Detection of Genuine Multipartite Entanglement in Quantum Network Scenario

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by  Biswajit Paul

Detection of Genuine Multipartite Entanglement in Quantum Network Scenario
Biswajit Paul,1,∗ Kaushiki Mukherjee,2,† Sumana Karmakar,3,‡ Debasis
Sarkar,3,§ Amit Mukherjee,4,¶ Arup Roy,4,∗∗ and Some Sankar Bhattacharya4,††
1Department of Mathematics, South Malda College, Malda, West Bengal, India
2Department of Mathematics, Government Girls’ General Degree College, Ekbalpore, Kolkata, India.
3Department of Applied Mathematics, University of Calcutta, 92, A.P.C. Road, Kolkata-700009, India.
4Physics 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 experimentalist
over 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
(DIEWs) for genuine tripartite entanglement have been proposed where witnesses are capable of
testing genuine entanglement without precise description of Hilbert space dimension and measured
operators i.e apparatus are treated as black boxes. Here we design a protocol for enhancing the
possibility of identifying genuine tripartite entanglement in a device independent manner. We con-
siderthreemixedtripartitequantumstatesnoneofwhosegenuineentanglementcanbedetectedby
applying standard DIEWs, but their genuine tripartite entanglement can be detected by applying
the same when distributed in some suitable entanglement swapping network.
7
1 PACSnumbers: 03.65.Ud,03.67.Mn
0
2
I. I. INTRODUCTION backs. Such an alternative method is provided by using
n
some specific Bell-type inequalities [16]. Bell inequality
a
J wasfirstdesignedbyJohnBelltoexplaintheincompati-
Entanglement is one of the most intriguing and
5 bility of quantum predictions with local-realism[16]. Till
mostfundamentallynon-classicalphenomenainquantum
1 datevarioustypesofBellinequalitieshavebeendesigned
physics. A bipartite quantum state without entangle-
forthepurposeofdetectionofnonlocalityofcorrelations
ment is called separable. A multipartite quantum state
] whereanyprecisecontrolofthedevicebytheexperimen-
h that is not separable with respect to any bi-partition is
p saidtobegenuinelymultipartiteentangled[1]. Thistype talist is not needed. Now presence of entanglement is a
- necessaryresourceforgenerationofnonlocalcorrelations.
t of entanglement is important not only for research con-
n InthiscontextsomespecifictypeofBellinequalitieshave
cerning the foundations of quantum theory but also in
a been proposed to detect entanglement, more specifically
quantum information protocols and quantum tasks such
u genuine multipartite entanglement(GME) certified from
q as extreme spin squeezing [2], high sensitivity in some
statistical data only. To be precise, if the value of a
[ general metrology tasks [3], quantum computing using
Bellexpressioninmultipartitescenarioexceedsthevalue
clusterstates[4],measurement-basedquantumcomputa-
1 obtained due to measurements on biseparable quantum
tion [5] and multiparty quantum network [6–9]. Several
v states, then the presence of genuine entanglement can
experiments have been conducted so far for generation
4 be guaranteed. This technique to detect genuine multi-
1 of genuine multipartite entanglement [10–12]. However,
partite entanglement in device-independent manner was
1 detection of this kind of resource in an experiment turns
first introduced in [17–20] followed by an extensive for-
4 out to be quite difficult. Experimental demonstration of
0 genuinemultipartiteentanglementisgenerallyperformed malization by Bancal et al. [21]. In particular they
. introduced the term Device-Independent Entanglement
1 with one of the two following techniques: tomography of
Witness(DIEW)ofgenuinemultipartiteentanglementfor
0 the full quantum state [13, 14], or evaluation of an en-
such Bell expressions. Later, Pal [22] and Liang et al.
7 tanglementwitness[1]. Butbothofthesetechniquesface
1 [23] developed other DIEWs for detecting genuine mul-
somecommondrawbacksviz. requirementofprecisecon-
: tipartite entanglement. Throughout the paper, we refer
v trol(bytheexperimentalist)overthesystemsubjectedto
to the procedure of detecting genuine entanglement as
i measurements and sensitivity of these techniques to sys-
X device-independent entanglement detection(DIED) and
tematic errors [15].
r the entanglement detected in device-independent way as
a However, there exists a way to avoid this sort of draw- device-independent entanglement(DIE).
Entanglementswapping[24]asaresourcehasbeenused
in a number of quantum information processing tasks
∗Electronicaddress: [email protected] such as entanglement concentration or distillation, pu-
†Electronicaddress: kaushiki mukherjee@rediffmail.com rification, speeding up the distribution of entanglement,
‡Electronicaddress: [email protected] correctionofamplitudeerrorsdevelopedduetopropaga-
§Electronicaddress: [email protected],[email protected]
tion, activation of nonlocality etc. Entanglement swap-
¶Electronicaddress: [email protected]
∗∗Electronicaddress: [email protected] ping in tripartite scenario mainly involves a network of
††Electronicaddress: [email protected] three parties say, Alice, Bob and Charlie. The proce-
2
dure of entanglement swapping was first generalized for tripartite bi-separable state ρ where ρ =
bi bi
multi-party scenario in [25]. In the present paper, we
(cid:88) (cid:79) (cid:88) (cid:79) (cid:88) (cid:79)
address the following questions: consider some tripartite p ρA ρBC+ p ρB ρAC+ p ρC ρAB.
λ λ λ µ µ µ ν ν ν
states whose genuine entanglement cannot be detected λ µ ν
by applying some standard DIEWs [17, 19, 21, 23], now (1)
(cid:80) (cid:80) (cid:80)
is it possible to find some suitable entanglement swap- Here 0 ≤ pλ, pµ,pν ≤ 1 and λpλ+ µpµ+ νpν =
ping process, after which the genuine entanglement of 1. To be precise, if there exists a state of the form
swapped state can be detected by those DIEWs? We given by Eq.(1) in some Hilbert space H and some
answer this question affirmatively and have designed a suitable local measurement operators M , M and
a|x b|y
protocol based on entanglement swapping procedure by M (without loss of generality these operators can be
c|z
whichgenuineentanglementofthetripartitestateresult- considered to be projection operators satisfying the re-
(cid:80)
ing from multiple swapping can be detected in a device striction Ma|xMa(cid:48)|x =δa,a(cid:48)Ma|x and aMa|x =I) such
independentway,i.e. withoutanyreferenceofthedevice that:
involved. Precisely speaking, this new protocol enhances (cid:79) (cid:79)
p(abc|xyz)=tr[M M M ρ ] (2)
the regime of DIED for tripartite quantum states. In a|x b|y c|z bi
this context another important question is whether one
If the correlations are not biseparable, then the state
canenhancedetectionofgenuineentanglementinasemi-
used is surely a genuine tripartite entangled state.
deviceindependentway(correspondingtophenomenonof
Such a conclusion can be drawn independent of the
quantumsteering). Wealsoanswerthisquestionaffirma-
corresponding Hilbert space dimension. Equivalently,
tively.
biseparable quantum correlations can also be decom-
The rest of this paper has been organized as follows:
posed as,
section II deals with some mathematical preliminaries
and a brief overview of some standard DIEWs. In sec-
(cid:88)
tion III we design the protocol involving entanglement P(abc|xyz)= Pk(ab|xy)Pk(c|z)+
Q Q
swapping procedure followed by detailed discussion on
k
enhancement of entanglement detection by using some
standardDIEWsinsectionIV.Insection.Vwehaveused
(cid:88) (cid:88)
this protocol to enhance genuine entanglement in a semi Pk(ac|xz)Pk(b|y)+ Pk(bc|yz)Pk(a|x) (3)
Q Q Q Q
deviceindependentway. Finallyweconcludewithabrief k k
discussionregardingtheimportanceofthisworkandpos-
wherePk(ab|xy)andPk(c|z)denotearbitrarytwoparty
sible further extensions. Q Q
andonepartyquantumcorrelationsrespectively. Sothey
are of the form: Pk(ab|xy) = tr[Mk a|x(cid:78)Mk ρk ]
Q a|x b|y AB
and Pk(c|z) = tr[Mk ρk] for some unnormalized
Q c|z C
II. BACKGROUND quantum states ρkAB, ρkC and measurement operators
Mk ,Mk ,Mk .
a|b b|y c|z
LetQ denotesthesetoftripartitequantumcorrelations
A. Notion of DIEWs 3
and Q denotes the set of biseparable quantum corre-
2|1
lations. Clearly, Q ⊆ Q . The set Q being convex,
2|1 3 2|1
Violation of Bell inequality by quantum mechanical can be characterized by linear inequalities. DIEWs of
systemsalwaysindicatespresenceofentanglement. Thus genuine tripartite entanglement correspond to those in-
aBellinequalitycanbeconsideredasasuitablecandidate equalities(Bell inequalities) that separate the sets of Q
3
fordetectingthepresenceofentanglementinadevicein- and Q . Now as Q has infinite number of extremal
2|1 2|1
dependent way unlike the standard procedures like state points so there exist many such DIEWs separating gen-
tomography or use of entanglement witnesses where ex- uine entanglement from bi-separable entanglement. In
perimentalist needs to trust the experimental apparatus. recenttimesmanysuchDIEWsaredesignedfordetecting
This is because detection of entanglement using Bell in- genuine tripartite entanglement in a device independent
equalitysolelydependsonthestatisticaldata. Tocharac- way [17–23]. As already mentioned in the introduction,
terizegenuineentanglementinadevice-independentway Bancal [21] was the first to formalize the concept of de-
for tripartite scenario where each of the three subsys- vice independent detection of entanglement introducing
tems,oneofmpossiblemeasurementscanbeperformed, the term DIEW for detecting genuine multipartite en-
yieldingoneoftwopossibleoutcomes. Themeasurement tanglement. In this context, one can consider the DIEW
settings are denoted by x, y, z ∈ {0,1,2,...m−1} and providedbytheMerminpolynomial[26]asthemostsim-
theiroutputsbya,b,c∈{-1,1}forAlice,BobandChar- pleexamplefordetectinggenuinetripartiteentanglement
lie respectively. The experiment is thus characterized by [17]. In[19]Uffink,designedanothernonlinearBell-type
the joint probability distribution p(abc|xyz). The corre- inequality which has been extensively used for this pur-
lations P(abc|xyz) can be categorized as bi-separable if pose. In recent times, Bancal et al. gave more efficient
they can be reproduced through the measurements on a 3-settings Bell inequality which can be used as a DIEW
3
todetectgenuinetripartiteentanglement[21]. Morethan measurementon2nd particleofρ (ρ2)and1st particleof
1 1
3 setting DIEWs are also provided in [22]. However in ρ (ρ1); A performs Bell basis measurement on 3rd par-
2 2 3
our present topic of discussion, we restrict our search for ticle of ρ (ρ3) and 2nd particle of ρ (ρ2). After all the
1 1 2 2
not more than 3-settings Bell inequalities due to obvi- three parties have performed Bell basis measurement on
ous computational complexity. More recently another 2 their respective particles, they communicate the results
settings DIEW was designed by Liang et al.[23](see Ap- amongthemselves,asaresultofwhichρ isgeneratedat
4
pendix.A). As detection of genuine nonlocality by any the end of the preparation stage. Clearly ρ varies with
4
Bell inequality implies genuine entanglement [27, 28] so theoutputoftheBellmeasurements. Thefinalstateρ is
4
it is a DIEW for detecting genuine entanglement. How- obtained from the initial states ρ (i = 1,2,3) by means
i
evertheconverseisnotnecessarilytrue. Afterdiscussing of post-selecting on particular results of local measure-
aboutDIEWandtheiradvantagesoverusualprocedures ments, in particular Bell basis measurements performed
of detecting entanglement experimentally, we are now in on these states(ρ (i = 1,2,3)). For example, let us con-
i
a position to use them for our purpose of detecting gen- sider the case when all the parties obtain outputs corre-
uine entanglement in an entanglement swapping proto- sponding to |ψ±(cid:105) = |01(cid:105)√±|10(cid:105). If the output of all mea-
col. This in turn helps to enhance the chance of genuine 2
surements correspond to |ψ+(cid:105)(|ψ−(cid:105)), the resultant state
tripartite entanglement being detected in a device inde- ρ+(ρ−) is given by:
pendent way. But before that we illustrate our multiple 4 4
entanglement swapping scenario. ρ± = (cid:104)ψ±|A2 ⊗(cid:104)ψ±|A3 ⊗(cid:104)ψ±|A4
4
(ρ ⊗ρ ⊗ρ )
1 2 3
|ψ±(cid:105)A2 ⊗|ψ±(cid:105)A3 ⊗|ψ±(cid:105)A4 (4)
III. MULTIPLE ENTANGLEMENT SWAPPING
PROCEDURE
So preparation stage of this protocol can be consid-
ered as a particular instance of Stochastic Local Oper-
Considerthemultipleentanglementswappingnetwork ation and Classical Communication (SLOCC). After ρ±
4
given in Fig.1. It is a network of six space-like separated is generated and shared among the parties in the prepa-
observers. Three tripartite quantum states ρi(i=1,2,3) ration stage, each of the three parties A1, A5 and A6
are used in the network. State ρ1 is shared among the performs projective measurement on the state ρ± in the
parties Ai(i = 1,2,3) such that jth particle(ρj1) of ρ1 is measurement stage. Now if the correlations ge4nerated
withpartyAj(j =1,2,3)respectively. Stateρ2 isshared from ρ± exhibit violation of any DIEW under the con-
4
among A2, A3 and A4 with the specification that jth textthattheinitialstatesρi(i=1,2,3)failtorevealthe
qubit(ρj2) is sent to party Aj+1(j =1,2,3). The remain- same, thenthatguaranteesenhancementofDIEDinour
ing state ρ3 is shared among A4, A5 and A6 such that protocol.
party A holds jth(j = 1,2,3) particle of ρ (ρj). So
j+3 3 3
each of the three parties A , A and A holds two parti-
2 3 4
cles: A2 holdsρ21 andρ12;A3 holdsρ31 andρ22;A4 holdsρ32 IV. ENHANCEMENT OF
and ρ1. Now in the preparation stage, each of the three DEVICE-INDEPENDENT ENTANGLEMENT
3
parties A (i = 1,2,3) performs Bell basis measurements DETECTION POSSIBILITY
i
on two of the three particles that each of them holds:
A1 performs Bell basis measurement on 3rd particle of Inthissectionwedealwiththeprocedureofenhancing
ρ2(ρ32) and 1st particle of ρ3(ρ13); A2 performs Bell basis DIED of tripartite quantum states in terms of expand-
ing the set of states by using the multiple entanglement
swappingprotocoldescribedinFig.1. Forthisweprovide
anexplicitexample. Initiallyweconsiderthreetripartite
quantum states ρ (i=1,2,3) with some restricted range
i
ofstateparameters,foreachofwhichnoneoftheDIEWs
proposed in the literature[21, 23, 26] can detect genuine
entanglement. These states, after being used in the mul-
tipleentanglementswappingnetwork(Fig.1), generatesa
state ρ whose genuine entanglement can be detected in
4
a device-independent manner.
Let the three initial states be given by:
ρ =p|ψ (cid:105)(cid:104)ψ |+(1−p)|001(cid:105)(cid:104)001| (5)
1 f f
with |ψ (cid:105) = cosθ|000(cid:105)+sinθ|111(cid:105), 0 ≤ θ ≤ π and 0 ≤
f 4
p≤1;
FIG. 1: Swapping scheme
ρ =p |ψ+(cid:105)(cid:104)ψ+|+(1−p )|010(cid:105)(cid:104)010| (6)
2 1 m m 1
4
with |ψm+(cid:105)= |000(cid:105)√+2|111(cid:105) and 0≤p1 ≤1;
ρ =p|ψ (cid:105)(cid:104)ψ |+(1−p)|100(cid:105)(cid:104)100| (7)
3 l l
with|ψ (cid:105)=sinθ|000(cid:105)+cosθ|111(cid:105). Noweachofthethree
l
parties A , A and A performs Bell basis measurement
2 3 4
ontheirrespectiveparticle. Asalreadystatedbefore,the
outputstatedependsontheoutputsoftheBellmeasure-
ments performed. For instance when |ψ±(cid:105) = |01(cid:105)√±|10(cid:105) is
2
obtained as the output odd number of times, a resultant
state ρ± is obtained which after correcting phase term is
4
given by:
ρ± =p |ψ±(cid:105)(cid:104)ψ±|+(1−p )|100(cid:105)(cid:104)100| (8)
4 f m m f FIG. 2: (color online)The shaded region denotes the
where |ψm±(cid:105) = |000(cid:105)√±2|111(cid:105) and pf = 12+ppccooss22θθ. Clearly wrhaincghethoef p3a-sraetmtientgesrsBoefllsitnaeteqsuaρl1ityan(dEqρ.3((9)p)1i≤s v23io)lafoterd
ρ±4 is independent of p1, but the probability of obtaining only after distributing them in the multiple
ρ±4 directly depends on it. Here reader must note that entanglement swapping network as described in Sec.III,
ρ±4 can also be generated for some other combination of i.e. neither of the three initial states violate Eq.(9)
swappingnetworkstogetherwithsomedifferentarrange- whereas the final state violates it. Hence this region
ment of particles in between the parties Ai(1,...,6) and gives the range where DIED is enhanced.
for different outputs of the Bell measurement. To detect
DIE of each of the states ρ (i = 1,2,3,4) we obtain the
i
condition for which they violate each of the DIEWs(see
Appendix.A) given in [17, 19, 21, 23]. Among all, the There exists a range of the state parameters where the
3-settings Bell inequality initial states ρ (i = 1,2,3) do not violate Bancal’s 3-
i
√ settingsBellinequality,butafterdistributingtheminthe
3 multiple entanglement swapping network, final states ρ±
((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) 4
2 0 0 0 2 0 0 1 1 0 2 1 0 violatesit. Therangeofstateparametersinwhichdetec-
tion of DIE is enhanced by this entanglement swapping
protocol(see Fig.2) is given by: p ≤ 2 and
−(cid:104)A0B2C0(cid:105)−(cid:104)A1B2C0(cid:105)−(cid:104)A1B0C1(cid:105)−(cid:104)A2B0C1(cid:105) 1 3
1 2
−(cid:104)A B C (cid:105)−(cid:104)A B C (cid:105)−(cid:104)A B C (cid:105) <p≤ . (12)
0 1 1 1 1 1 0 2 1
cos2θ+1 3sin2θ
+(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)+
2 2 1 0 0 2 1 0 2 0 1 2 The restrictions imposed on the state parameters(Fig.2)
indicate that DIED is enhanced at the end of the swap-
(cid:104)A B C (cid:105)+(cid:104)A B C (cid:105)+(cid:104)A B C (cid:105))≤9 (9) ping procedure. In this context, it is interesting to note
2 1 2 1 2 2 2 2 2
that the probability of success of this protocol psucc is
given by Bancal et al.[21] is the most efficient DIEW for given by
each of ρ (i = 1,2,3,4)(see Table.I). Here (cid:104)A B C (cid:105)
i α β γ
designate the expected value of the product of three ±1
1
observables, Aα, Bβ, Cγ. psucc = 2pp1[1+pcos2θ]sin2θ.
Now the initial states ρ (i = 1,2,3) do not violate
i
Inrecenttimestherehasbeenexperimentalimplementa-
Eq.(9) if and only if(see Appendix.A)
tion of DIED [29] and also experimental demonstration
2 of entanglement swapping [30, 31]. So our procedure of
p ≤
1 3 enhancing DIED method can also be demonstrated ex-
perimentallywithinthescopeofcurrenttechnology. The
and
factthatthisexplicitexampleshowsenhancementofthe
2 possibilityofentanglementdetectioninadeviceindepen-
p≤ (10)
3sin2θ dent manner indicates that for any genuinely entangled
tripartitestate(ρ,say)itmaybepossibletodesignasuit-
The condition of violation of Eq.(9) for the final states able swapping protocol via which entanglement of the fi-
ρ±4 is given by(see Appendix.A): nalstateresultingfromtheprotocolusingmanycopiesof
the initial state(ρ), can be detected even when the same
1
p> (11) cannot be detected for ρ itself.
cos2θ+1
5
DIEW Violation by ρ Violation by ρ Violation by ρ Enhanced range
1 2 3
Mermin[26] p>√2s1in2θ p1>√12 p>√2s1in2θ, (2√2−2)1cos2θ+1<p≤√2s1in2θ
Uffink[19] p>√2s1in2θ p1>√12 p>√2s1in2θ, (2√2−2)1cos2θ+1<p≤√2s1in2θ
Bancal et al.[21] p> 2 p >2 p> 2 , 1 <p≤ 2
3si√n2θ 1 √3 3si√n2θ √cos2θ+1 3sin2θ
Liang et al.[23] p>53sin22θ p1>352 p>53sin22θ, 53sin22θ<p≤(5√2−2)1cos2θ+1
3
TABLE I: The condition of violation of each of the DIEWs given in [19, 21, 23, 26] for each of the
states(ρ (i=1,2,3)) are enlisted here. These conditions restrict the state parameters of the corresponding states
i
such that these in turn give the enhanced region for detection of DIE in the swapping procedure. Moreover
comparison of these restrictions(for each of these states) in turn clearly justifies our claim that the DIEW given by
Bancal et al.[21] emerges to be efficient tool for the detection of genuine entanglement in a device independent way.
V. ENHANCEMENT OF SEMI
DEVICE-INDEPENDENT ENTANGLEMENT
DETECTION POSSIBILITY
Cavalcanti et. al. in [32] have provided an inequality
which detect genuine entanglement in semi device inde-
pendent way. The inequality looks like:
1−0.1831((cid:104)A B (cid:105)+(cid:104)A Z(cid:105)+(cid:104)B Z(cid:105))−0.2582((cid:104)A B X(cid:105)
3 3 3 3 1 1
−(cid:104)A B Y(cid:105)−(cid:104)A B Y(cid:105)−(cid:104)A B X(cid:105))≥0 (13)
1 2 2 1 2 2
Following the same procedure as that for device in-
dependent case, it is observed that the initial states
ρ (i = 1,2,3) do not violate Cavalcanti et al. inequality
i
if and only if
FIG. 3: Shaded region gives the restrictions imposed on
the state parameters for which enhancement of DIED is
1.1831
p≥ observed via the multiple swapping procedure under the
0.7324+1.0328sin[2θ] restriction of p ≥0.670236 over the state parameter p
1 1
of ρ .
2
and
p ≥0.670236 (14)
1 VI. CONCLUSION
Now,theconditionofviolationoftheinequality(Eq.(13))
In a nutshell, our present topic of discussion may be
for the final state ρ is given by:
4 considered as a contribution in the field of device in-
dependent entanglement detection which minimizes the
2pcos2[θ]
p> (15) requirement of precise control over measurement devices
1+pcos[2θ]
by an experimentalist in an experimental detection of
entanglement. More precisely, in our work we have
Thusthereexistsarangeofthestateparameters(p,p1) shown that it is possible to enhance device independent
where the initial states ρi(i = 1,2,3) do not violate the detection of genuine tripartite entanglement in some
inequalityEq.(13),butafterdistributingtheminthenet- suitablemeasurementcontext. Forourpurpose, wehave
work and executing the protocol, final states ρ4 violates considered four DIEWs given by Mermin[26], Uffink[19],
it. The range of state parameters in which enhancement Bancal[21] and Liang et. al.[23], out if which the
of detection of semi-device independent entanglement is DIEW given by Bancal et.al.[21] emerges to be the most
observed by our protocol is given by: p1 ≤0.670236 and efficient. We have designed a state preparation protocol
(prior to receiving final measurements), particularly an
2pcos2[θ] 1.1831 entanglement swapping procedure involving six distant
<p< (16)
1+pcos[2θ] 0.7324+1.0328sin[2θ] observers via which genuine tripartite entanglement of
the resultant(swapped) state, generated by using three
which indicates a clear advantage as shown in Fig.3. initial tripartite states(whose entanglement cannot be
6
detected by the standard DIEWs), can be detected by edgefruitfuldiscussionswithProf. GuruprasadKar. We
these standard DIEWs(used for testing entanglement of also thank Tamal Guha and Mir Alimuddin for useful
the initial states) after performing the state preparation. discussions. AM acknowledges support from the CSIR
project 09/093(0148)/2012-EMR-I.
Acknowledgment-We would like to gratefully acknowl-
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[2] A.S.SørensenandK.Mølmer,Phys.Rev.Lett.86,4431 P.H. Souto Ribeiro, and S. P. Walborn, Nat. Commun.
(2001). 6, 7941 (2015).
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Wieczorek, H. Weinfurter, L. PezzAˆ´e, and A. Smerzi, Theor. 48, 465302 (2015).
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5188 (2001). Appendix A: Condition for violation of DIEWs
[5] H.J. Briegel et al., Nat. Phys. 5, 19 (2009).
[6] M. Murao, D. Jonathan, M.B. Plenio and V. Vedral,
We are now going to enlist the DIEWs which are used
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astoolsforDIEDinmaintext. Tostartwithonecancon-
[7] M. Hillery, V. Buzek, A. Berthiaume, Phys. Rev. A. 59,
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(2001). ample for detecting genuine tripartite entanglement [17]:
[9] Z. Zhao, Y.A. Chen, A.N. Zhang, T. Yang, H.J. Briegel
√
and J.W. Pan, Nature 430, 54 (2004). 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)|≤2 2
1 0 0 0 1 0 0 0 1 1 1 1
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[11] W-B. Gao et al. Nature Phys. 6, 331-335 (2010).
In [19] Uffink, designed another nonlinear Bell-type in-
[12] T. Monz et al. Phys. Rev. Lett. 106, 130506 (2011).
equalitywhichhasbeenextensivelyusedforthispurpose:
[13] J. B. Altepeter, D. F. V. James and P. G. Kwiat, Lect.
Notes Phys. 649, 113-145 (2004).
[14] A. I. Lvovsky, Rev. Mod. Phys. 81, 299-332 (2009).
(cid:104)A B C +A B C +A B C −A B C (cid:105)2+
[15] D. Rosset, R. Ferretti-Sch¨obitz, J-D. Bancal, , N. Gisin 1 0 0 0 1 0 0 0 1 1 1 1
and Y-C. Liang, Phys. Rev. A 86, 062325 (2012).
[16] J.S. Bell, Physics 1, 195 (1964).
(cid:104)A B C +A B C +A B C −A B C (cid:105)2 ≤8 (A2)
[17] M. Seevinck, J. Uffink, Phys. Rev. A 65, 012107 (2001). 1 1 0 0 1 1 1 0 1 0 0 0
[18] K. Nagata, M. Koashi, N. Imoto Phys. Rev. Lett. 89,
260401 (2002). In recent times, Bancal et al.[21] have provided a more
[19] J. Uffink, Phys. Rev. Lett. 88, 230406 (2002). efficient 3-settings Bell inequality(already discussed in
[20] M. Seevinck, G. Svetlichny, Phys. Rev. Lett. 89, 060401 details in main text) which can be used as a DIEW to
(2002). detect genuine tripartite entanglement:
[21] J.-D.Bancal,N.Gisin,Y.-C.LiangandS.Pironio,Phys. More than 3 setting DIEWs are also provided in [22].
Rev. Lett. 106, 250404 (2011). However in our present topic of discussion, we restrict
[22] K.F.Pa´landT.V´ertesi,Phys.Rev.A83,062123(2011).
our search for only 3-settings Bell inequalities due
[23] Y.-C. Liang, D. Rosset, J.-D. Bancal, G. Pu¨tz, T. J.
to obvious computational complexity. More recently
Barnea, and N. Gisin, Phys. Rev. Lett. 114, 190401
another 2 settings DIEW has been designed by Liang et
(2015).
al.[23] :
[24] M.Zukowski,A.Zeilinger,M.A.Horne,andA.K.Ekert,
Phys. Rev. Lett. 71, 4287 (1993).
[25] S. Bose, V. Vedral, and P. L. Knight, Phys. Rev. A 57,
1
822 (1998). ((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)+
4 0 0 0 0 1 0 0 0 1 0 1 1
[26] N.D. Mermin, Phys. Rev. Lett. 65, 1838 (1990).
[27] G. Svetlichny, Phys. Rev. D 35, 3066 (1987).
[28] J.-D. Bancal, J. Barrett, N. Gisin, and S. Pironio, Phys. √
(cid:104)A B C (cid:105)+(cid:104)A B C (cid:105)+(cid:104)A B C (cid:105)−3(cid:104)A B C (cid:105))≤ 2.
Rev. A 88, 014102 (2013). 1 0 0 1 1 0 1 0 1 1 1 1
[29] J. T. Barreiro, J.-D. Bancal, P. Schindler, D. Nigg, M. (A3)
Hennrich, T. Monz, N. Gisin, and R. Blatt, Nat Phys 9, Now we present the detailed proofs of the results stated
559 (2013). in the main text. To obtain the condition of violation of
[30] J. W. Pan, D. Bouwmeester, H. Weinfurter, and A. eachoftheDIEWs(Eqs.(A1,A2,9,A3))intermsofstate
Zeilinger, Phys. Rev. Lett. 80, 3891 (1998). parameters for each of the initial states ρ (i = 1,2,3)
i
[31] A. M. Goebel, C.Wagenknecht, Q. Zhang, Y.-A. Chen, and final state ρ , we apply the same method as used in
4
K. Chen, J. Schmiedmayer, and J.-W. Pan, Phys. Rev.
[33]. First we find the condition of violation(in terms of
Lett. 101, 080403 (2008).
state parameters) of the DIEW given in Eq.(A1) for the
7
initialstateρ . Weconsiderthefollowingmeasurements: (cid:126)x = (sinθa cosφa ,sinθa sinφa ,cosθa ), (cid:126)y =
1 0 0 0 0 0
A0 = (cid:126)x.σ(cid:126)1 or A1 = (cid:126)x´.σ(cid:126)1 on 1st qubit, B0 = (cid:126)y.σ(cid:126)2 or (sinαb0cosβb0,sinαb0sinβb0,cosαb0) and (cid:126)z =
B1 = (cid:126)y´.σ(cid:126)2 on 2nd qubit, and C0 = (cid:126)z.σ(cid:126)3 or C1 = (cid:126)z´.σ(cid:126)3 (sinζc0cosηc0,sinζc0sinηc0,cosζc0) and similarly,
on 3rd qubit, where (cid:126)x,(cid:126)x´,(cid:126)y,(cid:126)y´ and (cid:126)z,(cid:126)z´ are unit vectors we define, (cid:126)x´,(cid:126)y´ and (cid:126)z´ by replacing 0 in the indices by
and σ are the spin projection operators that can be 1. Then the value of the operator M (Eq.(A1)) with
i
written in terms of the Pauli matrices. Representing respect to the state ρ1 (Eq.(5)) gives:
the unit vectors in spherical coordinates, we have,
M(ρ ) = −cosαb (−1 + p + pcos2θ)(cosζc cosθa + cosζc cosθa ) − sinαb (p sin2θ )(cos(βb + ηc +
1 1 0 1 1 0 1 1 1 1 1
φa )sinζc sinθa +cos(βb +ηc +φa )sinζc sinθa )+cosαb (−1+p+pcos2θ)(cosζc cosθa −cosζc cosθa )+
0 1 0 1 0 1 0 1 0 0 0 1 1
sinαb (psin2θ)(cos(βb +ηc +φa )sinζc sinθa −cos(βb +ηc +φa )sinζc sinθa ). (A4)
0 0 0 0 0 0 0 1 1 1 1
HenceinordertogetmaximumvalueofS(ρ ),wehaveto ifwemaximizethelastequationwithrespecttoαb and
1 0
performmaximizationover12measurementangles. Now αb , we have
1
(cid:112)
M(ρ )≤ ((X)(cosζc cosθa +cosζc cosθa ))2+(Y)2(A sinζc sinθa +A sinζc sinθa )2
1 0 1 1 0 110 1 0 101 0 1
(cid:112)
+ ((X)(cosζc cosθa −cosζc cosθa ))2+(Y)2(A sinζc sinθa −A sinζc sinθa )2 (A5)
0 0 1 1 000 0 0 011 1 1
Where X = −1+p+pcos2θ, Y = psin2θ, and A = ment angles θa , ζc and θa , ζc that the right hand
ijk 0 0 1 1
cos(βb +ηc +φa )(i,j,k ∈ {0,1}). The last inequal- side of Eq.(A5) gives maximum value when θa = ζc
i j k 0 0
ity is obtained by using the inequality xcosθ+ysinθ ≤ and θa =ζc . Hence Eq.(A5) takes the form:
1 1
(cid:112)
x2+y2. Itisclearfromthesymmetryofthemeasure-
(cid:112)
M(ρ )≤ ((X)(2cosθa cosθa ))2+(Y sinθa sinθa )2(A +A )2+
1 0 1 0 1 110 101
(cid:113)
((X)(cos2θa −cos2θa ))2+(Y)2(A sin2θa −A sin2θa )2 (A6)
0 1 000 0 011 1
(cid:113)
Again we maximize it with respect to θa1. Critical point + X2cos4θa +Y2(A sin2θa −A )2
0 or π gives the maximum value depending on values of 0 000 0 011
2
the state parameters. For the critical point 0, Eq.(A6)
becomes (cid:112) (cid:113)
≤ 4(Y sinθa )2+ X2cos4θa +Y2(sin2θa +1)2
0 0 0
(cid:113)
(cid:112)
M(ρ )≤ (2Xcosθa )2+ sin4θa (X2+Y2) (A7)
1 0 0
≤4|Y| (A9)
where we have chosen A2 = 1. Maximizing over θa ,
000 0
we get The second inequality in Eq.(A9) is obtained from the
first by setting A =1, A =1, A =1 and A =
2X2+Y2 110 101 000 011
M(ρ )≤ √ (A8) −1.Thefinalinequalityisachievedwhenθa = π . Two
1 X2+Y2 sets of measurement angles which realize th0e two2 values
tthheemotahxeirmcurimticbaelinpgoionbttaπ2i,nEedq.f(oAr6c)ostaθkae0s=th√eXf|oX2r+m|Y:2. For ζ√2cXX022++=YY22c(oEs−q1.((A√8X)|X2)+|aYn2d),4|θYa|1(=Eq.α(Ab19)=), aζrce1 θ=a00=, βαbbi0 ==
ηc = φa = 0 (i = 0, 1) and θa = αb = ζc = π(i =
(cid:112) i i i i i 2
M(ρ )≤ (Y sinθa )2(A +A )2 0, 1) , βb = ηc = φa = 0, βb = −ηc = −φa = π
1 0 110 101 0 0 0 1 1 1 2
8
respectively. Hence from Eq.(A8) and Eq.(A9), we have However among the four DIEWs given by Mer-
min(Eq.(A1)), Uffink(Eq.(A2)), Bancal et al.(Eq.(9))
2X2+Y2 and Liang et al.(Eq.(A3)), the one given by Bancal et
M(ρ )≤max[√ ,4|Y|]. (A10)
1 X2+Y2 al. turns out to be the most efficient for this purpose.
The DIEW based on Bancal et al. polynomial (Eq.(9))
√
Clearly, √2X2+Y2 ≤ 2 < 2 2 for any value of p ∈ [0,1] can thus detect genuine tripartite entanglement in
and 0≤θ ≤X2+π4Y.2So the initialstate ρ1 violates the DIEW a device-independent way in ρ1 for√p > 3si2n2θ(see
based on Mermin expression (Eq.(A1)) if TableIV.). As 2 < √ 1 < 3 2 , so the DIEW
3sin2θ 2sin2θ 5sin2θ
√ based on Bancal et al. polynomial (Eq.(9)) is the
4|Y|=4|p|sin2θ >2 2. (A11) most efficient DIEW for the state ρ to detect genuine
1
tripartite entanglement among all the standard DIEWs
The last inequality is considered as the condition of vio-
considered in Eqs.((A1), (A2), (9), (A3)). Similarly by
lation of the DIEW based on Mermin expression for the
comparing the range of violation of p (for the state ρ )
1 2
initial state ρ . We have applied the same method over
1 andp(forthestateρ ,ρ ),onecancheckthatBancalet
3 4
other states ρ (i = 2, 3, 4)to find the condition of viola-
i al. Bell inequality is the best DIEW for the other states
tionoftheDIEWbasedonMerminexpression. Forother
ρ (i = 2, 3, 4) to detect genuine tripartite entanglement
i
DIEWS (Eqs.(A2), (9), (A3)), we have made analysis in
compared to other standard DIEWs.
similar manner so as to obtain the condition of violation
for each of states ρ . All the conditions are summarized
i
in Table.I.