based on Mermin expression for theinitial state ρ . We have applied the same method overother states ρ i (i = 2, 3, 4)to find the condition of viola-tion of the DIEW based on Mermin expression. For otherDIEWS (Eqs.(A2), (9), (A3)), we have made analysis insimilar manner so as to obtain the condition of violationfor each of states ρ i . All the conditions are summarizedin Table.I. However among the four DIEWs given by Mer-min(Eq.(A1)), Uffink(Eq.(A2)), Bancal et al.(Eq.(9))and Liang et al.(Eq.(A3)), the one given by Bancal etal. turns out to be the most efficient for this purpose.The DIEW based on Bancal et al. polynomial (Eq.(9))can thus detect genuine tripartite entanglement ina device-independent way in ρ for p >
23 sin 2 θ (seeTableIV.). As
23 sin 2 θ < √ θ < √
25 sin 2 θ , so the DIEWbased on Bancal et al. polynomial (Eq.(9)) is themost efficient DIEW for the state ρ to detect genuinetripartite entanglement among all the standard DIEWsconsidered in Eqs.((A1), (A2), (9), (A3)). Similarly bycomparing the range of violation of p (for the state ρ )and p (for the state ρ , ρ ), one can check that Bancal etal. Bell inequality is the best DIEW for the other states ρ ii