Maximum tri-partite Hardy's nonlocality respecting all bi-partite principles
Subhadipa Das, Manik Banik, M. D. Rajjak Gazi, Ashutosh Rai, Samir Kunkri, Ramij Rahaman
MMaximum tri-partite Hardy’s nonlocality respecting all bi-partite principles
Subhadipa Das, ∗ Manik Banik, † MD Rajjak Gazi, ‡ Ashutosh Rai, § Samir Kunkri, ¶ and Ramij Rahaman
4, 5, ∗∗ S.N. Bose National Center for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata-700098, India Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata-700108, India Mahadevananda Mahavidyalaya, Monirampore, Barrakpore, North 24 Parganas-700120, India The Institute of Mathematical Sciences (IMSc.), CIT Campus, Taramani, Chennai-600113, India Institute of Theoretical Physics and Astrophysics, University of Gdansk, 80-952 Gdansk, Poland
The set of multiparty correlations that respect all bi-partite principles has been conjectured tobe same as the set of time-ordered-bi-local correlations. Based on this conjuncture we find themaximum value of success probability of tri-partite Hardy’s correlation respecting all bi-partitephysical principles. Unlike in quantum mechanics, the no-signaling principle does not reveal anygap in Hardy’s maximum success probability for bi-partite and tri-partite system. Informationcausality principle is shown to be successful in qualitatively revealing this quantum feature and thisresult is independent of the conjecture mentioned above.
PACS numbers: 03.65.Ud
I. INTRODUCTION
The outcome of local measurements on spatially sepa-rated parts of a composite quantum system can be non-classically (nonlocally) correlated. Violation of the Bell-CHSH inequality [1, 2] is a witness of this nonlocal featurein such correlations. The value of Bell-CHSH expression,exceeding the classical bound 2, then qualifies as a mea-sure of nonlocality. This nonlocality within the quantummechanics is limited by the Cirel’son bound 2 √ no signaling (NS) principle, nolocality under-lying these correlations can achieve any value up to thealgebraic maximum 4 for the Bell-CHSH expression (e.g.PR-box correlation achieves the value 4 [4]). So the nat-ural questions arises; what are the physical principles,other than NS, that can distinguish quantum correla-tions from post-quantum no-signaling correlations? Thisfundamental question has been addressed in several re-cent works proposing novel physical principles, like, nonontrivial communication complexity [5, 6], macroscopiclocality [7] and information causality [8], for explainingthe boundary defining quantum correlations. In partic-ular, the application of the principle of noviolation ofinformation causality (IC) has produced very interestingresults, like explaining the Cirel’son’s bound and show-ing that in a bipartite scenario any correlation goingbeyond the Cire’lson bound is unphysical. IC principleis a generalization of no-signaling condition—while rela- ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] ∗∗ Electronic address: [email protected] tivistic causality (the no-signaling principle) states that aparty cannot extract more information then the commu-nicated (say, m ) number of cbits; information causalityfurther restricts free choice to decode deterministically asingle m -cbit information, from different possible m -cbitinformations potentially encoded within the communi-cated amount of m -cbit message. Applications of the ICprinciple in the study of both bi-partite and tri-partitecorrelations has produced some interesting results [9–12].On the other hand, IC or any other bi-partite principlehas been shown to be insufficient for witnessing all mul-tipartite post-quantum correlations [13, 14]. Thus, mul-tipartite generalization of IC, or some other genuinelymultipartite physical principle(s) are necessary to char-acterize all quantum correlations. Studying some simpleclass of multipartite correlations, like Hardy’s set [15–19],can give useful insight about the strength and weaknessof such principles [10, 11, 14].Like Bell-CHSH nonlocality test, Lucian Hardy firstproposed [15] an elegant argument for witnessing non-local correlations without any use of inequality. Fortwo qubit states subjected to local projective measure-ments, the maximum success probability of Hardy’s non-locality argument has been shown to be (5 √ − / ≈ . √ − / (= 0 . a r X i v : . [ qu a n t - ph ] D ec physical principles like NS condition and IC condition.Under the NS condition, optimal success probability forHardy’s nonlocality is , both for bi-partite system andtri-partite system [19]. Thus, in contrast to the quan-tum mechanical feature, under the NS condition thereis no gap between Hardy’s maximum success probabil-ity for bi-partite and tri-partite system. On the otherhand, under the IC principle, it has been shown that themaximum success probability of Hardy’s argument forbi-partite system is bounded above by 0 .
207 [10]. Thestudy for the bound on Hardy’s success probability fortri-partite system under IC condition has not yet beenstudied. The problem is very intriguing as informationcausality is a bi-partite principle and it is highly nontriv-ial to exhaust the IC condition under all bi-partitionswith all possible wirings.In this work we show that the maximum value ofHardy’s success for tri-partite correlation satisfying everybi-partite principle is . Then, we argue that in partic-ular IC principle successfully reveals a quantum featureviz. a gap between Hardy’s maximum success probabilityfor bi-partite and tri-partite systems. Moreover, the gapbetween two bounds is decisive, as for tri-partite systemwe achieve a lower bound through a probability distribu-tion which is time-ordered-bi-local (TOBL) [20–22] andhence it not only satisfies IC but satisfies any bi-partiteinformation principles discovered or not discovered. Onthe other hand, for the bipartite case, the upper boundon maximum success probability was derived by applyinga necessary condition for respecting the IC principle.The paper is organized as follows. In section (II) weoverview the properties of the sets of a tripartite two in-put two output probability distributions. In section (III),we describe Hardy’s non-locality conditions, and reviewsome results in this context. In section (IV), we brieflydescribe TOBL correlations. In section (V), we derivethat maximum success of tri-partite Hardy’s nonlocalityrespecting all bi-partite information principle, and thenin section (VI), we argue that, in the context of maxi-mum Hardy’s success probability, IC reproduces a quan-tum like feature while NS fails to do so. Finally we giveour conclusions in section (VII). II. TRIPARTITE NO-SIGNALINGCORRELATIONS
The set of tripartite no-signaling correlations with bi-nary input and binary output for each party is a con-vex set in a 26-dimensional space [20]. Let P ( abc | xyz )denotes the tri-joint probabilities, where x, y, z ∈ { , } denote inputs and a, b, c ∈ { , } denote outputs of threeparties respectively. Normalization and positivity of jointprobabilities are expressed by following conditions: (cid:88) abc P ( abc | xyz ) = 1 ∀ x, y, z, (1) P ( abc | xyz ) ≥ ∀ a, b, c, x, y, z. (2) The set of no-signaling distributions is obtained by fur-ther imposing the condition that the choice of measure-ment by any one party cannot affect the outcome dis-tributions of the remaining two parties and vice-versa,i.e., (cid:88) c p ( abc | xyz ) = (cid:88) c p ( abc (cid:48) | xyz (cid:48) ) , ∀ a, b, x, y, z, z (cid:48) (3)and other similar conditions obtained by permuting theparties. The set of these no-signaling correlations forms apolytope with 53 ,
856 extremal points belonging to 46 dif-ferent classes among which 45 contains non-local points[20]. The remaining class contains all deterministic lo-cal points and correspond to extreme points of the localpolytope; i.e., the set of correlations with a local model p ( abc | xyz ) = (cid:88) λ p λ p ( a | x, λ ) p ( b | y, λ ) p ( c | z, λ ) (4)where p λ is the distribution of some random variable λ shared by the parties. Since, in general quantum cor-relations respects the no-signaling condition, set of alltripartite quantum correlations with two possible out-comes at each local site are obviously contained withinthe tripartite no-signaling polytope. Then, a correlationis quantum if it can be expressed as p ( abc | xyz ) = tr ( ρM ax ⊗ M by ⊗ M cz ) (5)where ρ is some quantum state and indexed M are themeasurement operators (in general positive operator val-ued measure POVM) associated with the outcomes a, b, c of measurements x, y, z respectively. Post-quantum no-signaling correlations are those that cannot be written inthe above forms (i.e. eqn-(4) or (5)). III. TRIPARTITE HARDY’S CORRELATIONS
A tripartite two-input-two-output Hardy’s correlationis defined by some restrictions on a certain choice of 5out of 64 joint probabilities in the correlation matrix.Any tripartite Hardy’s correlation can be expressed byfollowing five conditions: P ( A = i, B = j, C = k ) = q > P ( A (cid:48) = l, B = m, C = k ) = 0 (6b) P ( A = i, B (cid:48) = m, C = k ) = 0 (6c) P ( A = i, B = j, C (cid:48) = n ) = 0 (6d) P ( A (cid:48) = l ⊕ , B (cid:48) = m ⊕ , C (cid:48) = n ⊕
1) = 0 (6e)where, { A, A (cid:48) } , { B, B (cid:48) } and { C, C (cid:48) } are the respec-tive outcomes for the choice of local observables for threeparties, say, Alice, Bob and Charlie, and i, j, k, l, m, n ∈{ , } .One can prove that Hardy’s correlations are non-local.To show this, let us suppose that these correlations arelocal, i.e., these correlations can be simulated from pre-shared (hidden) local variables. Now, consider the subsetof those hidden variables λ shared between the three par-ties such that the first condition holds. Then, the second,the third, and the fourth condition, together with the firstcondition implies that P ( A (cid:48) = l ⊕ , B (cid:48) = m ⊕ , C (cid:48) = n ⊕ > q , the value of the joint probability appear-ing in the first condition is the success probability of theHardy’s argument. In quantum mechanics, for tri-partitesystem the maximum value of q is 0 .
125 [14]. On theother hand in a general no-signaling theory q attains amaximum value 0 . q , ofa bipartite two-input-two-output Hardy’s argument are:(i) Under no-signaling constraint max ( q ) = 0 .
5, whichis same as for the tripartite correlations [19], (ii) Underthe information causality principle max ( q ) is boundedfrom above by 0 .
207 [10], and (iii) In quantum mechanics, max ( q ) = 0 .
09 [18].
IV. TIME-ORDERED BI-LOCALCORREALIONS
A tripartite no-signaling probability distribution P ( abc | xyz ) belongs to TOBL [20–22] if it can be writ-ten as P ( abc | xyz ) = (cid:88) λ p λ P ( a | x, λ ) P B → C ( bc | yz, λ ) (7)= (cid:88) λ p λ P ( a | x, λ ) P B ← C ( bc | yz, λ ) (8)and analogously for B | AC and C | AB , where p λ is thedistribution of some random variable λ , shared by theparties. The distributions P B → C and P B ← C respect theconditions P B → C ( b | y, λ ) = (cid:88) c P B → C ( bc | yz, λ ) (9) P B ← C ( c | z, λ ) = (cid:88) b P B ← C ( bc | yz, λ ) (10)From these equations it is clear that the distributions P B → C allow signaling from Alice to Bob and P B ← C allowsignaling from Bob to Alice. If a tripartite no-signalingprobability distribution P ( abc | xyz ) belongs to the set ofTOBL distributions, all possible bipartite distributionsderived by applying any local wirings on P ( abc | xyz ) arelocal, i.e., the probability distribution P ( abc | xyz ) in gen-eral respects any bi-partite physical principles, therefore,any TOBL correlation must also respect the informationcausality principle. TABLE I: Tripartite no-signaling probability distribution P ( abc | xyz ) with maximum Hardy’s success 1 / xyz \ abc |
000 001 010 011 100 101 110 111000
14 14
14 14
14 14
14 14
14 14
14 14
14 14
101 0
14 14
14 14
14 14 V. MAXIMUM TRI-PARTITE HARDY’SNONLOCALITY RESPECTING ALL BI-PARTITEPRINCIPLES
Without loss of generality we consider following tri-partite Hardy’s correlation: P (110 | > P (110 | P (100 | P (010 | P (111 | P ( abc | xyz ) belonging to TOBL set respectsall bi-partite information principles. On maximizing suc-cess probability P (110 | . A probability distri-bution which attains this maximum value for Hardynonlocality is given in TABLE I.We now construct a TOBL model (TABLES (II-VII))for the distribution P ( abc | xyz ) given in TABLE I. Proba-bility distribution appearing in the TOBL decompositionfor the bi-partition A | BC , are such that for a given λ Al-ice’s outcome a depends only her measurement settings x . Also, for given λ , P B → C ( b | y, λ ) is independent of z but for B → C , c depends on both y and z . Similarly,for given λ , P B ← C ( c | z, λ ) is independent of y but for B ← C , b depends on both y and z . Let a x , b y and c z denote the outcomes for Alice, Bob and Charlie for therespective inputs x , y and z . For all TABLES (II-VII)in TOBL decompositions, the outputs are deterministicand the weights p λ are same. And the outcome assign-ments of A for A | B → C and A | B ← C are same andsimilar is true for other two bipartition. We summarizethe above result in the following proposition. Proposition:
The maximum success probability forHardy’s nonlocality argument for tri-partite time-ordered-bi-local (TOBL) correlations is . Now, in view of the conjuncture made in [21], TOBLset constitute the largest set of correlations that remainsconsistent under wirings and classical communication
TABLE II: TOBL model A | B → C . λ p λ a a b b c c c c A | B ← C . λ p λ a a b b b b c c B | A → C . λ p λ b b a a c c c c B | A ← C . λ p λ b b a a a a c c A → B | C . λ p λ a a b b b b c c A ← B | C . λ p λ a a a a b b c c prior to the inputs protocols when some parties collabo-rate (i.e., the set of correlations respecting all bi-partiteprinciples). Based on this conjuncture, the value inthe above proposition is the maximum value of Hardy’snonlocality for tri-partite correlations satisfying all bi-principles. VI. QUANTUM LIKE FEATURE UNDER IC
When two distance observers are involved, IC princi-ple has been proved to be more efficient to single outthe physically allowed correlations in comparison to NSprinciple. But when more than two distance observersare involved both IC and NS principles have been provedto be insufficient to witness physical correlations [12–14].However, here we argue that IC, in spite of being a bi-partite principle like NS, exhibits a qualitative quantumlike feature when set of multipartite correlations are con-sidered. Interestingly, this feature cannot be reproducedjust by considering the NS principle.According to our proposition stated in the section (V),as the maximum success probability of Hardy’s argumentfor tri-partite system satisfying any bi-partite informa-tion principle can not be less than , it is even more truewhen IC principle is the only constraint. On the otherhand, for bi-partite system, applying a necessary condi-tion for respecting the IC principle, the maximum successprobability of Hardy’s nonlocality argument have beenshown to be bounded above by 0 .
207 [10]. Therefore,there is a gap between maximum success probabilities ofHardy’s nonlocality argument for bi-partite system andtri-partite system under the constraint of IC principleand this gap is decisive in the sense that it may increasebut cannot decrease. If we consider the scenario withinquantum correlations then from the results of [18] and[14] we observed a genuine gap between maximum suc-cess probabilities for Hardy’s nonlocality argument for bi-partite quantum system and tri-partite quantum system(in [18] ([14]) it has been shown that maximum successprobabilities for Hardy’s nonlocality argument for bi(tri)-partite system is 0 . ) ). Interestingly, the set of NScorrelations both for bi-partite and tri-partite scenarioexhibit Hardy’s nonlocality argument with same maxi-mum success probability 0 . VII. CONCLUSION
Time-ordered-bi-local correlation is an important char-acterization to study nonlocality in a multiparty scenario.We studied the tripartite Hardy’s nonlocality by span-ning the set of all TOBL correlations. We obtain themaximum value of Hardy’s success probability within thisset to be . Assuming the conjuncture made in [21] tobe true, this value is the maximum success probabilityof tripartite Hardy’s correlation respecting all bi-partiteprinciples. In case, if the conjuncture does not hold thederived value is a lower bound on the maximum successprobability for tripartite Hardy’s correlation satisfying allbi-partite principles. We argue that though IC principlecan not reproduce various quantum features quantita-tively like maximum success probability for Hardy’s non-locality in quantum mechanics, still it could reproduce aqualitative interesting quantum feature like revealing agap between bi-partite and tri-partite cases for the max-imum success probability of Hardy’s nonlocality argu-ment; this result holds independent of the above men-tioned conjuncture. Thus though IC, being a bi-partiteprinciple, is insufficient for the reproduction of physicallyallowed correlations for multipartite case [13], it can stillbe a useful in reproducing many quantum like featureseven when three or more spatially separated observersare involved. Acknowledgments
It is a pleasure to thank Guruprasad Kar for manystimulating discussion. SK acknowledges Sibasish Ghoshfor fruitful discussions during his recent visit to IMSC.SD and AR acknowledges support from the DST projectSR/S2/PU-16/2007. RR acknowledges partial supportby TEAM program of Foundation for Polish Science andERC grant QOLAPS. [1] J. S. Bell, Speakable and Unspeakable in Quantum Me-chanics: Collected papers on quantum philosophy, Cam-bridge University Press.[2] J.F. Clauser, M.A. Horne, A. Shimony, and R.A. Holt,Phys. Rev. Lett. , 880 (1969).[3] B. S. Cirel’son, Lett. Math. Phys. , 93 (1980).[4] S. Popescu and D. Rohrlich, Foundations of Physics, ,379 (1994).[5] W. van Dam, Nonlocality and Communication complex-ity, Ph.D. thesis,University of Oxford (2000).[6] W. van Dam, arXiv:0501159 [quant-ph] (2005).[7] M. Navascues and H.Wunderlich, Proc. Roy. Soc. Lond.A, 466881 (2009).[8] M. Pawlowski, T. Paterek, D. Kaszlikowski, V. Scsrani,A. Winter and M. Zukowski, Nature , 1101 (2009).[9] J. Allcock, N. Brunner, M. Pawlowaski, and V. Scarani,Phys. Rev. A , 040103(R)(2009).[10] A. Ahanj, S. Kunkri, A. Rai, R. Rahaman, and P. S.Joag, Phys. Rev. A , 032103 (2010).[11] MD. R. Gazi, A. Rai, S. Kunkri, and R. Rahaman, J.Phys. A: Math. Theor. , 452001 (2010). [12] T. H. Yang, D. Cavalcanti, M. L. Almeida, C. Teo andV. Scarani, New J. Phys. , 013061.[13] R. Gallego, L. Erik Wurflinger, A. Acin and M. Navas-cues, Phys. Rev. Lett , 210403 (2011).[14] S. Das, M. Banik, A. Rai, MD R. Gazi and S. Kunkri,arXiv:1209.3490 [quant-ph] (2012).[15] L. Hardy, Phys. Rev. Lett. , 2981 (1992).[16] L. Hardy, Phys. Rev. Lett. , 1665 (1993).[17] S. Kunkri, S. K. Chaudhary, A. Ahanj, and P. Joag,Phys. Rev. A , 022346 (2006).[18] R. Rabelo, L. Y. Zhi and V. Scarani, Phys. Rev. Lett. , 180401 (2012).[19] S. K. Choudhary et.al. , Quantum Information and Com-putation, Vol. 10 , No. 9 and 10 (2010) 08590871.[20] S. Pironio, J. D. Bancal and V. Scarani, J. Phys. A:Math. Theor.44