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Revealing Genuine Steering under Sequential Measurement Scenario Amit Mukherjee,1,∗ Arup Roy,1 Some Sankar Bhattacharya,1 Biswajit Paul,2 Kaushiki Mukherjee,3 and Debasis Sarkar4 1Physics and Applied Mathematics Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata 700108 , India. 2Department of Mathematics, South Malda College, Malda, West Bengal, India 3Department of Mathematics, Government Girls’ General Degree College, Ekbalpore, Kolkata, India. 4Department of Applied Mathematics, University of Calcutta, 92, A.P.C. Road, Kolkata-700009, India. Genuine steering is still not well understood enough in contrast to genuine entanglement and nonlocality. Here we provide a protocol which can reveal genuine steering under some restricted operationscomparedtotheexistingwitnessesofgenuinemultipartitesteering. Ourmethodhasan impression of some sort of ‘hidden’ protocol in the same spirit of hidden nonlocality, which is well understoodinbipartitescenario. Wealsointroduceagenuinesteeringmeasurewhichindicatesthe enhancement of genuine steering in the final state of our protocol compared to the initial states. PACSnumbers: 03.65.Ud,03.67.Mn 7 I. I. INTRODUCTION quantum inseparability, intermediate in between entan- 1 glementandBellnonlocality. Consideringpurequantum 0 states these three notions are equivalent whereas in 2 Einstein-Podolsky-Rosen steering, the phenomenon general they are inequivalent in case of mixed states n that was first discussed by Schrodinger and afterwards [19]. However in the context of comparison of steering a considered as a notion of quantum nonlocality, has nonlocality with that of Bell-nonlocality, it is interesting J gained significant attention in recent days[1–3]. This to mention that analogous to hidden nonlocality[20, 21], 7 quantum phenomenon, which has no classical analogue, existence of hidden steering has been proved in [19] for 1 is observed if one of two distant observers, sharing an bi-partite scenario. Just as in the case of exploiting entangled state, can remotely steer the particle of the nonlocality beyond Bell scenarios via the notion of ] h other distant observer by performing measurements hidden nonlocality[20, 21], hidden steering refers to rev- p on his/her particle only. The experimental criteria elation of steering nonlocality under suitable sequential t- for analyzing the presence of bipartite steering, first measurements. In this context an obvious interest grows n investigated in [4], was formalized in ref where the regarding analysis of the same for multipartite scenario. a authorsgeneralizedthisconceptforarbitrarysystems[5]. Duetoincreaseincomplexityasoneshiftsfrombipartite u q Till date there has been a lot of analysis regarding to multipartite system, till date there has been limited [ various features of steering nonlocality such as methods attempts to understand the feature of multipartite of detection[6] and quantification of steering[7, 8], steering phenomenon. Analogous to both entanglement 1 steering of continuous variable systems[9], loop-hole and Bell-nonlocality the concept of genuine steering v 7 free demonstration of steering[10], applications as a has been established in recent days. In this context 1 resource of nonlocal correlations in the field of quantum it may be mentioned that unlike Bell-nonlocality and 5 information protocols, exploiting the relation of steering entanglement, due to asymmetric nature of steering 4 with incompatibility of quantum measurements[11, 12] nonlocality the notion of genuine steering nonlocality 0 and its ability to detect bound entanglement[13], etc. lacks uniqueness. However genuine steering was first 1. Apart from its foundational richness, EPR steering do introducedin[22]wheretheauthorsprovidedthecriteria 0 have multi-faceted applications in practical tasks such for detecting genuineness in steering scenario for both 7 as semi-device independent scenario [14] where only continuous as well as discrete variable systems. Later 1 one party can trust his or her apparatus but the other two other notions of genuine steering were introduced : v party’s apparatus is not trusted. In that situation the in [23] mainly for tripartite framework where two i presence of steerable state provides a better chance to parties measurements are fully specified i.e one party X allow secure key distribution[15]. Even for some other can control remaining. In this context, the author has ar tasks such as randomness certification[16], entanglement also designed genuine steering inequalities to detect assisted sub-channel discrimination[17], and secure genuine tripartite steering. Now speaking of genuine teleportation through continuous-variables steerable steering nonlocality, it may be interesting to explore states[18] are found to be useful. the possibility of exploiting the same via some suitable Being a notion of nonlocality there exists a hierarchy sequential measurement protocol. according to which steering is defined as a form of To be precise, our present topic of discussion will continueinthedirectionofanalyzinghiddengenuinetri- partite steering nonlocality in the framework introduced in [23]. For present topic of discussion we will follow ∗Electronicaddress: [email protected] terms and terminologies used in [23]. We will design 2 a protocol involving a sequence of measurements such his measurement devices and hence are uncharacterized. that initially starting from tripartite states which may Aliceandbobshouldhaveorthonormalmeasurementset- not be genuinely steerable, the protocol may generate a tings. Ifcorrelationsarisingduetomeasurementsonany genuinely steerable state. Interestingly the initial states given quantum state(ρ) violate this inequality(Eq.(2)), which will be used in the protocol do have a bilocal thenthatguaranteesgenuinelysteerableofρfromChar- model [24]. lie to Alice and Bob. Analogously genuine steerability of ρ from Bob to Charlie and Alice and that from Alice The paper has organized as follows. In section[II] we to Charlie and Bob can be guaranteed respectively by have introduced the notion of steering both in bi partite violation of the following criteria: as well as genuine multipartite scenario. Then in sec- √ tion[III]wehavepresentedsuitablesequentialoperations (cid:104)CHSHBCx1+CHSHB(cid:48) Cx0(cid:105)N2×L2H×S? ≤2 2. (3) to achieve the final state. Section[IV] contains our main √ results then discussion. (cid:104)CHSH y +CHSH(cid:48) y (cid:105)NLHS ≤2 2. (4) AC 1 AC 0 2×2×? Terms CHSH , CHSH(cid:48) , CHSH , CHSH(cid:48) BC BC AC AC II. BACKGROUND have analogous definitions. Hence a state is genuinely steerable from one party to the remaining two parties if In this section we are basically going to include a brief itcanviolateatleastoneofthesethreecriteria(Eqs.2,3,4). detailing of the mathematical tools that will be used in We now discuss about some relevant tools for measuring our work. genuine multipartite entanglement and genuine steering. A. Genuine tripartite steering B. Genuine multipartite concurrence Firstly we discuss the criteria of detecting genuine We briefly now describe C , a measure of gen- GM steering[23]. Correlations P(a,b,c|x,y,z) shared be- uine multipartite entanglement. For pure n-partite tween three parties, say Alice, Bob and Charlie are said states(|ψ(cid:105)),thismeasureisdefinedas[26]: C (|ψ(cid:105)):= GM (cid:112) to be genuinely steerable[23] from one party, say Char- min 2(1−Π (|ψ(cid:105))) where Π (|ψ(cid:105)) is the purity of jth j j j lie to remaining two parties Alice and Bob, if those are bipartition of |ψ(cid:105). The expression of C for X states GM inexplicable in the following form: is given in [27]. For tripartite X states, (cid:88) C =2max {0,|γ |−w } (5) P(a,b,c|x,y,z)= q [P(a,b|x,y,ρ (λ))]P(c|z,λ) GM i i i λ AB (cid:80) (cid:112) λ with w = a b where a , b and γ (j =1,2,3,4) (cid:88) i j(cid:54)=i j j j j j + p P(a|x,ρλ)P(b|y,ρλ)P(c|z,λ). (1) are the elements of the density matrix of tripartite X λ a b state: λ where P(a,b|x,y,ρ (λ)) denotes the nonlocal proba- AB a 0 0 0 0 0 0 γ  bility distribution arising from two-qubit state ρλ , and 1 1 AB 0 a 0 0 0 0 γ 0 P(a|x,ρλ) and P(b|y,ρλ) are the distributions arising  2 2  A B  0 0 a 0 0 γ 0 0 from qubit states ρλ and ρλ.  3 3  A B  0 0 0 a γ 0 0 0 Here Charlie performs uncharacterized measurement  4 4   0 0 0 γ ∗ b 0 0 0 whereas both Alice and Bob have access to qubit mea-  4 4   0 0 γ ∗ 0 b 0 0 surements. The tripartite correlation will be called gen-  3 3   0 γ ∗ 0 0 0 0 b 0 uinely unsteerable if it is explained by 1 where ρ (λ) 2 2 AB γ ∗ 0 0 0 0 0 0 b is called hidden state for Alice and Bob side. In [23], 1 1 theauthordesignedadetectioncriteriaoftripartitegen- uine steering(Svetlichny steering), based on Svetlichny C. Genuine steering measure inequality[25]. The detection criterion is given in the form of a Bell-type inequality: Firstwedefinegenuinesteeringmeasurewhichisanal- √ (cid:104)CHSH z +CHSH(cid:48) z (cid:105)NLHS ≤2 2. (2) ogoustothebi-partitesteeringmeasurefirstdescribedin AB 1 AB 0 2×2×? [28]. This measure is given by the following quantity: walhenertefaCcHetsSHdeAfiBninangdBCelHl-CSHHSA(cid:48)HBpsotalyntdopfoerfotwroAilniceequainvd- Sgen(ρ)=max{0,SSnm(aρx)−−11} (6) Bob and {z ,z } are measurements on Charlie’s part. n 0 1 Here NLHS stands for nonlocal hidden state whereas whereSmax = max S (ρ) and S (ρ) = max S (ρ,η) n ρ n n η n 2×2×?impliesthatonlytwoparties(AliceandBob)have with the maximization taken over all measurement set- access to qubit measurements but Charlie does not trust tings η and 0≤S (ρ)≤1. gen 3 After giving a brief detailing of our mathematical tools, B. Measurement Stage we now proceed with our results. To start with, we de- signthesequentialmeasurementprotocolbasedonwhich • In the measurement stage, all the three parties can we observe the enhanced revelation of genuine steering. perform any projective measurement in arbitrary directions. But in this stage they are not allowed to communicate among themselves. III. REVEALING MULTIPARTITE GENUINE • After measurements they can generate a tripartite STEERING correlation so that they can verify that this correlation can violate the genuine steering inequality. The protocol that we propose here is a We refer to this protocol of sequential measurements by SLOCC(Stochastic Local Operation and Classical the three parties sharing n states as a sequential mea- Communication) protocol which consists of two stages: surement protocol (SMP). Preparation Stage and Measurement Stage. We name Having sketched the protocol we now give examples of this protocol as Sequential Measurement Protocol. A somefamiliesoftripartitestateswhichwhenusedinthis detailed sketch of the protocol is given below: network, reveal genuine steering for some members of Sequential Measurement Protocol: Three spatially these families. Such an observation is supported with an separated parties(say, A ;i = 1,2,3) are involved in this i increase in the amount of genuine steering, guaranteed protocol. n number of tripartite quantum states can be by the measure of steering S (ρ)(Eq.(6)). gen distributed among them. None of these states violate genuine steering inequality[23]. As each party holds one Let the three initial states be given by: particle from each of the n tripartite states hence each of the parties holds n number of particles. ρ =p |ψ (cid:105)(cid:104)ψ |+(1−p )|001(cid:105)(cid:104)001| (7) 1 1 f f 1 with |ψ (cid:105) = cosθ |000(cid:105)+sinθ |111(cid:105), 0 ≤ θ ≤ π and f 1 1 1 4 0≤p ≤1; 1 A. Preparation Stage ρ =p |ψ (cid:105)(cid:104)ψ |+(1−p )|010(cid:105)(cid:104)010| (8) 2 2 m m 2 • In the preparation stage, every party can perform with |ψm(cid:105)= |000(cid:105)√+|111(cid:105) and 0≤p2 ≤1; some joint measurement on their respective n−1 2 particles and then broadcast the results to others. ρ =p |ψ (cid:105)(cid:104)ψ |+(1−p )|100(cid:105)(cid:104)100| (9) 3 3 l l 3 with |ψ (cid:105) = sinθ |000(cid:105) + cosθ |111(cid:105),0 ≤ θ ≤ π and • At the end of measurements by all the three par- l 3 3 1 4 0 ≤ p ≤ 1. In this context it may be noted that the ties, a tripartite quantum state shared among A , 1 1 three initial states have Svetlichny bi-local model under A and A is generated. Clearly this final state is 2 3 projectivemeasurementforthefollowingrestrictedrange always prepared depending upon the measurement of state parameters: resultsobtainedbythepartiesinthepreviousstep. • For first state(ρ ) : p ≤ 1 ; 1 1 (1+sin[2θ1]) • Second state(ρ ) : p ≤ 1 ; 2 2 2 • Third state(ρ ) : p ≤ 1 . 3 3 (1+sin[2θ3]) EachofthethreepartiesA ,A andA performsBellba- 1 2 3 sis measurements on their respective particles. Depend- ing on a particular output of all the measurements(here |ψ±(cid:105)= |01(cid:105)√±|10(cid:105)), a resultant state ρ± is obtained which 2 4 after correcting the phase term is given by: p |φ(cid:105)(cid:104)φ|+(1−p )sin2θ |100(cid:105)(cid:104)100| ρ = 3 3 1 (10) 4 sin2θ +p cos2θ sin2θ 1 3 1 3 where |φ(cid:105)=cosθ sinθ |000(cid:105)+sinθ cosθ |111(cid:105). 1 3 1 3 Clearly ρ is independent of p and p . Interestingly, 4 1 2 ρ can also be generated for some other combination of 4 FIG. 1: Schematic diagram for preparation and mesurement sequential operations on some different arrangement of stage. particles between the parties A (1,2,3) and for different i 4 output of Bell measurement. For the initial states ρ whereas that of ρ is given by i 4 (i = 1,2,3), the amount of genuine entanglement are given by p sin2θ sin2θ Cρ4 = 3 1 3 . (12) GM 2(sin2θ +p cos2θ sin2θ ) Cρ1 =p sin2θ , 1 3 1 3 GM 1 1 Eq.(11) indicates that the initial states ρ (i=1,2,3) are i Cρ2 =p genuinely entangled for any nonzero value of the state GM 2 parameters . and The maximum value of the genuine steering operators(S )(Eq.2) under projective measurements, i Cρ3 =p sin2θ (11) for state ρ (i=1,2,3) is given by: GM 3 3 i 1 (cid:112) S =max[2p sin2θ ,√ ((1−p −p Cos[2θ ])2+(p Sin[2θ ])2], 1 1 1 1 1 1 1 1 2 respectivelywhereasthatforthefinalstateρ ,itisgiven 4 by 1 (cid:112) S =max[2p ,√ ((1−p )2+(p )2)] 2 2 2 2 2 and 1 (cid:112) S =max[2p sin2θ ,√ ((1−p +p Cos[2θ ])2+(p Sin[2θ ])2)|] 3 3 3 3 3 3 3 3 2 (13) p sin2θ sin2θ S =max[ 3 1 3 , 4 sin2θ +p cos2θ sin2θ 1 3 1 3 √ (cid:112) 2 (1−p +p Cos[2θ ]−Cos[2θ ])2+(p Sin[2θ ]Sin[2θ ])2 3 3 3 1 3 1 3 . (14) 2−2(1−p )Cos[2θ ]−p Cos[2(θ −θ )]−p Cos[2(θ +θ )] 3 1 3 1 3 3 1 3 It is clear from the maximum value of genuine steer- inequalities(Eqs.2,3,4). ing operator(Eqs.(13), (14)) and the measure of entanglement (Eqs.(11), (12)) of both initial states and final state,thateachofthemdoesnotviolategenuinesteering inequalities(Eqs.2,3,4) for Cρi ≤ 1(i=1,2,3,4). GM 2 But when used in our protocol(Sec.III), they can gen- Thus to observe genuine steering revelation there erate a state ρ (with Cρ4 > 1) which exhibits gen- should exist some fixed values of the parameters of the 4 GM 2 uine steering by violating genuine steering inequalities three initial Sveltlichny bi-local states with Cρi ≤ 1 such that the final state can have CGρ4M > 12. GInMterest2- fsoterepri3n≥g f0o.r33p535∈7.[0T.3h3is55g7u,a0r.a8n3t4e2e6s].reSveolaintiiotnialolyf geeancuhinoef ingly we get such states from the families of the initial these three states are unable to exhibit genuine steer- states ρ (Eq.(7)), ρ (Eq.(8)) and ρ (Eq.(9)). 1 2 3 ing but after the sequential measurements are taken into For example, let θ = 0.1, p ≤ 0.509 , p ≤ 1, account they can violate that genuine steering inequal- 1 1 2 2 θ = 0.1 and p ∈ [0,0.83426]. Then each of the ity. Now a pertinent question would be whether one can 3 3 initial states have Svetlichny bi-local model (moreover quantify this revelation of genuine steering as observed one can show that these models are NS local[24]) and in our protocol. We deal with this question in the next 2 Cρi ≤ 1. Thus they do not violate genuine steering sub-section. GM 2 5 C. Enhancement of the Genuine Steering measure ious practical tasks. So apart from its theoretical importance, revelation of such a resource under any protocol In this part we show that the prescribed protocol in- thatallowsonlyclassicalcommunicationandsharedran- deed enhances a measure of genuine steering in the re- domness is of immense practical importance. Motivated sulting state. The amount of genuine steering for the by that we have attempted to design a SLOCC protocol three initial states are: which demonstrates revelation of ‘hidden’ genuine steering. Our discussion in a restricted sense guarantees the S (ρ )=max{0,2p sin2θ −1}, fact that under suitable measurements by the parties in- gen 1 1 1 volved in the network, our protocol is sufficient to show genuine steering even from some quantum states which Sgen(ρ2)=max{0,2p2−1}, havebi-localmodels. Howeverunderourprotocoleachof the parties having two particles perform Bell basis measurements and the remaining parties perform projective S (ρ )=max{0,2p sin2θ −1} (15) gen 3 3 3 measurements. In brief, this protocol enables one to go beyondthescopeofexistingwitnessesofgenuinesteering whereasforthefinalstatesthegenuinesteerablequantity and thus demonstrate genuine steering for a larger class takes the form: of multipartite states. In this context, it will be interest- p sin2θ sin2θ ing to consider more generalized measurement settings S (ρ )=max{0, 3 1 3 −1} (16) gen 4 sin2θ +p cos2θ sin2θ by the parties which may be yielding better results. 1 3 1 3 If we take p = p and θ = θ then for any values of 1 3 1 3 p andθ thefinalstateismoregenuinelysteerablethan 1 1 the initial ones. V. ACKNOWLEDGEMENT IV. CONCLUSION WewouldliketothankProf. GuruprasadKarforuse- Genuine steering nonlocality, being a weaker notion of fuldiscussions. AMacknowledgesupportfromtheCSIR genuine nonlocality is considered to be a resource in var- project 09/093(0148)/2012-EMR-I. [1] E. Schrödinger, Proc. Cambridge Philos. 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