Tripartite entanglement detection through tripartite quantum steering in one-sided and two-sided device-independent scenarios
C. Jebaratnam, Debarshi Das, Arup Roy, Amit Mukherjee, Some Sankar Bhattacharya, Bihalan Bhattacharya, Alberto Riccardi, Debasis Sarkar
TTripartite entanglement detection through tripartite quantum steering in one-sided and two-sideddevice-independent scenarios
C. Jebaratnam, ∗ Debarshi Das, † Arup Roy, Amit Mukherjee, Some SankarBhattacharya, Bihalan Bhattacharya, Alberto Riccardi, and Debasis Sarkar ‡ S. N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 098, India Centre for Astroparticle Physics and Space Science (CAPSS),Bose Institute, Block EN, Sector V, Salt Lake, Kolkata 700 091, India Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India. Optics & Quantum Information Group, The Institute of Mathematical Sciences,HBNI, C.I.T Campus, Tharamani, Chennai 600 113, India Dip. Fisica and INFN Sez. Pavia, University of Pavia, via Bassi 6, I-27100 Pavia, Italy Department of Applied Mathematics, University of Calcutta, 92, A.P.C. Road, Kolkata-700009, India. (Dated: July 19, 2018)In the present work, we study tripartite quantum steering of quantum correlations arising from two localdichotomic measurements on each side in the two types of partially device-independent scenarios: 1-sideddevice-independent scenario where one of the parties performs untrusted measurements while the other twoparties perform trusted measurements and 2-sided device-independent scenario where one of the parties per-forms trusted measurements while the other two parties perform untrusted measurements. We demonstrate thattripartite steering in the 2-sided device-independent scenario is weaker than tripartite steering in the 1-sideddevice-independent scenario by using two families of quantum correlations. That is these two families of quan-tum correlations in the 2-sided device-independent framework detect tripartite entanglement through tripartitesteering for a larger region than that in the 1-sided device-independent framework. It is shown that tripartitesteering in the 2-sided device-independent scenario implies the presence of genuine tripartite entanglement of2 × × PACS numbers: 03.65.Ud, 03.67.Mn, 03.65.Ta
I. INTRODUCTION
Multipartite entanglement is a resource for quantum infor-mation and computation when quantum networks are consid-ered. Therefore, detecting the presence of multipartite en-tanglement in quantum networks is an important problem inquantum information science. In particular, a genuinely mul-tipartite entangled state (which is not separable with respect toany partitions) [1] is important not only for quantum founda-tional research but also in various quantum information pro-cessing tasks, for example, in the context of extreme spinsqueezing [2], high sensitive metrology tasks [3, 4]. Gener-ation and detection of this kind of resource state is found to bedi ffi cult as the detection process deals with tomography andevaluation via constructing entanglement witness which re-quire precise experimental control over the system subjectedto measurements. But there is an alternative way to certifythe presence of entanglement by observing the violation ofBell inequality [5] as entanglement is necessary ingredient toobserve the violation. Motivated by this fact, a number ofmultipartite Bell type inequalities [6–10, 12] have been pro-posed to detect the genuine multipartite entanglement. To bespecific, if the value of any Bell expression, in a Bell experi-ment, exceeds the value of the same expression obtained dueto measurements on biseparable quantum states, then the pres- ∗ [email protected] † [email protected] ‡ [email protected] ence of genuine entanglement is guaranteed. This kind of re-search was first initiated in [6, 7] but it took a shape by Bancalet. al. [10] where they have constructed device-independententanglement witness (DIEW) of genuine multipartite entan-glement for such Bell expressions.The concept of quantum steering was first pointed out bySchrodinger [13] in the context of Einstein-Podolsky-Rosenparadox (EPR) [11], which has no classical analogue. Quan-tum steering as pointed out by Schrodinger occurs when oneof the two spatially separated observers prepares genuinelydi ff erent ensembles of quantum states for the other distantobserver by performing suitable quantum measurements onher / his side. Wiseman et. al. [14] gave the formal definitionof quantum steering from the foundational as well as quan-tum information perspective. Quantum steering is certifiedby the violation of steering inequalities. A number of steer-ing inequalities have been proposed to observe steering [15].Violation of such steering inequalities certify the presence ofentanglement in a one-sided device-independent way.In Refs. [17, 18], the notion of steering has been gener-alized for multipartite scenarios and multipartite steering in-equalities have been derived to detect multipartite entangle-ment in asymmetric networks where some of the parties’ mea-surements are trusted while the other parties’ measurementsare uncharacterized. These studies did not examine genuinemultipartite steering, in which the nonlocality, in the formof steering, is necessarily shared among all observers. Gen-uine multipartite steering has been proposed in [19, 20]. InRefs. [21, 22], genuine tripartite steering inequalities havebeen derived to detect genuine tripartite entanglement in a par- a r X i v : . [ qu a n t - ph ] J u l tially device-independent way. Characterization of multipar-tite quantum steering through semidefinite programming hasalso been performed [21–23].In the present work, we study tripartite steering (which isanalogous to standard Bell nonlocality) and genuine tripartitesteering of quantum correlations arising from two local mea-surements on each side in the two types of partially device-independent scenarios: 1-sided device-independent scenariowhere one of the parties performs untrusted measurementswhile the other two parties perform trusted measurements and2-sided device-independent scenario where one of the partiesperforms trusted measurements while the other two partiesperform untrusted measurements.In the 1-sided device-independent framework, we study tri-partite steering and genuine tripartite steering of two fami-lies of quantum correlations in the following scenarios: oneof the parties performs two dichotomic black-box measure-ments and the other two parties perform incompatible qubitmeasurements that demonstrate Bell nonlocality [16] in one ofthe types or perform incompatible measurements that demon-strate EPR steering without Bell nonlocality [17, 24] in theother type. The first family of quantum correlation consid-ered by us is called Svetlichny family as it can be obtained byperforming the non-commuting measurements that lead to theviolation of Svetlichny inequality and it violates Svetlichnyinequality in a particular region. On the other hand, the sec-ond family of quantum correlation considered by us is calledMermin family as it can be obtained by performing the non-commuting measurements that lead to the violation of Mer-min inequality and it violates Mermin inequality in a particu-lar region, but it does not violate Svetlichny inequality in anyregion. We demonstrate in which range these two families de-tect tripartite and genuine tripartite steering in the aforemen-tioned 1SDI scenarios, respectively.We also explore in which range the Svetlichny family andMermin family detect tripartite steering and genuine tripartitesteering in the 2-sided device-independent framework.Our study demonstrates that tripartite steering in the 2-sided device-independent framework is weaker than tripar-tite steering in the 1-sided device-independent framework.In other words, tripartite steering in the context of 2-sideddevice-independent framework detect tripartite entanglementfor a larger region than that in the context of 1-sided device-independent framework. We demonstrate that tripartite steer-ing in the 2-sided device-independent scenario implies thepresence of genuine tripartite entanglement of 2 × × × × II. TRIPARTITE NONLOCALITY
We consider a tripartite Bell scenario where three spatiallyseparated parties, Alice, Bob and Charlie, perform two di-chotomic measurements on their subsystems. The correla-tion is described by the conditional probability distributions: P ( abc | A x B y C z ), here x , y , z ∈ { , } and a , b , c ∈ { , } . Thecorrelation exhibits standard tripartite nonlocality (i.e., Bellnonlocality) if it cannot be explained by a fully local hiddenvariable (LHV) model, P ( abc | A x B y C z ) = (cid:88) λ p λ P λ ( a | A x ) P λ ( b | B y ) P λ ( c | C z ) , (1)for some hidden variable λ with probability distribution p λ ; (cid:80) λ p λ =
1. The Mermin inequality (MI) [25], (cid:104) M (cid:105) : = (cid:104) A B C + A B C + A B C − A B C (cid:105) LHV ≤ , (2)is a Bell-type inequality whose violation implies that thecorrelation cannot be explained by a fully local hiddenvariable model as in Eq. (1). Here (cid:104) A x B y C z (cid:105) = (cid:80) abc ( − a ⊕ b ⊕ c P ( abc | A x B y C z ).If a correlation violates a MI, it does not necessarily implythat it exhibits genuine tripartite nonlocality [6, 10]. In Ref.[6], Svetlichny introduced the strongest form of genuine tri-partite nonlocality (see Ref. [10] for the other two forms ofgenuine nonlocality). A correlation exhibits Svetlichny non-locality if it cannot be explained by a hybrid nonlocal-LHV(NLHV) model, P ( abc | A x B y C z ) = (cid:88) λ p λ P λ ( a | A x ) P λ ( bc | B y C z ) + (cid:88) λ q λ P λ ( ac | A x C z ) P λ ( b | B y ) + (cid:88) λ r λ P λ ( ab | A x B y ) P λ ( c | C z ) , (3)with (cid:80) λ p λ + (cid:80) λ q λ + (cid:80) λ r λ =
1. The bipartite probabilitydistributions in this decomposition can have arbitrary nonlo-cality.Svetlichny derived Bell-type inequalities to detect thestrongest form of genuine tripartite nonlocality [6]. For in-stance, one of the Svetlichny inequalities (SI) reads, (cid:104) S (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) ≤ . (4)Quantum correlations violate the SI up to 4 √
2. AGreenberger-Horne-Zeilinger (GHZ) state [26] gives rise tothe maximal violation of the SI for a di ff erent choice of mea-surements which do not demonstrate GHZ paradox [27].In the seminal paper [25], the MI was derived to demon-strate standard tripartite nonlocality of three-qubit correla-tions arising from the genuinely entangled states. For thispurpose, noncommuting measurements that do not demon-strate Svetlichny nonlocality was used. Note that when aGreenberger-Horne-Zeilinger (GHZ) state [26] maximally vi-olates the MI, the measurements that give rise to it exhibit theGHZ paradox [27]. III. DEFINITIONS OF TRIPARTITE EPR STEERING
Before we define tripartite EPR steering, let us review thedefinition of bipartite EPR steering in the following 1-sideddevice-independent scenario. Two spatially separated parties,Alice (who is the trusted party) and Bob (who is the un-trusted party) share an unknown bipartite system describedby the density matrix ρ AB in C d A ⊗ C d B with the dimensionof Alice d A is known and the dimension of Bob d B is un-known. On this shared state, Bob performs black-box mea-surements (positive operator valued measurement, or in short,POVM) with the measurement operators { M b | y } b , y ( M b | y ≥ ∀ b , y ; (cid:80) b M b | y = I ∀ y ), here y and b denote the measurementchoices and measurement outcomes of Bob, respectively, toprepare the set of conditional states on Alice’s side. The abovesteering scenario is characterized by the set of unnormalizedconditional states on Alice’s side { σ Ab | y } b , y , which is called anassemblage. Each element in this assemblage is given by σ Ab | y = Tr B ( ⊗ M b | y ρ AB ).Wiseman et. al. [14] provided an operational definition ofsteering. According to this definition, Bob’s measurements inthe above scenario demonstrates steerability to Alice i ff the as-semblage certifies entanglement. The assemblage which doesnot certify entanglement, i.e., does not imply steerability fromBob to Alice has a local hidden state (LHS) model as follows:for all b , y , each element σ Ab | y in the assemblage admits thefollowing decomposition: σ Ab | y = (cid:88) λ q λ P λ ( b | B y ) ρ λ A , (5)where λ denotes classical random variable which occurs withprobability q λ ; (cid:80) λ q λ = P λ ( b | B y ) are some conditionalprobability distributions and the quantum states ρ λ A are calledlocal hidden states which satisfy ρ λ A ≥ ρ λ A =
1. Sup-pose Alice performs positive operator valued measurements(POVM) with measurement operators { M a | x } a , x ( M a | x ≥ ∀ a , x ; (cid:80) a M a | x = I ∀ x ) on the assemblage to detect steerabil-ity through the violation of a steering inequality. Then thescenario is characterized by the set of conditional probabilitydistributions, P ( ab | A x B y ) = Tr (cid:16) M a | x σ Ab | y (cid:17) . (6)The above quantum correlation P ( ab | A x B y ) detects steerabil-ity if and only if it cannot be explained by a LHS-LHV modelof the form, P ( ab | A x B y ) = (cid:88) λ q λ P ( a | A x , ρ λ A ) P λ ( b | B y ) ∀ a , x , b , y , (7)with (cid:80) λ q λ =
1. Here P ( a | A x , ρ λ A ) are the distributions arisingfrom the local hidden states ρ λ A . On the other hand, the quantum correlation P ( ab | A x B y )demonstrates Bell nonlocality if and only if it cannot be ex-plained by a LHV-LHV model of the form, P ( ab | A x B y ) = (cid:88) λ q λ P λ ( a | A x ) P λ ( b | B y ) ∀ a , x , b , y , (8)with (cid:80) λ q λ =
1. The quantum correlation that does not have aLHV-LHV model also implies steering, on the other hand, thequantum correlation that does not have a LHS-LHV modelmay not imply Bell nonlocality since certain local correla-tions may also detect steering in the given 1-sided device-independent scenario.Let us now focus on the definition of tripartite steering. Inthe tripartite scenario, there are two types of partially device-independent scenarios where one can generalize bipartite EPRsteering. These two scenarios are called 1-sided device-independent (1SDI) and 2-sided device-independent (2SDI)scenarios [23].
A. Tripartite steering in SDI scenario
We will consider the following 1-sided device-independent(1SDI) scenario (depicted in FIG. 1): Three spatially sepa-rated parties share an unknown tripartite quantum state ρ ABC in C ⊗ C ⊗ C d on which Charlie performs black-box mea-surements (POVMs). Suppose M c | z denote the unknown mea-surement operators of Charlie ( M c | z ≥ ∀ c , z ; (cid:80) c M c | z = I ∀ z ). Then, the scenario is characterized by the set of (unnor-malized) conditional two-qubit states on Alice and Bob’s side { σ ABc | z } c , z , each element of which is given as follows: σ ABc | z = Tr C ( ⊗ ⊗ M c | z ρ ABC ) . (9)Alice and Bob can do local state tomography to determine theabove assemblage prepared by Charlie.Analogous to the operational definition of bipartite EPRsteering, we will now provide the operational definition oftripartite steering in the above 1SDI scenario. The assemblage σ ABc | z given by Eq. (9) is called steerable ifi) the assemblage prepared on Alice and Bob’s side cannotbe reproduced by a fully separable state, in C ⊗ C ⊗ C d , ofthe form, ρ ABC = (cid:88) λ p λ ρ λ A ⊗ ρ λ B ⊗ ρ λ C , (10)with (cid:80) λ p λ =
1; andii) entanglement between Charlie and Alice-Bob is de-tected.In the genuine steering scenario, Charlie demonstrates gen-uine tripartite EPR steering to Alice and Bob if the assemblageprepared on Alice and Bob’s side cannot be reproduced by abiseparable state in C ⊗ C ⊗ C d , ρ ABC = (cid:88) λ p λ ρ λ A ⊗ ρ λ BC + (cid:88) λ q λ ρ λ AC ⊗ ρ λ B + (cid:88) λ r λ ρ λ AB ⊗ ρ λ C , (11) × × d z ∈{ } x ∈{ } a ∈{ } y ∈{ } b ∈{ } c ∈{ }σ c ∣ zAB × × d FIG. 1. Schematic diagram of our 1SDI tripartite steering scenario:Alice, Bob and Charlie share a 2 × × d quantum state. Charlie per-forms two dichotomic black-box measurements to produce assem-blages σ ABc | z (9) on Alice and Bob’s side. On this assemblage, Aliceand Bob perform two dichotomic measurements producing the jointprobability distributions P ( abc | A x B y C z ) (here a , b , c denotes the out-comes and x , y , z denotes the measurement choices) to check whetherCharlie demonstrates steerability to them through the violation of asteering inequality by P ( abc | A x B y C z ). In case of the scenario con-sidered in Section IV, Alice and Bob perform incompatible qubitmeasurements that demonstrate Bell nonlocality of certain two-qubitstates [16]; for instance, the singlet state. On the other hand, in caseof the scenario considered in Section V, they perform incompatiblequbit measurements that demonstrate EPR steering without Bell non-locality of certain two-qubit states [17, 24]; for instance, the singletstate. with (cid:80) λ p λ + (cid:80) λ q λ + (cid:80) λ r λ = { M a | x } a , x and { M b | y } b , y , respectively, for detecting tripartite steering. Here M a | x ≥ ∀ a , x ; (cid:80) a M a | x = I ∀ x ; and M b | y ≥ ∀ b , y ; (cid:80) b M b | y = I ∀ y . Then the scenario is characterized by theset of conditional probability distributions, P ( abc | A x B y C z ) = Tr (cid:16) M a | x ⊗ M b | y σ ABc | z (cid:17) , (12)where M a | x and M b | y are the measurement operators of Aliceand Bob, respectively. Suppose the above quantum correlation P ( abc | A x B y C z ) detects tripartite steerability. Then, it cannotbe explained by a fully LHS-LHV model of the form, P ( abc | A x B y C z ) = (cid:88) λ q λ P ( a | A x , ρ λ A ) P ( b | B y , ρ λ B ) P λ ( c | C z ) , (13)with (cid:80) λ q λ =
1. Here P ( a | A x , ρ λ A ) and P ( b | B y , ρ λ B ) are thedistributions arising from the local hidden states ρ λ A and ρ λ B which are in C , respectively. It should be noted that if a quantum correlation does not have an fully LHS-LHV model(13), then it does not necessarily imply that it detects tripartitesteering from Charlie to Alice-Bob [17]. The correlation P ( abc | A x B y C z ) detects tripartite steerability if and only ifi) P ( abc | A x B y C z ) does not have a fully LHS-LHV model asin Eq. (13); andii) entanglement between Charlie and Alice-Bob is detected.The quantum correlation P ( abc | A x B y C z ) that detects tripar-tite steering also detects genuine tripartite steering if it can-not be explained by the following steering LHS-LHV (StLHS)model: P ( abc | A x B y C z ) = (cid:88) λ r λ P ( ab | A x B y , ρ λ AB ) P λ ( c | C z ) + (cid:88) λ p λ P ( a | A x , ρ λ A ) P Q λ ( bc | B y C z ) + (cid:88) λ q λ P ( b | B y , ρ λ B ) P Q λ ( ac | A x C z ) , (14)with (cid:80) λ p λ + (cid:80) λ q λ + (cid:80) λ r λ =
1. Here, P ( a | A x , ρ λ A ) and P ( b | B y , ρ λ B ) are the distributions arising from the qubit states ρ λ A and ρ λ B on Alice’s side and Bob’s side, respectively, P λ ( c | C z ) is the distribution on Charlie’s side arising fromblack-box measurements performed on a d dimensional quan-tum state and P Q λ ( bc | B y C z ) and P Q λ ( ac | A x C z ) are the distri-butions that can be produced from a 2 × d quantum states;and P ( ab | A x B y , ρ λ AB ) can be reproduced by two-qubit quantumstates ρ λ AB shared between Alice and Bob. Note that in themodel given in Eq. (14), the bipartite distributions at each λ level may have Bell nonlocality or steering without Bell non-locality [17, 24]. Equivalently, the quantum correlation thatdetects genuine tripartite steering cannot be reproduced by abiseparable state in C ⊗ C ⊗ C d . B. Tripartite steering in SDI scenario
We will consider the following 2-sided device-independent(2SDI) scenario (depicted in FIG. 2): Three spatially sepa-rated parties share an unknown tripartite quantum state ρ ABC in C ⊗ C d ⊗ C d on which Bob and Charlie performs local black-box measurements (POVMs). Suppose { M b | y } b , y and { M c | z } c , z denote the unknown measurement operators of Bob and Char-lie, respectively. Here M b | y ≥ ∀ b , y ; (cid:80) b M b | y = I ∀ y and M c | z ≥ ∀ c , z ; (cid:80) c M c | z = I ∀ z . Then, the scenario is character-ized by the set of (unnormalized) conditional qubit states onAlice’s side { σ Abc | yz } b , c , y , z . The each element in this assemblageis given as follows: σ Abc | yz = Tr BC ( ⊗ M b | y ⊗ M c | z ρ ABC ) . (15)Alice can do local state tomography to determine the aboveassemblage prepared by Charlie.We will now provide the operational definition of tripar-tite steering in the above 2SDI scenario. The assemblage { σ Abc | yz } b , c , yz is called steerable if it cannot be reproduced by × × d z ∈{ } y ∈{ } b ∈{ } x ∈{ } a ∈{ } c ∈{ }σ bc ∣ yzA × d × d FIG. 2. Schematic diagram of our 2SDI tripartite steering scenario:Alice, Bob and Charlie share a 2 × d × d quantum state. Bob andCharlie perform two dichotomic black-box measurements to produceassemblages σ Abc | yz (15) on Alice’s side. On this assemblage, Aliceperforms two dichotomic measurements producing the joint proba-bility distributions P ( abc | A x B y C z ) (here a , b , c denotes the outcomesand x , y , z denotes the measurement choices) to check whether the as-semblages σ Abc | yz prepared by Bob and Charlie demonstrate steerabil-ity through the violation of a steering inequality by P ( abc | A x B y C z ). a fully separable state in C ⊗ C d ⊗ C d of the form, ρ ABC = (cid:88) λ p λ ρ λ A ⊗ ρ λ B ⊗ ρ λ C , (16)with (cid:80) λ p λ = σ Abc | yz cannot be reproduced by a biseparable stateas given by Eq. (11) in C ⊗ C d ⊗ C d .Suppose in our tripartite 2SDI scenario, the trusted partyAlice performs POVMs having elements { M a | x } a , x for detect-ing tripartite steering. Here M a | x ≥ ∀ a , x ; (cid:80) a M a | x = I ∀ x .Then the scenario is characterized by the set of conditionalprobability distributions, P ( abc | A x B y C z ) = Tr (cid:16) M a | x σ Abc | yz (cid:17) , (17)where M a | x are the measurement operators of Alice. Supposethe above quantum correlation P ( abc | A x B y C z ) cannot be ex-plained by a fully LHS-LHV model of the form, P ( abc | A x B y C z ) = (cid:88) λ q λ P ( a | A x , ρ λ A ) P λ ( b | B y ) P λ ( c | C z ) , (18)with (cid:80) λ q λ = P ( a | A x , ρ λ A ) are the distributions arisingfrom the local hidden states ρ λ A which are in C ). Then, itdetects tripartite steerability. The quantum correlation P ( abc | A x B y C z ) that detects tripar-tite steering in our 2SDI scenario also detects genuine tripar-tite steering if it cannot be explained by the following steeringLHS-LHV (StLHS) model: P ( abc | A x B y C z ) = (cid:88) λ r λ P Q λ ( ab | A x B y ) P λ ( c | C z ) + (cid:88) λ p λ P ( a | A x , ρ λ A ) P λ ( bc | B y C z ) + (cid:88) λ q λ P λ ( b | B y ) P Q λ ( ac | A x C z ) , (19)with (cid:80) λ p λ + (cid:80) λ q λ + (cid:80) λ r λ =
1. Here, P ( a | A x , ρ λ A ) are the dis-tributions arising from the qubit states ρ λ A and, P λ ( b | B y ) and P λ ( c | C z ) are the distribution on Bob’s and Charlie’s sides, re-spectively, arising from black-box measurements performedon a d dimensional quantum state and P Q λ ( ab | A x B y ) and P Q λ ( ac | A x C z ) are the distribution that can be produced froma 2 × d quantum state; and P λ ( bc | B y C z ) can be reproduced bya d × d quantum state. Note that in the model given in Eq. (19),the bipartite distributions at each λ level may have Bell non-locality or steering without Bell nonlocality [17, 24]. Equiv-alently, the quantum correlation that detects genuine tripartitesteering in our 2SDI cannot be reproduced by a biseparablestate in C ⊗ C d ⊗ C d .In the next Section we study in which range two one-parameter families of quantum correlations obtained from lo-cal dichotomic measurements on tripartite quantum states de-tect tripartite quantum steering in our 1SDI and 2SDI scenar-ios. IV. DETECTION OF TRIPARTITE STEERING WITHSVETLICHNY FAMILY
The Svetlichny family of tripartite correlations is definedas: P VS vF ( abc | A x B y C z ) = + ( − a ⊕ b ⊕ c ⊕ xy ⊕ yz ⊕ xz √ V , (20)where 0 ≤ V ≤
1, which can be obtained from the noisy three-qubit GHZ state, ρ = V | Φ GHZ (cid:105)(cid:104) Φ GHZ | + (1 − V ) /
8, where | Φ GHZ (cid:105) = √ ( | (cid:105) + | (cid:105) ), for the measurements that giverise to the maximal violation of the SI; for instance, A = σ x , A = σ y , B = σ x − σ y √ , B = σ x + σ y √ , C = σ x and C = σ y .The noisy three-qubit GHZ state is genuinely entangled i ff V > .
429 [29]. The Svetlichny family certifies genuine en-tanglement in a fully device independent way for V > √ , as itviolates the SI in this range. The Svetlichny family has a fullylocal hidden variable (LHV) model when V ≤ √ [10]. Thisimplies that in this range, it can also arise from a separablestate in the higher dimensional space [28]. A. SDI scenario
We consider a tripartite 1SDI steering scenario where Char-lie performs two dichotomic black-box measurements to pre-pare conditional two-qubit states on Alice and Bob’s side onwhich Alice and Bob perform pair of incompatible qubit mea-surements that demonstrate Bell nonlocality of certain two-qubit states; for instance, the singlet state. Now we are goingto present a Lemma which is useful to find out in which rangesthe Svetlichny family detects genuine tripartite steering andtripartite steering in the context of the above 1SDI scenario.
Lemma 1.
In our SDI scenario mentioned above theSvetlichny family has a steering LHS-LHV model as in Eq.(14) in the range < V ≤ √ and has a fully LHS-LHV modelas in Eq. (13) i ff < V ≤ √ .Proof. See Appendix A. (cid:3)
The above Lemma implies the following two propositions.
Proposition 1.
The Svetlichny family detects genuine tripar-tite steering i ff V > √ in the context of our SDI scenario.Proof.
Since the Svetlichny family violates the Svetlichny in-equality for V > √ , it certifies genuine tripartite entangle-ment in a fully device independent way in that range. Hence,it is followed that the Svetlichny family certifies genuine tri-partite entanglement in our 1SDI scenario as well for V > √ . The Svetlichny family, therefore, does not have a steer-ing LHS-LHV model as in Eq.(14) in our 1SDI scenario for V > √ . On the other hand, following Lemma 1 we can statethat the Svetlichny family has a steering LHS-LHV model asin Eq. (14) in our 1SDI scenario in the range 0 < V ≤ √ .Hence, the Svetlichny family detects genuine tripartite steer-ing i ff V > √ in the context of our 1SDI scenario. (cid:3) Proposition 2.
The Svetlichny family detects tripartite steer-ing i ff V > √ in the context of our SDI scenario.Proof.
Svetlichny family detects entanglement between Char-lie and Alice-Bob for V > √ as it violates the Svetlichnyinequality in this range. Moreover, the steering LHS-LHVmodel given in the proof of Lemma 1 for the Svetlichny fam-ily implies that for V ≤ √ , it can be reproduced by a 2 × × d biseparable state of the form, ρ ABC = (cid:88) λ = r λ ρ λ AB ⊗ | λ (cid:105)(cid:104) λ | , (21)with (cid:80) λ r λ =
1. Therefore, the Svetlichny family detectsentanglement between Charlie and Alice-Bob i ff V > √ .On the other hand, in the context of our 1SDI scenario, theSvetlichny family does not have a fully LHS-LHV model asin Eq. (13) following Lemma 1 for V > √ . Combiningthese two facts we can state that the Svetlichny family de-tects entanglement between Charlie and Alice-Bob and doesnot have a fully LHS-LHV model as in Eq. (13) in the range V > √ following Lemma 1. Hence, in the context of our1SDI scenario, the Svetlichny family detects tripartite steer-ing i ff V > √ . (cid:3) From the Propositions 1 and 2 we observe the followingtwo salient features: 1) in our 1SDI scenario, the Svetlichnyfamily does not detect tripartite steering in the range √ < V ≤ √ despite it does not have a fully LHS-LHV model inthis range and 2) the ranges in which the Svetlichny familydetects tripartite steering and genuine tripartite steering in our1SDI scenario are the same. B. SDI scenario
We now consider a tripartite 2SDI steering scenario whereBob and Charlie perform two dichotomic black-box measure-ments to prepare conditional single qubit states on Alice’sside on which Alice performs two mutually unbiased qubitmeasurements. We are now interested in which ranges theSvetlichny family detects genuine tripartite steering and tri-partite steering in the context of this 2SDI scenario.
Proposition 3.
The Svetlichny family detects genuine tripar-tite steering in our SDI scenario i ff V > √ .Proof. Note that the Svetlichny family can be reproduced bya 2 × × d dimensional biseparable state of the form given inEq. (21) for V ≤ √ . This implies that it does not detect gen-uine tripartite entanglement in the range V ≤ √ in our 2SDIscenario. On the other hand, the Svetlichny family detectsgenuine tripartite entanglement for V > √ in the fully deviceindependent scenario as it violates the Svetlichny inequalityin this range. Hence, the Svetlichny family detects genuinetripartite entanglement for V > √ in our 2SDI scenario aswell. The Svetlichny family, therefore, detects genuine tripar-tite steering in our 2SDI scenario i ff V > √ . (cid:3) Proposition 4.
The Svetlichny family detects tripartite steer-ing in our SDI scenario for V > .Proof. In Ref. [17], it has been shown that the violation ofthe following inequality (Eq. (22) in [17] with N (Number ofparties) = T (Number of trusted parties) = (cid:104) S (cid:105) × ? × ? LHS ≤ √ , (22)detects tripartite steering in our 2SDI scenario. Here, S is the Svetlichny operator given in the Svetlichny inequal-ity (4), 2 × ? × ? indicates that Alice performs qubit measure-ments while Bob and Charlie perform black-box measure-ments. Note that the Svetlichny family violates the abovesteering inequality for V > . Thus, the Svetlichny familydetects tripartite steering for V > in the 2SDI scenario. (cid:3) From the aforementioned Propositions we observe the fol-lowing two salient features: 1) the ranges in which theSvetlichny family detects tripartite steering and genuine tri-partite steering in our 2SDI scenario are di ff erent and 2)Svetlichny family detects more tripartite entangled states tobe tripartite steerable in the 2SDI scenario than in the caseof 1SDI scenario. Now we are going to make the followingimportant observation. V √ Tripartite steering in the 2SDI scenario andviolation of biseparability inequality
Tripartite steering in the 1SDI scenario, genuine steering in both the 1SDI and 2SDI scenarios and genuine nonlocality.Fully LHV model
FIG. 3. Regions of the parameter V in which the Svetlichny family isgenuinely nonlocal, detects genuine steering and tripartite steering,has a fully LHV model and violates biseparability inequality. Observation 1.
Quantum violation of the tripartite steeringinequality (22) by × × systems certifies genuine entangle-ment in that × × systems, even if genuine nonlocality orgenuine steering is not detected.Proof. We consider the following Svetlichny biseparabilityinequality: (cid:104) S (cid:105) × × − sep ≤ √ , (23)whose violation detects genuine tripartite entanglement in2 × × (cid:104) S (cid:105) × × denotes the Svetlichny operator with the measure-ment observables on each side being incompatible qubit mea-surements. Note that quantum violation of tripartite steeringinequality (22) by 2 × × × × (cid:3) We have illustrated the above results with the Svetlichnyfamily in Fig. 3.
V. DETECTION OF TRIPARTITE STEERING WITHMERMIN FAMILY
The Mermin family of tripartite correlations is defined as P VMF ( abc | A x B y C z ) = + ( − a ⊕ b ⊕ c ⊕ xy ⊕ yz ⊕ xz δ x ⊕ y ⊕ , z V , (24)where 0 < V ≤
1, which can be obtained from the noisythree-qubit GHZ state for the measurements that give rise tothe GHZ paradox; for instance, A = σ x , A = σ y , B = σ x , B = σ y , C = σ x and C = − σ y . The Mermin family is Bellnonlocal for V > as it violates the MI given in Eq. (2). Thisimplies that it certifies tripartite entanglement for V > . Inthat range, the Mermin family is not genuinely nonlocal sinceit has a NLHV model as in Eq. (3) [33]. However, it certifiesgenuine tripartite entanglement for V > √ in a fully device-independent way since it violates the Mermin inequality morethan 2 √ V √ Tripartite steering in the 2SDI scenario,Bell nonlocality, violation of biseparability inequality, hybrid nonlocal LHV model.
Tripartite steering in the 1SDI scenario and genuine steering in both the 1SDI and 2SDI scenarios.Fully LHV model
FIG. 4. Regions of the parameter V in which the Mermin familydetects Bell-nonlocality, genuine steering and tripartite steering, hasfully LHV model and hybrid nonlocal LHV model, and violates thebiseparability inequality. A. SDI scenario
We consider a tripartite 1SDI steering scenario where Char-lie performs two dichotomic black-box measurements to pre-pare conditional two-qubit states on Alice and Bob’s side onwhich Alice and Bob perform pair of incompatible qubit mea-surements that demonstrate EPR steering without Bell nonlo-cality. Now we present a Lemma which is useful to find outin which ranges the Mermin family detects genuine tripartitesteering and tripartite steering in the context of the above 1SDIscenario.
Lemma 2.
In our SDI scenario mentioned above the Merminfamily has a steering LHS-LHV model as in Eq. (14) in therange < V ≤ √ and has a fully LHS-LHV model i ff < V ≤ √ .Proof. See Appendix C for the proof. (cid:3)
The above Lemma implies the following two propositions.
Proposition 5.
The Mermin family detects genuine tripartitesteering i ff V > √ in the context of SDI scenario.Proof.
Since the Mermin family violates the Mermin inequal-ity more than 2 √ V > √ , it certifies genuine tripartiteentanglement in the fully device independent scenario in thatrange [34]. Hence, the Mermin family certifies genuine tri-partite entanglement in our 1SDI scenario as well for V > √ .This implies that, for V > √ , it does not have a steering LHS-LHV model as in Eq. (14) in our 1SDI scenario. On the otherhand, following Lemma 2 we can state that the Mermin fam-ily has a steering LHS-LHV model as in Eq. (14) in the range0 < V ≤ √ . The Mermin family, therefore, detects genuinetripartite steering i ff V > √ . (cid:3) Proposition 6.
The Mermin family detects tripartite steeringi ff V > √ in the context of our SDI scenario.Proof.
Mermin family detects entanglement between Charlieand Alice-Bob for V > √ as it violates the Mermin inequal-ity more than 2 √ V ≤ √ , it can be reproduced bya 2 × × d biseparable state of the form given by Eq. (21).Therefore, the Mermin family detects entanglement betweenCharlie and Alice-Bob i ff V > √ . On the other hand, inthe context of our 1SDI scenario, the Mermin family does nothave a fully LHS-LHV model as in Eq. (13) for V > √ fol-lowing Lemma 2. Combining these two facts we can state thatthe Mermin family detects entanglement between Charlie andAlice-Bob and does not have a fully LHS-LHV model as inEq. (13) in the range V > √ in our 1SDI scenario. Hence, inthe context of our 1SDI scenario, the Mermin family detectstripartite steering i ff V > √ . (cid:3) From the Propositions 5 and 6, we observe the follow-ing two salient features: 1) in our 1SDI scenario, the Mer-min family does not detect tripartite steering in the range √ < V ≤ √ despite it does not have a fully LHS-LHVmodel in this range and 2) the ranges in which the Merminfamily detects tripartite steering and genuine tripartite steer-ing in our 1SDI scenario are the same. B. SDI scenario
We will now study tripartite steering of the Mermin familyin our 2SDI scenario where Bob and Charlie perform two di-chotomic black-box measurements to prepare conditional sin-gle qubit states on Alice’s side on which Alice performs twomutually unbiased qubit measurements. We are interested tofind out in which ranges the Mermin family detects genuinetripartite steering and tripartite steering in the context of this2SDI scenario.
Proposition 7.
The Mermin family detects genuine tripartitesteering in our SDI scenario i ff V > √ .Proof. Note that the Mermin family can be reproduced by a2 × × d dimensional biseparable state of the form given inEq. (21) for V ≤ √ . This implies that it does not detectgenuine tripartite entanglement in the range V ≤ √ in our2SDI scenario. On the other hand, the Mermin family detectsgenuine tripartite entanglement for V > √ in the fully deviceindependent scenario as the Mermin family violates the Mer-min inequality more than 2 √ V > √ . The Mermin family, there-fore, detects genuine tripartite steering in our 2SDI scenarioi ff V > √ . (cid:3) Proposition 8.
The Mermin family detects tripartite steeringin our SDI scenario for V > .Proof. In Ref. [17], it has been shown that the violation of thefollowing inequality (Eq. (21) in [17] with N = T = (cid:104) M (cid:105) × ? × ? LHS ≤ , (25) detects tripartite steering in our 2SDI scenario. Here, M is theMermin operator given in the Mermin inequality (2), 2 × ? × ?indicates that Alice performs qubit measurements while Boband Charlie perform black-box measurements. Note that theMermin family violates the above steering inequality for V > . Thus, the Mermin family detects tripartite steering for V > in the 2SDI scenario. (cid:3) From aforementioned Propositions we observe the follow-ing two salient features: 1) the ranges in which the Merminfamily detects tripartite steering and genuine tripartite steeringin our 2SDI scenario are di ff erent and 2) the Mermin familydetects more tripartite entangled states to be tripartite steer-able in the 2SDI scenario than in the case of 1SDI scenario.Now we want to state the following important observation. Observation 2.
Quantum violation of the tripartite steeringinequality (25) by × × systems certifies genuine entangle-ment in that × × systems, even if genuine nonlocality orgenuine steering is not detected.Proof. In Ref. [37], it was shown that the Mermin inequalitydetect genuine entanglement of three-qubit systems in the sce-nario where all three parties perform two mutually unbiasedqubit measurements. This implies that the violation of the in-equality (25) implies the presence of genuine entanglementif all three parties performs qubit measurements in mutuallyunbiased bases. Similar to the derivation of Svetlichny bisep-arability inequality presented in Appendix B, one can obtainfollowing the Mermin biseparability inequality: (cid:104) M (cid:105) × × − sep ≤ , (26)whose violation detects genuine tripartite entanglement of2 × × (cid:104) M (cid:105) × × denotes the Mermin operatorwith the measurement observables on each side being incom-patible qubit measurements. Note that quantum violation oftripartite steering inequality (25) by 2 × × × × (cid:3) We have illustrated the above results with the Mermin fam-ily in Fig. 4.
VI. CONCLUSION
In this work, we have studied tripartite EPR steering ofquantum correlations arising from two local measurementson each side in the two types of partially device-independentscenarios: 1-sided device-independent scenario where oneof the parties performs untrusted measurements while theother two parties perform trusted measurements and 2-sideddevice-independent scenario where one of the parties per-forms trusted measurements while the other two parties per-form untrusted measurements.We have studied tripartite steering and genuine tripartitesteering in the 1-sided device-independent framework in thefollowing scenarios: one of the parties performs two di-chotomic black-box measurements and the other two partiesperform incompatible qubit measurements that demonstrateBell nonlocality [16] in one of the types or perform incom-patible measurements that demonstrate EPR steering withoutBell nonlocality [17, 24] in the other type. In the context ofthese two scenarios, we have studied tripartite steering of twofamilies of quantum correlations called Svetlichny family andMermin family, respectively. We have shown that the rangesin which these families detect tripartite steering and genuinetripartite steering are the same.On the other hand, in the 2-sided device-independentframework, the ranges in which the Svetlichny family andMermin family detect tripartite steering and genuine tripar-tite steering are di ff erent. These studies reveal that tripartitesteering in the 2-sided device-independent scenario is weakerthan tripartite steering in the 1-sided device-independent sce-nario. That is the Svetlichny family and Mermin family in the2-sided device-independent framework detect tripartite entan-glement for a larger region than that in the 1-sided device-independent framework. Using biseparability inequality, ithas been demonstrated that tripartite steering in the 2-sideddevice-independent framework implies the presence of gen-uine tripartite entanglement of 2 × × × d × d sys-tems in the 2-sided device-independent framework. Further,the author argued that the violation of one of these inequali-ties imply genuine entanglement if one assumes only dimen-sion of the trusted parties to be qubit dimension. However, thepresent study demonstrates that the violation of these inequal-ities do not detect genuine entanglement in this context, on theother hand, the violation of those inequalities may imply gen-uine entanglement in the scenario where the dimensions of allthree parties are assumed to be qubit dimension. ACKNOWLEDGEMENT
Authors are thankful to the anonymous referee for drawingtheir attention to Ref. [38]. Authors are thankful to Prof. Gu-ruprasad Kar and Dr. Nirman Ganguly for fruitful discussions.CJ is thankful to Prof. Paul Skrzypczyk for useful discussionsduring the 657.WE-Heraeus Seminar “Quantum Correlationsin Space and Time”. DD acknowledges the financial supportfrom University Grants Commission (UGC), Government ofIndia. BB, CJ and DS acknowledge the financial support fromproject SR / S2 / LOP-08 / Appendix A: Proof for Lemma 1
We consider the following classical simulation scenario todemonstrate in which range the Svetlichny family has a steer-ing LHS-LHV model as in Eq. (14) and a fully LHS-LHVmodel as in Eq. (13) in our 1SDI scenario considered in Sec-tion IV:
Scenario 1.
Charlie generates his outcomes by using clas-sical variable λ which he shares with Alice-Bob. Alice andBob share a two-qubit system for each value of λ and performpair of incompatible qubit measurements that demonstrateBell nonlocality of certain two-qubit states; for instance, thesinglet state. < V ≤ √ , the Svetlichny family given by Eq.(20)can be written as P VS vF ( abc | A x B y C z ) = (cid:88) λ = r λ P ( ab | A x B y , ρ λ AB ) P λ ( c | C z ) (A1)where r = r = r = r = , and P ( c | C z ) = P D , P ( c | C z ) = P D , P ( c | C z ) = P D , P ( c | C z ) = P D ,here, P αβ D ( c | C z ) = (cid:40) , c = α z ⊕ β , otherwise (A2)Here, α, β ∈ { , } . The four bipartite distributions P ( ab | A x B y , ρ λ AB ) in Eq. (A1) are given as follows:1. For λ =
0, it is given by, P ( ab | A x B y , ρ AB ) = abxy
00 01 10 1100 + √ V − √ V − √ V + √ V
14 14 14 14
14 14 14 14 − √ V + √ V + √ V − √ V , (A3)where each row and column corresponds to a fixed mea-surement ( xy ) and a fixed outcome ( ab ) respectively.Throughout the paper we will follow the same conven-tion. Note that, each of the probability distributionsmust satisfy 0 ≤ P ( ab | A x B y , ρ AB ) ≤
1, which impliesthat 0 < V ≤ √ .This joint probability distribution at Alice and Bob’sside can be reproduced by performing measurements ofthe observables corresponding to the operators A = σ x , A = σ y ; and B = σ x − σ y √ , B = σ x + σ y √ on the two-qubitstate given by, | ψ (cid:105) = cos θ | (cid:105) + (1 − i ) sin θ √ | (cid:105) , (A4)with sin 2 θ = √ V ; 0 ≤ θ ≤ π . | (cid:105) and | (cid:105) are theeigenstates of σ z corresponding to the eigenvalues + − λ =
1, it is given by, P ( ab | A x B y , ρ AB ) = − √ V + √ V + √ V − √ V
414 14 14 1414 14 14 141 + √ V − √ V − √ V + √ V . (A5)Note that, each of the probability distributions mustsatisfy 0 ≤ P ( ab | A x B y , ρ AB ) ≤
1, which implies that0 < V ≤ √ . This joint probability distribution at Alice and Bob’sside can be reproduced by performing measurements ofthe observables corresponding to the operators A = σ x , A = σ y ; and B = σ x − σ y √ , B = σ x + σ y √ on the two-qubitstate given by, | ψ (cid:105) = cos θ | (cid:105) − (1 − i ) sin θ √ | (cid:105) , (A6)with sin 2 θ = √ V ; 0 ≤ θ ≤ π .3. For λ =
2, it is given by, P ( ab | A x B y , ρ AB ) =
14 14 14 141 + √ V − √ V − √ V + √ V + √ V − √ V − √ V + √ V
414 14 14 14 . (A7)Note that, each of the probability distributions mustsatisfy 0 ≤ P ( ab | A x B y , ρ AB ) ≤
1, which implies that0 < V ≤ √ .This joint probability distribution at Alice and Bob’sside can be reproduced by performing measurements ofthe observables corresponding to the operators A = σ x , A = σ y ; and B = σ x − σ y √ , B = σ x + σ y √ on the two-qubitstate given by, | ψ (cid:105) = cos θ | (cid:105) + (1 + i ) sin θ √ | (cid:105) , (A8)with sin 2 θ = √ V ; 0 ≤ θ ≤ π .4. For λ =
3, it is given by, P ( ab | A x B y , ρ AB ) =
14 14 14 141 − √ V + √ V + √ V − √ V − √ V + √ V + √ V − √ V
414 14 14 14 . (A9)Note that, each of the probability distributions mustsatisfy 0 ≤ P ( ab | A x B y , ρ AB ) ≤
1, which implies that0 < V ≤ √ .This joint probability distribution at Alice and Bob’sside can be reproduced by performing measurements ofthe observables corresponding to the operators A = σ x , A = σ y ; and B = σ x − σ y √ , B = σ x + σ y √ on the two-qubitstate given by, | ψ (cid:105) = cos θ | (cid:105) − (1 + i ) sin θ √ | (cid:105) , (A10)with sin 2 θ = √ V ; 0 ≤ θ ≤ π .1Here it can be easily checked that the aforementioned ob-servables corresponding to the operators A = σ x , A = σ y ;and B = σ x − σ y √ , B = σ x + σ y √ used to reproduce the joint prob-ability distributions at Alice and Bob’s side can demonstratenonlocality of the singlet state given by, | ψ − (cid:105) = √ ( | (cid:105)−| (cid:105) ).Hence, the Svetlichny family given by Eq.(20) has a steeringLHS-LHV model as in Eq. (14) in the range 0 < V ≤ √ inScenario 1.In the steering LHS-LHV model given for theSvetlichny family as in Eq.(A1), the bipartite distribu-tions P ( ab | A x B y , ρ λ AB ) belong to the BB84 family up to localreversible operations (LRO) , P BB ( ab | A x B y ) = + ( − a ⊕ b ⊕ x . y δ x , y W W = √ V is a real number such that 0 < W ≤ ff W > . This implies that for W ≤ , it can be reproduced by a two-qubit separable state.Therefore, the bipartite distributions P ( ab | A x B y , ρ λ AB ) in Eq.(A1) has a LHS-LHS decomposition for V ≤ √ . This im-plies that the Svetlichny family can be reproduced by a fullyLHS-LHV model, P VS vF ( abc | A x B y C z ) = (cid:88) λ q λ P ( a | A x , ρ λ A ) P ( b | B y , ρ λ B ) P λ ( c | C z ) , (A12)for V ≤ √ in Scenario 1. Here, P ( a | A x , ρ λ A ) and P ( b | B y , ρ λ B )are the distributions arising from the local hidden states ρ λ A and ρ λ B which are in C , respectively.In Ref. [17], it has been shown that violation of the follow-ing inequality (Eq. (22) in [17] with N (Number of parties) = T (Number of trusted parties) = (cid:104) S (cid:105) × × ? LHS ≤ , (A13)detects non-existence of fully LHS-LHV model in Scenario1. Here, S is the Svetlichny operator given in the Svetlichnyinequality (4), 2 × × ? indicates that Alice and Bob performqubit measurements while Charlie performs black-box mea-surements. Note that the Svetlichny family violates the aboveinequality for V > √ . Thus, the Svetlichny family does nothave fully LHS-LHV model in the region V > √ in Sce-nario 1. Hence, we can conclude that the Svetlichny familyhas fully LHS-LHV model i ff < V ≤ √ in Scenario 1. Appendix B: Derivation of the Svetlichny biseparabilityinequality
Here we derive a biseparability inequality that detectgenuine entanglement of three-qubit systems by using the LRO is designed [31] as follows: Alice may relabel her inputs: x → x ⊕ a → a ⊕ α x ⊕ β ( α, β ∈ { , } ); Bob can perform similar operations. Svetlichny operator in the scenario where each party performsincompatible qubit measurements. In this scenario, the tri-partite correlations that can be reproduced by a biseparablethree-qubit state has the following nonseparable LHS-LHS(NSLHS) model: P ( abc | A x B y C z ) = (cid:88) λ p λ P ( a | A x , ρ λ A ) P ( bc | B y C z , ρ λ BC ) + (cid:88) λ q λ P ( ac | A x C z , ρ λ AC ) P ( b | B y , ρ λ B ) + (cid:88) λ r λ P ( ab | A x B y , ρ λ AB ) P ( c | C z , ρ λ C ) , (B1)with (cid:80) λ p λ + (cid:80) λ q λ + (cid:80) λ r λ =
1. Here, P ( a | A x , ρ λ A ), P ( b | B y , ρ λ B )and P ( c | C z , ρ λ C ) are the distributions which can be repro-duced by the qubit states ρ λ A , ρ λ B and ρ λ C , respectively, and P λ ( bc | B y C z , ρ λ BC ), P λ ( ac | A x C z , ρ λ AC ) and P λ ( ab | A x B y , ρ λ AB ) canbe reproduced by the 2 × ρ λ BC , ρ λ AC and ρ λ AB , respec-tively. Note that in the model given by Eq. (B1), the bipartitedistributions at each λ level may have nonseparability.The Svetlichny operator can be rewritten as follows: S = CHS H AB C + CHS H (cid:48) AB C . (B2)Here, CHS H AB = A B + A B + A B − A B is the canon-ical CHSH (Clauser-Horne-Shimony-Holt) operator [16] and CHS H (cid:48) AB = − A B + A B + A B + A B is one of its equiv-alents. Note that the expectation value of the Svetlichny oper-ator for the correlation which has the nonseparable LHS-LHSmodel as given in Eq. (B1) have the following form: (cid:88) λ p λ (cid:104) A (cid:105) ρ λ A (cid:104) CHS H BC (cid:105) ρ λ BC + (cid:88) λ p λ (cid:104) A (cid:105) ρ λ A (cid:104) CHS H (cid:48) BC (cid:105) ρ λ BC + (cid:88) λ q λ (cid:104) CHS H AC (cid:105) ρ λ AC (cid:104) B (cid:105) ρ λ B + (cid:88) λ q λ (cid:104) CHS H (cid:48) AC (cid:105) ρ λ AC (cid:104) B (cid:105) ρ λ B + (cid:88) λ r λ (cid:104) CHS H AB (cid:105) ρ λ AB (cid:104) C (cid:105) ρ λ C + (cid:88) λ r λ (cid:104) CHS H (cid:48) AB (cid:105) ρ λ AB (cid:104) C (cid:105) ρ λ C . (B3)Let us now argue that the above quantity is upper bounded by2 √
2. Consider the first line of the decomposition given in Eq.(B3). Suppose Bob and Charlie’s correlation at each λ levelof this line detects nonseparability. Then ± (cid:104) CHS H BC (cid:105) ρ λ BC ±(cid:104) CHS H (cid:48) BC (cid:105) ρ λ BC ≤ √
2. Suppose Bob and Charlie’s corre-lation at each λ level has a LHS-LHS model. Then also ± (cid:104) CHS H BC (cid:105) ρ λ BC ± (cid:104) CHS H (cid:48) BC (cid:105) ρ λ BC ≤ √
2. In a similar way,considering the second line of the decomposition given in Eq.(B3), one can show that ± (cid:104)
CHS H AC (cid:105) ρ λ AC ± (cid:104) CHS H (cid:48) AC (cid:105) ρ λ AC ≤ √
2; and considering the third line of the decompositiongiven in Eq. (B3), one can show that ± (cid:104)
CHS H AB (cid:105) ρ λ AB ±(cid:104) CHS H (cid:48) AB (cid:105) ρ λ AB ≤ √
2. Therefore, any convex combinationof the three above mentioned expression should be upperbounded by 2 √
2. Hence, we can conclude that in the Sce-nario where each party performs incompatible qubit measure-ments, the Svetlichny operator is upper bounded by 2 √ (cid:104) S (cid:105) × × − sep ≤ √ , (B4)serves as the biseparability inequality whose violation detectsgenuine tripartite entanglement of 2 × × (cid:104) S (cid:105) × × denotes the Svetlichny operator with the measure-ment observables on each side being incompatible qubit mea-surements. Appendix C: Proof for Lemma 2
We consider the following classical simulation scenario todemonstrate in which range the Mermin family has a steeringLHS-LHV model as in Eq. (14) and a fully LHS-LHV modelas in Eq. (13) in the 1SDI scenario considered in Section V:
Scenario 2.
Charlie generates his outcomes by using clas-sical variable λ which he shares with Alice-Bob. Alice andBob share a two-qubit system for each λ and perform pairof incompatible qubit measurements that demonstrate EPRsteering without Bell nonlocality of certain two-qubit states[17, 24]; for instance, the singlet state. Following the steering LHV-LHS model of the Merminfamily mentioned in Ref. [35], we can write down the fol-lowing steering LHS-LHV model of the Mermin family in therange 0 < V ≤ √ , P VMF ( abc | A x B y C z ) = (cid:88) λ = r λ P ( ab | A x B y , ρ λ AB ) P λ ( c | C z ) , (C1)as it is invariant under the permutations of the parties. Here, r = r = r = r = , and P ( c | C z ) = P D , P ( c | C z ) = P D , P ( c | C z ) = P D , P ( c | C z ) = P D .The bipartite distributions in the model (C1) are given asfollows:1. For λ =
0, it is given by P ( ab | A x B y , ρ AB ) = abxy
00 01 10 1100 + V − V − V + V + V − V − V + V + V − V − V + V − V + V + V − V , (C2)where each row and column corresponds to a fixed mea-surement ( xy ) and a fixed outcome ( ab ) respectively.This joint probability can be reproduced by performingthe projective qubit measurements of the observablescorresponding to the operators A = σ x , A = σ y ; and B = σ x , B = σ y on the two-qubit state given by, | ψ (cid:105) = cos θ | (cid:105) + (1 + i ) sin θ √ | (cid:105) , (C3) where, 0 ≤ θ ≤ π with sin 2 θ = √ V ; | (cid:105) and | (cid:105) arethe eigenstates of σ z corresponding to the eigenvalues + − λ =
1, it is given by P ( ab | A x B y , ρ AB ) = − V + V + V − V − V + V + V − V − V + V + V − V + V − V − V + V , which can be reproduced by performing the projectivequbit measurements of the observables correspondingto the operators A = σ x , A = σ y ; and B = σ x , B = σ y on the two-qubit state given by, | ψ (cid:105) = cos θ | (cid:105) − (1 + i ) sin θ √ | (cid:105) , (C4)where, 0 ≤ θ ≤ π with sin 2 θ = √ V .3. For λ =
2, it is given by P ( ab | A x B y , ρ AB ) = − V + V + V − V + V − V − V + V + V − V − V + V + V − V − V + V , which can be reproduced by performing the projectivequbit measurements of the observables correspondingto the operators A = σ x , A = σ y ; and B = σ x , B = σ y on the two-qubit state given by | ψ (cid:105) = cos θ | (cid:105) − (1 − i ) sin θ √ | (cid:105) , (C5)where, 0 ≤ θ ≤ π with sin 2 θ = √ V .4. For λ =
3, it is given by P ( ab | A x B y , ρ AB ) = + V − V − V + V − V + V + V − V − V + V + V − V − V + V + V − V , which can be reproduced by performing the projectivequbit measurements of the observables correspondingto the operators A = σ x , A = σ y ; and B = σ x , B = σ y on the two-qubit state given by | ψ (cid:105) = cos θ | (cid:105) + (1 − i ) sin θ √ | (cid:105) , (C6)where, 0 ≤ θ ≤ π with sin 2 θ = √ V .Note that | sin 2 θ | ≤ ≤ θ ≤ π ), which implies that V ≤ √ . It can be easily checked that the aforementionedobservables corresponding to the operators A = σ x , A = σ y ;and B = σ x , B = σ y used to reproduce the joint probability3distributions at Alice and Bob’s side can demonstrate EPRsteering without Bell nonlocality of the singlet state given by, | ψ − (cid:105) = √ ( | (cid:105) − | (cid:105) ). Hence, the Mermin family given byEq.(24) has a steering LHS-LHV model as in Eq.(14) in therange 0 < V ≤ √ in Scenario 2.In the steering LHS-LHV model given for the Mer-mim family as in Eq.(C1), the bipartite distributions P ( ab | A x B y , ρ λ AB ) belong to the CHSH family up to local re-versible operations [31], P CHS H ( ab | A x B y ) = + ( − a ⊕ b ⊕ xy √ W , (C7)where W = √ V is a real number such that 0 < W ≤ < V ≤ √ . In Ref. [32], it has been that the CHSH familycertifies two-qubit entanglement i ff W > . This implies thatfor W ≤ , it can be reproduced by a two-qubit separablestate. Therefore, the bipartite distributions P ( ab | A x B y , ρ λ AB ) inEq. (C1) has a LHS-LHS decomposition for V ≤ √ . Thisimplies that the Mermin family can be reproduced by a fullyLHS-LHV model, P VMF ( abc | A x B y C z ) = (cid:88) λ q λ P ( a | A x , ρ λ A ) P ( b | B y , ρ λ B ) P λ ( c | C z ) , (C8) for V ≤ √ in Scenario 2. Here, P ( a | A x , ρ λ A ) and P ( b | B y , ρ λ B )are the distributions arising from the local hidden states ρ λ A and ρ λ B which are in C , respectively.In Ref. [17], it has been shown that violation of the follow-ing inequality (Eq. (21) in [17] with N = T = (cid:104) M (cid:105) × × ? LHS ≤ √ , (C9)detects non-existence of fully LHS-LHV model in Scenario2. Here M is the Mermin operator given in the Mermin in-equality (2), 2 × × ? indicates that Alice and Bob performqubit measurements while Charlie performs black-box mea-surements. Note that the Mermin family violates the aboveinequality for V > √ . Thus, the Mermin family does nothave fully LHS-LHV model in the region V > √ in Sce-nario 2. Hence, we can conclude that the Mermin family hasfully LHS-LHV model i ff < V ≤ √ in Scenario 2. [1] O. Guhne, and G. Toth, Entanglement detection , Physics Re-ports , 1 (2009).[2] A. S. Sorensen, and K. Molmer,
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