arXiv:1808.00318v3 [quant-ph] 3 Feb 2019 age ytm.I h eia ae [17] paper entanglement without seminal cality the In systems. tangled in [12–16]. nonclassicality manner quantify device-independent they a as technologies quantum foun- in applicatio the have tests from nonlocality Apart Bell implications, theory. probabilitie dational realistic by local satisfied any be from must arising [3–11]—the that inequalities inequalities Bell-type of are family violate states they Entangled [2]. for states nonlocal entangled from arises [1], well- ity most nonlocal- nonlocality—Bell The quantum of properties. manifestation known nonlocal possess can separated, sawy osbeuigLOCC. using discrimination possible any exact always for states is product whether orthogonal was of asked set they known question the The communica- state classical (LOCC). which and tion in operations local possible, using as in, well is as system identify, to states. product is orthogonal task known The several of was one observers, separated in by prepared held sys- parts quantum two of a consisting that tem, con- Suppose they problem: particular, following In the sidered nonlocality. Bell from fundamen- way different a tally in properties nonlocal exhibit can states uct ∗ § ‡ † [email protected] [email protected] [email protected] [email protected] olclpoete,hwvr r o etitdol oen- to only restricted not are however, properties, Nonlocal physically are which of parts systems, quantum Composite eur nage eore cosalbipartitions. all across resources implemen local entangled that require implies bases product such of existence for construction the nitnusal,btw hwta h ovrede o al not does converse on the bases product that orthogonal show no we postmeas is but the it of indistinguishable, if orthogonality preserving irreducible while locally set be the loc from to of defined notion is p the states we through quantum Here, systems multiparty communication. in classical nonlocality and operations po not local is of it i.e., indistinguishable; to locally bee said are is states have states the product may of set parts a particular, In whose properties. systems, unentangled even However, unu olclt suulyascae ihetnlds entangled with associated usually is nonlocality Quantum .N oeNtoa etrfrBscSine,BokJ,Sect JD, Block Sciences, Basic for Center National Bose N. S. ninIsiueo cec dcto n eerhKolkata Research and Education Science of Institute Indian eateto ahmtc,Ida nttt fSineEdu Science of Institute Indian Mathematics, of Department Bennett , eateto hsc n etrfrAtoatcePhysics Astroparticle for Center and Physics of Department togQatmNnoaiywtotEntanglement without Nonlocality Quantum Strong d rni aps oenetII ehmu 600 Odisha, 760010, Berhampur ITI, Government Campus, Transit oeIsiue N8,Sco ,Bdangr okt 70009 Kolkata Bidhannagar, V, Sector 80, EN Institute, Bose 3 = tal. et civstemnmmdmninncsayfrsc rdc s product such for necessary dimension minimum the achieves C d hwdta prod- that showed ⊗ unu nonlo- Quantum C d osuhoBandyopadhyay Somshubhro ⊗ C d aoahHalder Saronath for rsyAgrawal Sristy ai Banik Manik d ns 3 = s sbet pial itnus h ttsb n sequence any by states the distinguish optimally to ssible , xii qatmnnoaiywtotetnlmn”if entanglement” without nonlocality “quantum exhibit 4 osbeol yetnldmaueet ntejitsys- joint the on becomes measurements tem. certainty entangled with by elimination only possible state where prod- states, nonorthogonal also of uct 41]) feature [40, us nonlocal Refs. Let another see (also yet [39] labs. reveals theorem different PBR recent in the that prepared though note even been parts, have their on may sys- LOCC the they of of sequence state any the whole than tem about the on information measurement more a reveals that Here, system sense the entanglement. in distinguished without is orthogonal) nonlocality nonlocality are (ex- exhibit states optimally the LOCC be say if by cannot now only We that and states [38]. if product actly, true of be set to any conjec- shown that The recently only discrimination. was measure- state ture LOCC for that suboptimal conjectured are nonorthog- three ments they of [35] set states, earlier specific product a years For onal few 37]). a [36, Wootters Refs. see and (also Peres by out pointed and 24–30] 31–34]. 21, 19, 20, [18, explored [18, properties systems their multiparty and bipar- [18–23] both in tite found [17]. were LOCC examples using such more possible Subsequently, not is discrimination exact which etda rhgnlpoutbss(P)on (OPB) basis product orthogonal pre- an authors with the sented system surprisingly, the But of alone. state measurements the it local about therefore, learn and to rules), possible be of should set known prepa some local admit (following states ration product because yes be to ought tion htaelclyirdcbei l iattos where bipartitions, all in irreducible locally are that aino utprysprbemaueetmay measurement separable multiparty a of tation rmn tts uhast ydfiiin slocally is definition, by set, a Such states. urement h eut fPrsWotr 3]adBennett and [35] Peres-Wootters of results The first fact, in was, result of kind this of possibility The o nuto ugssta h nwrt h bv ques- above the to answer the that suggests intuition Now osbet oal lmnt n rmr states more or one eliminate locally to possible t lirdcblt.Asto utpryorthogonal multiparty of set A irreducibility. al ae ytervoain fBl-yeinequalities. Bell-type of violations their by tates † eetasrne aietto fti idof kind this of manifestation stronger a resent ‡ ashl.W rvd h rteape of examples first the provide We hold. ways ∗ rII ihnaa,Klaa709,India 700098, Kolkata Bidhannagar, III, or oapr etBna 426 India 741246, Bengal West Mohanpur, , rprdsprtl,cnso nonlocal show can separately, prepared n ainadRsac Berhampur, Research and cation § n pc Science, Space and ,India 1, India ae oeit The exist. to tates C 3 tal. et ⊗ C 3 [17] for - 2

n initiated a plethora of studies on more general local state dis- quantum states on H = i=1 Hi with n ≥ 2 and dim Hi ≥ crimination problems—the task of optimal discrimination of 2, i = 1,...,n, is locally irreducible if it is not possible to multiparty states, not necessarily product, by means of LOCC eliminate one or more statesN from the set by orthogonality- [17, 18, 21–23, 25–30, 32–34, 37, 38, 42–72]. It was found preserving local measurements. that in some cases, e.g., a set of two pure states, LOCC can A locally indistinguishable set in general is not locally irre- indeed accomplish the task as efficiently as global measure- ducible except when it contains three orthogonal pure states. ments [51, 52], whereas in some other cases, e.g., orthogonal entangled bases [53, 55–58, 61, 63, 65, 68, 70] they cannot, Proposition 1. Any set of three locally indistinguishable or- n and we call such states locally indistinguishable. Locally in- thogonal pure states on H = i=1 Hi with n ≥ 2 and distinguishable states have found useful applications in quan- dim Hi ≥ 2, i =1,...,n, is locally irreducible. N tum cryptography primitives such as data hiding [73–76] and Since any two orthogonal pure states can be exactly dis- quantum secret sharing [77]. tinguished by LOCC [51], a locally reducible set containing In this paper, we report new nonlocal properties of mul- three orthogonal pure states must be locally distinguishable. tiparty orthogonal product states—within the framework of But this contradicts thefactthat theset is known to be locally local state discrimination, but considering instead a more ba- indistinguishable. This proves the proposition. sic problem—quantum state elimination using orthogonality- We will now describe a sufficient condition for local irre- preserving local measurements (a measurement is orthogo- ducibility. The formalism was originally developed [54] (also nality preserving if the the post-measurement states remain see Refs. [43, 44]) for local indistinguishability. We begin by orthogonal). The motivation stemmed from the observation defining a nontrivial measurement [54]. that some sets of orthogonal states on a composite Hilbert space are locally reducible; i.e., it is possible to locally elimi- Definition 2. A measurement is nontrivial if not all the nate one or more states from the set while preserving orthog- POVM elements are proportional to the identity operator. onality of the postmeasurement states. For such sets, the task Otherwise, the measurement is trivial. of local state discrimination is therefore reduced to that of a The crux of the argument [54] was that, in any local pro- subset of states. tocol one of the parties must go first, and whoever goes While a locally distinguishable set is locally reducible (triv- first must be able to perform some nontrivial orthogonality- ially), the opposite is not true in general. In the following ex- preserving measurement (NOPM). This fits naturally into amples, the locally indistinguishable sets are locally reducible our scenario for the following reasons. The measurement to a union of two or more disjoint subsets, each of which can should be orthogonality preserving because we require that be addressed individually. any measurement outcome must leave the postmeasurement 2 4 (a) Consider the entangled orthogonal basis on C ⊗ C : states mutually orthogonal, possibly eliminating some states but not all (unless it correctly identifies the input right away). |00i ± |11i |02i ± |13i (1) It is also essential that the measurement is nontrivial be- |01i ± |10i |03i ± |12i . cause a trivial measurement despite satisfying (trivially) the orthogonality-preserving conditions, gives us no information Here, Bob performs a local measurement to distinguish the about the state. The sufficient condition follows by noting subspaces spanned by {|0i , |1i} and {|2i , |3i}. Depending that, if none of the parties can perform a local NOPM, the upon the outcome, Alice and Bob end up with a state belong- states must be locally irreducible. ing to one of the two subsets (left or right). Note that, neither Following Ref. [54] we now discuss how to apply this con- subset is locally distinguishable [53] (in fact, neither is locally dition when a set contains only orthogonal pure states. The reducible—see Proposition 2). basic idea is to check whether an orthogonality-preserving C3 C3 (b) Consider the orthogonal basis on ⊗ : POVM on any of the subsystems is trivial or not. If it is triv- ial for all subsystems, the states are locally irreducible. |00i ± |11i |02i |20i |22i k (2) Let S = {|ψii} =1 be a set of orthogonal pure states on |01i ± |10i |12i |21i . n i H = i=1 Hi, where n ≥ 2, and dim Hi ≥ 2, i =1,...,n. i Here, if the unknown state is one of the product states, it Consider a POVM πα , α = 1, 2,... that may be per- N can always be correctly identified, and if it is not, all product formed on the ith subsystem. The POVM elements πi are  α states can be locally eliminated. In the latter case, Alice and positive operators summing up to identity and correspond Bob will end up with one of the four Bell states (local protocol to the measurement outcomes. Further, each element admits i i i i given in Appendix A). Note that, unlike the previous example, the Krauss form: πα = Mα†Mα, where Mαs are the Krauss here not all the subsets are locally indistinguishable. operators. The probability that an input state |ψxi ∈ S yields I i I The above examples give rise to the following question: the outcome α is pα = ψx 1 ⊗···⊗ πα ⊗···⊗ n ψx Are all locally indistinguishable sets locally reducible? The with the corresponding postmeasurement state given by 1 i I1 ⊗···⊗ M ⊗···⊗ I |ψ i. Since we require the answer is no. As will be shown, some of the well-known lo- √pα α n x cally indistinguishable sets are not locally reducible. First, we POVM to be orthogonality preserving, for all pairs of states
have the following definition. {|ψxi , |ψyi} , x 6= y and all outcomes α, the conditions I i I Definition 1. (Locally irreducible set) A set of orthogonal ψx 1 ⊗···⊗ πα ⊗···⊗ n ψy =0 (3)

3 need to be satisfied. To use the above conditions effectively, a genuinely entangled orthogonal basis (the basis vectors are i we represent each POVM element πα, α = 1, 2,... by a entangled in every bipartition) is a promising candidate be- di × di matrix (in the computational basis) and solve for the cause in any bipartition, the states are not only locally indis- matrix elements by choosing suitable pairs of vectors (also tinguishable but also none can be correctly identified with a expressed in the computational basis of H). This can bedone nonzero probability using LOCC [55]. However, we find that exactly in many problems of interest. Now if we find that the the GHZ basis, which is genuinely entangled and locally irre- i conditions (3) are satisfied only if πα is proportional to the ducible [Proposition 3], is locally reducible in all bipartitions. identity for all α, then the measurement is trivial. This means Proposition 4. The three-qubit GHZ basis given in proposi- the ith party cannot begin a LOCC protocol, and if this is true tion 3 is locally reducible in all bipartitions. for all i, then none of the parties can go first. Therefore, S is locally irreducible. We will use this condition extensively in The proof is simple. Note that, one can always perform a our proofs. joint measurement on any two qubits to distinguish the sub- The OPB on C3 ⊗ C3 [17] is locally irreducible. This fol- spaces spanned by {|00i , |11i} and {|01i , |10i}. Thus in any lows from the proof showing that the states are locally indis- bipartition, the whole set can be locally reduced to two dis- tinguishable [54]. We now show that the Bell basis and the joint subsets, each of which is locally equivalent to the Bell three-qubit GHZ basis are locallyirreducible (both are locally basis (the proof can be extended mutatis mutandis for a N- indistinguishable [53, 55]) using the method just described. qubit GHZ basis with the identical conclusion). Proposition 4 gives rise to an interesting question: Can Proposition 2. The two-qubit Bell basis (unnormalized): multiparty orthogonal product states be locally irreducible |00i ± |11i , |01i ± |10i is locally irreducible. in all bipartitions? If such sets exist, then they would clearly The proof is by contradiction. Suppose that the Bell basis demonstrate quantum nonlocality stronger than what we is locally reducible. Then, either Alice or Bob must be able to presently understand. begin the protocol by performing some local NOPM. Without First we observe that such product states cannot be found loss of generality assume that Bob goes first. Bob’s general in systems where one of subsystems has dimension two: If measurement can be represented by a set of 2 × 2 POVM el- the system contains a qubit, then the set is locally distin- guishable in the bipartition qubit|rest because orthogonal a00 a01 2 ements πα = written in the {|0i , |1i} basis. product states on C ⊗ Cd, d ≥ 2 are known to be lo- a10 a11   cally distinguishable [20]. So they can only exist, if at all, Since this measurement is orthogonality preserving, for any n pair of Bell states the conditions (3) must hold. By choos- on H = i=1 Hi, n ≥ 3, where dim Hi ≥ 3 for every i. Thus the minimum dimension corresponds to a three-qutrit ing suitable pairs, it is easy to show that πα must be propor- N tional to the identity (details in Appendix B). As the argument system. holds for all outcomes, all of Bob’s POVM elements are pro- We checked all the known examples (to the best of our portional to the identity. This means Bob cannot go first, and knowledge) of locally indistinguishable multiparty orthogo- from the symmetry of the Bell states, neither can Alice. This nal product states, but did not find any with the desired prop- completes the proof. erty. Some were ruled out by the dimensionality constraint, and the rest turned out to be either locally distinguishable Proposition 3. The three-qubit GHZ basis (unnormalized): [20, 24, 26, 28, 29, 43, 44, 48] or locally reducible [30] in one |000i ± |111i, |011i ± |100i, |001i ± |110i, |010i ± |101i, is or more bipartitions. locally irreducible. The main result of this paper lies in showing that multi- party orthogonal product states that are locally irreducible The proof is along the same lines as in the previous one in all bipartitions, exist. We call such sets strongly nonlocal. and is given in Appendix C (can be extended for a N-qubit GHZ basis). Definition 3. Consider a composite quantum system H = n We now come to the main part of the paper. Here we con- i=1 Hi with n ≥ 3 and dim Hi ≥ 3, i =1,...,n. Aset of sider the following question: Do there exist multiparty or- orthogonal product states |ψii = |αii1 ⊗ |βii2 ⊗ · · · ⊗ |γii N n thogonal sets that are locally irreducible in every bipartition? on H is strongly nonlocal if it is locally irreducible in every The motivation for asking this question is that many of the bipartition. properties of multiparty states in general are not preserved if We now give an example of an OPB on C3 ⊗ C3 ⊗ C3 we change the spatial configuration. For example, the three- and prove it strongly nonlocal. Note that, this construction party (A, B, and C) unextendible product basis (UPB) on achieves the minimum dimension required (as discussed ear- C2 C2 C2 ⊗ ⊗ [18] is locally indistinguishable (hence, nonlo- lier). We will use the notation |1i, |2i, |3i for the bases of Al- cal when all parts are separated) but can be perfectly distin- ice, Bob, and Charlie’s Hilbert spaces. Consider the following guished across all bipartitions A|BC, B|CA, and C|AB [18] OPB on C3 ⊗ C3 ⊗ C3: using LOCC (and therefore, not nonlocal in the bipartitions). In fact, one can also find sets of entangled states that are lo- |1i |2i |1 ± 2i |2i |1 ± 2i |1i |1 ± 2i |1i |2i cally distinguishable in one bipartition but not in others (see |1i |3i |1 ± 3i |3i |1 ± 3i |1i |1 ± 3i |1i |3i Appendix D). |2i |3i |1 ± 2i |3i |1 ± 2i |2i |1 ± 2i |2i |3i (4) So which sets of orthogonal states are expected to remain |3i |2i |1 ± 3i |2i |1 ± 3i |3i |1 ± 3i |3i |2i locally irreducible in all bipartitions? Intuition suggests that |1i |1i |1i |2i |2i |2i |3i |3i |3i , 4

1 1 9 where |1 ± 2i stands for √2 (|1i ± |2i) etc. Note that, the set 9 × 9 matrix in the {| i ,..., | i} basis of HBC : (4) is invariant under cyclic permutation of the parties A, B, a11 a12 a13 a14 a15 a16 a17 a18 a19 and C. We first show that the states are locally irreducible. a21 a22 a23 a24 a25 a26 a27 a28 a29 3 3 3   Lemma 1. The set of states given by (4) on C ⊗ C ⊗ C is a31 a32 a33 a34 a35 a36 a37 a38 a39 locally irreducible. a41 a42 a43 a44 a45 a46 a47 a48 a49   Πα = a51 a52 a53 a54 a55 a56 a57 a58 a59 . (8)   To prove the lemma, we first consider the following states a61 a62 a63 a64 a65 a66 a67 a68 a69   a71 a72 a73 a74 a75 a76 a77 a78 a79 |1i |2i |1 ± 2i |2i |1 ± 2i |1i |1 ± 2i |1i |2i a81 a82 a83 a84 a85 a86 a87 a88 a89 (5)   |1i |3i |1 ± 3i |3i |1 ± 3i |1i |1 ± 3i |1i |3i , a91 a92 a93 a94 a95 a96 a97 a98 a99     chosen from the whole set. For the above states it was shown The measurement must leave the postmeasurement states [26] that any 3×3 orthogonality-preserving POVM acting on mutually orthogonal. By choosing suitable pairs of vectors any subsystem must be proportional to the identity. Clearly, {|ψii , |ψj i}, i 6= j, we find that all the off-diagonal ma- this must also hold for the whole set (4) of which the states (5) trix elements aij , i 6= j must be zero if the orthogonality- form a subset because all the states belong to the same state preserving conditions hψi |I ⊗ Πα| ψj i = 0 are to be satis- space C3 ⊗C3 ⊗C3. Therefore, none of the parties can begin fied. Table I in Appendix F shows the complete analysis. Sim- a LOCC protocol by performing some local NOPM. Hence, ilarly, we find that the diagonal elements are all equal. For ex- the proof (for completeness, we have included the details in ample, by setting the inner product h1|h4 + 5|I ⊗ Πα|1i|4 − Appendix E). 5i = 0, we get a44 = a55. Table II (Appendix F) summarizes this analysis. As the diagonal elements of Πα areallequaland Theorem 1. The orthogonal product basis (4) is strongly non- the off-diagonal elements are all zero, Πα must be propor- local. tional to the identity. The argument applies to all measure- ment outcomes, and thus all POVM elements {Π } must be We need to show that the states (4) form a locally irre- α proportional to the identity. This means the POVM must be ducible set in any bipartition. To begin with, consider the trivial, and therefore, BC cannot go first. Thus the states (7) bipartition A|BC (H ⊗ H ). In this bipartition the states A BC form a locally irreducible set in the bipartition A|BC. Now (4) take the form from the symmetry of the states (4) [invariant under cyclic |1i |21 ± 22i |2i |11 ± 21i |1 ± 2i |12i permutation of the parties], it follows that the states (4) are |1i |31 ± 33i |3i |11 ± 31i |1 ± 3i |13i also locally irreducible in the bipartitions C|AB, and B|CA. |2i |31 ± 32i |3i |12 ± 22i |1 ± 2i |23i (6) This completes the proof of the theorem. |3i |21 ± 23i |2i |13 ± 33i |1 ± 3i |32i In Appendix G we have given an example of a strongly C4 C4 C4 |1i |11i |2i |22i |3i |33i . nonlocal OPB on ⊗ ⊗ along with the complete proof. We now discuss the question of local discrimination of Physically this means the subsystems B and C are treated strongly nonlocal product states using entanglement as a re- together as a nine-dimensional subsystem BC. For clarity, source. Note that, in our examples, the three-party separa- 3 ble measurements cannot be locally implemented even if any denote the elements of the basis {|iji}i,j=1 on HBC as: ∀i = 1, 2, 3, |1ii → |ii, |2ii → |i + 3i, and |3ii → |i + 6i and two share unlimited entanglement. So exact local implemen- rewrite the states (6) as: tation would require a resource state which must be entan- gled in all bipartitions (this also holds for product states that |1i |4 ± 5i |2i |1 ± 4i |1 ± 2i |2i are locally indistinguishable in all bipartitions [30] but not |1i |7 ± 9i |3i |1 ± 7i |1 ± 3i |3i strongly nonlocal). As to how much entanglement must one |2i |7 ± 8i |3i |2 ± 5i |1 ± 2i |6i consume, we do not have any clear answer. A teleportation (7) 3 3 |3i |4 ± 6i |2i |3 ± 9i |1 ± 3i |8i protocol can perfectly distinguish the states (4) using C ⊗C |1i |1i |2i |5i |3i |9i . maximally entangled states shared between any two pairs, but whether one can do just as well using cheaper resources We now show that any orthogonality-preserving local POVM (see Ref. [31]) is an intriguing question. performed either on A or BC must be trivial. Therefore, nei- The results in this paper also leave open other interesting ther Alice(A) norBob andCharlietogether(BC) can go first. questions. One may consider generalizing our constructions n Cd First, consider Alice. Recall that, Lemma 1 holds because on i=1 for n ≥ 4, and d ≥ 3. Another problem worth none of the parties can perform a local NOPM when all parts considering is whether incomplete orthogonal product bases are separated. Since in the bipartition A|BC Alice’s subsys- canN be strongly nonlocal, e.g., can we have a strongly non- tem is still separated from the rest, we conclude that Alice local UPB? Finally, one may ask, whether one can find en- cannot go first. tangled bases that are locally irreducible in all bipartitions. We now consider whether it is possible to initiate a local In view of Proposition 4, it seems that to satisfy “local ir- protocol by performing some NOPM on BC. Let the POVM reducibility in all bipartitions,” the structure of the states is {Πα} describe a general orthogonality-preserving measure- likely to be more important than their entanglement. Here, 2 2 2 ment on BC. Each POVM element Πα can be written as a even examples in the simplest case of C ⊗ C ⊗ C can help 5 us to understand this property better. the Department of Science and Technology, Government of India. S. B. is supported in part by SERB (Science and Engi- neering Research Board), DST, Govt. of India through Project ACKNOWLEDGMENTS No. EMR/2015/002373.

M. B. acknowledges support through an INSPIRE-faculty position at S. N. Bose National Center for Basic Sciences by

[1] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, M. Terhal, Unextendible product bases, uncompletable prod- Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014); Erratum Rev. uct bases and bound entanglement, Commun. Math. Phys. 238, Mod. Phys. 86, 839 (2014). 379 (2003). [2] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, [21] Y. Feng and Y.-Y. Shi, Characterizing locally indistinguishable Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009). orthogonal product states, IEEE Trans. Inf. Theory 55, 2799 [3] J. S. Bell, On the Einstein Podolsky Rosen Paradox, Physics 1, (2009). 195 (1964); On the Problem of Hidden Variables in Quantum [22] Y.-H. Yang, F. Gao, G.-B. Xu, H.-J. Zuo, Z.-C. Zhang, and Q.- Mechanics, Rev. Mod. Phys. 38, 447 (1966). Y. Wen, Characterizing unextendible product bases in qutrit- [4] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposed ququad system, Sci. Rep. 5, 11963 (2015). experiment to test local hidden-variable theories, Phys. Rev. [23] G.-B. Xu, Y.-H. Yang, Q.-Y. Wen, S.-J. Qin, and F. Gao, Locally Lett. 23, 880 (1969). indistinguishable orthogonal product bases in arbitrary bipar- [5] S.J. Freedman and J.F. Clauser, Experimental test of local tite quantum system, Sci. Rep. 6, 31048 (2016). hidden-variable theories, Phys. Rev. Lett. 28, 938 (1972). [24] J. Niset and N. J. Cerf, Multipartite nonlocality without entan- [6] A. Aspect, P. Grangier, and G. Roger, Experimental Tests of glement in many dimensions, Phys. Rev. A 74, 052103 (2006). Realistic Local Theories via Bell’s Theorem, Phys. Rev. Lett. [25] Y.-H. Yang, F. Gao, G.-J. Tian, T.-Q. Cao, and Q.-Y. Wen, Local 47, 460 (1981). distinguishability of orthogonal quantum states in a 2 ⊗ 2 ⊗ 2 [7] A. Aspect, J. Dalibard, and G. Roger, Experimental Test of Bell’s system, Phys. Rev. A 88, 024301 (2013). Inequalities Using Time-Varying Analyzers, Phys. Rev. Lett. [26] S. Halder, Several nonlocal sets of multipartite pure orthogonal 49, 1804 (1982). product states, Phys. Rev. A 98, 022303 (2018). [8] B. Hensen et al., Loophole-free Bell inequality violation us- [27] G.-B. Xu, Q.-Y. Wen, S.-J. Qin, Y.-H. Yang, and F. Gao, Quantum ing electron spins separated by 1.3 kilometres, Nature 526, 682 nonlocality of multipartite orthogonal product states, Phys. (2015). Rev. A 93, 032341 (2016). [9] J. Handsteiner et al., Cosmic Bell Test: Measurement Settings [28] G.-B. Xu, Q.-Y. Wen, F. Gao, S.-J. Qin, and H.-J. Zuo, Local from Milky Way Stars, Phys. Rev. Lett. 118, 060401 (2017). indistinguishability of multipartite orthogonal product bases, [10] W. Rosenfeld et al., Event-Ready Bell Test Using Entangled Quantum Inf. Process. 16, 276 (2017). Atoms Simultaneously Closing Detection and Locality Loop- [29] Y.-L. Wang, M.-S. Li, Z.-J. Zheng, and S.-M. Fei, The local in- holes, Phys. Rev. Lett. 119, 010402 (2017). distinguishability of multipartite product states, Quantum Inf. [11] BIG Bell Test Collaboration, Challenging local realism with hu- Processing 16, 5 (2017). man choices, Nature 557, 212 (2018). [30] Z.-C. Zhang, K.-J. Zhang, F. Gao, Q.-Y. Wen, and C. H. Oh, Con- [12] J. Barrett, L. Hardy, and A. Kent, No Signaling and Quantum struction of nonlocal multipartite quantum states, Phys. Rev. A Key Distribution, Phys. Rev. Lett. 95, 010503 (2005). 95, 052344 (2017). [13] A. Acín, N. Gisin, and L. Masanes, From Bell’s Theorem to Se- [31] S. M. Cohen, Understanding entanglement as resource: Locally cure Quantum Key Distribution, Phys. Rev. Lett. 97, 120405 distinguishing unextendible product bases, Phys. Rev. A 77, (2006). 012304 (2008). [14] N. Brunner, S. Pironio, A. Acin, N. Gisin, A. A. Methot, and V. [32] A. M. Childs, D. Leung, L. Mančinska, and M. Ozols, A frame- Scarani, Testing the Dimension of Hilbert Spaces, Phys. Rev. work for bounding nonlocality of state discrimination, Com- Lett. 100, 210503 (2008). mun. Math. Phys. 323, 1121 (2013). [15] S. Pironio, A. Acín, S. Massar, A. Boyer de la Giroday, D. N. [33] S. Croke and S. M. Barnett, Difficulty of distinguishing product Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. states locally, Phys. Rev. A 95, 012337 (2017). Manning, and C. Monroe, Random Numbers Certified by Bell’s [34] M. Demianowicz and R. Augusiak, From unextendible prod- Theorem, Nature 464, 1021 (2010). uct bases to genuinely entangled subspaces, Phys. Rev. A 98, [16] R. Colbeck and R. Renner, Free randomness can be amplified, 012313 (2018). Nat. Phys. 8, 450 (2012). [35] A. Peres and W. K. Wootters, Optimal Detection of Quantum [17] C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, Information, Phys. Rev. Lett. 66, 1119 (1991). P. W. Shor, J. A. Smolin, and W. K. Wootters, Quantum nonlo- [36] S. Massar and S. Popescu, Optimal extraction of informa- cality without entanglement, Phys. Rev. A 59, 1070 (1999). tion from finite quantum ensembles, Phys. Rev. Lett. 74, 1259 [18] C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, (1995). and B. M. Terhal, Unextendible Product Bases and Bound En- [37] W. K. Wootters, Distinguishing unentangled states with an un- tanglement, Phys. Rev. A 59, 1070 (1999). entangled measurement, Int. J. Quantum Inf. 4, 219 (2006). [19] B. Groisman and L. Vaidman, Nonlocal variables with product [38] E. Chitambar and Min-Hsiu Hsieh, Revisiting the optimal de- state eigenstates, J. Phys. A: Math. Gen. 34, 6881 (2001). tection of quantum information, Phys. Rev. A 88, 020302(R) [20] D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. (2013). 6

[39] M. F. Pusey, J. Barrett and T. Rudolph, On the reality of the Lett. 95, 080505 (2005). quantum state, Nature Physics 8, pages 475–478 (2012). [60] M. Hayashi, D. Markham, M. Murao, M. Owari, and S. Virmani, [40] S. Bandyopadhyay, R. Jain, J. Oppenheim, and C. Perry, Con- Bounds on entangled orthogonal state discrimination using lo- clusive exclusion of quantum states, Phys. Rev. A 89, 022336 cal operations and classical communication, Phys. Rev. Lett. (2014). 96, 040501 (2006). [41] Shane Mansfield, Reality of the quantum state: Towards a [61] R. Y. Duan, Y. Feng, Z. F. Ji, and M. S. Ying, Distinguishing stronger ψ-ontology theorem, Phys. Rev. A 94, 042124 (2016). arbitrary multipartite basis unambiguously using local opera- [42] S. De Rinaldis, Distinguishability of complete and unextendible tions and classical communication, Phys. Rev. Lett. 98, 230502 product bases, Phys. Rev. A 70, 022309 (2004). (2007). [43] Z.-C. Zhang, F. Gao, G.-J. Tian, T.-Q. Cao, and Q.-Y. Wen, Non- [62] J. Calsamiglia, J. I. de Vicente, R. Munoz-Tapia, E. Bagan, Lo- locality of orthogonal product basis quantum states, Phys. Rev. cal discrimination of mixed states, Phys. Rev. Lett. 105, 080504 A 90, 022313 (2014). (2010). [44] Z.-C. Zhang, F. Gao, S.-J. Qin, Y.-H. Yang, and Q.-Y. Wen, Non- [63] S. Bandyopadhyay, S. Ghosh and G. Kar, LOCC distinguishabil- locality of orthogonal product states, Phys. Rev. A 92, 012332 ity of unilaterally transformable quantum states, New J. Phys. (2015). 13, 123013 (2011). [45] Y.-L. Wang, M.-S. Li, Z.-J. Zheng, and S.-M. Fei, Nonlocality [64] S. Bandyopadhyay, More nonlocality with less purity, Phys. of orthogonal product-basis quantum states, Phys. Rev. A 92, Rev. Lett. 106, 210402 (2011). 032313 (2015). [65] A. Cosentino, Positive-partial-transpose-indistinguishable [46] Z.-C. Zhang, F. Gao, Y. Cao, S.-J. Qin, and Q.-Y. Wen, Local states via semidefinite programming, Phys. Rev. A 87 (1), indistinguishability of orthogonal product states, Phys. Rev. A 012321 32 (2013). 93, 012314 (2016). [66] S. Bandyopadhyay, M. Nathanson, Tight bounds on the distin- [47] X. Zhang, X. Tan, J. Weng, and Y. Li, LOCC indistinguishable guishability of quantum states under separable measurements, orthogonal product quantum states, Sci. Rep. 6, 28864 (2016). Phys. Rev. A 88, 052313 (2013). [48] Y.-L. Wang, M.-S. Li, S.-M. Fei, and Z.-J. Zheng, Con- [67] R. Y. Duan, Y. Feng, Y. Xin, and M. S. Ying, Distinguishability structing unextendible product bases from the old ones, of quantum states by separable operations, IEEE Trans. Inform. arXiv:1703.06542 [quant-ph] (2017). Theory 55, 1320 (2009). [49] X. Zhang, J. Weng, X. Tan, and W. Luo, Indistinguishability of [68] N. Yu, R. Duan, and M. Ying, Four Locally Indistinguish- pure orthogonal product states by LOCC, Quantum Inf. Pro- able Ququad-Ququad Orthogonal Maximally Entangled States, cessing 16, 168 (2017). Phys. Rev. Lett. 109, 020506 (2012). [50] X. Zhang, J. Weng, Z. Zhang, X. Li, W. Luo, and X. Tan, Locally [69] M. Nathanson, Testing for a pure state with local operations distinguishable bipartite orthogonal quantum states in a d ⊗ n and classical communication, Journal of Mathematical Physics system, arXiv:1712.08830 [quant-ph] (2017). 51 (2010) 042102. [51] J. Walgate, A. J. Short, L. Hardy, and V. Vedral, Local distin- [70] A. Cosentino and V. Russo, Small sets of locally indistinguish- guishability of multipartite orthogonal quantum states, Phys. able orthogonal maximally entangled states, Quantum Infor- Rev. Lett. 85, 4972 (2000). mation & Computation 14 (13-14), 1098-1106 (2014). [52] S. Virmani, M. F. Sacchi, M. B. Plenio, D. Markham, Optimal [71] S. Bandyopadhyay, A. Cosentino, N. Johnston, V. Russo, J. Wa- local discrimination of two multipartite pure states, Phys. Lett. trous, and N. Yu, Limitations on separable measurements by A. 288 (2001). convex optimization, IEEE Transactions on Information The- [53] S. Ghosh, G. Kar, A. Roy, A. Sen (De), and U. Sen, Distinguisha- ory, Vol 61, Issue 6, Pages: 3593-3604 (2015). bility of Bell states, Phys. Rev. Lett. 87, 277902 (2001). [72] S. X. Yu and C. H. Oh, Detecting the local indistinguishability [54] J. Walgate and L. Hardy, Nonlocality, asymmetry, and distin- of maximally entangled states, arXiv:1502.01274 (2015). guishing bipartite states, Phys. Rev. Lett. 89, 147901 (2002). [73] B. M. Terhal, D. P. DiVincenzo, and D. W. Leung, Hiding bits [55] M. Horodecki, A. Sen(De), U. Sen, and K. Horodecki, Local in- in Bell states, Phys. Rev. Lett. 86, 5807 (2001). distinguishability: more nonlocality with less entanglement, [74] D. P. DiVincenzo, D. W. Leung, and B. M. Terhal, Quantum data Phys. Rev. Lett. 90, 047902 (2003). hiding, IEEE Trans. Inf. Theory 48, 580 (2002). [56] S. Ghosh, G. Kar, A. Roy, and D. Sarkar, Distinguishability of [75] T. Eggeling, and R. F. Werner, Hiding classical data in multi- maximally entangled states, Phys. Rev. A 70, 022304 (2004). partite quantum states, Phys. Rev. Lett. 89, 097905 (2002). [57] H. Fan, Distinguishability and indistinguishability by local [76] W. Matthews, S. Wehner, A. Winter, Distinguishability of operations and classical communication, Phys. Rev. Lett. 92, quantum states under restricted families of measurements with 177905 (2004). an application to quantum data hiding, Comm. Math. Phys. [58] M. Nathanson, Distinguishing bipartite orthogonal states by 291, Number 3 (2009). LOCC: best and worst cases, Journal of Mathematical Physics [77] D. Markham and B. C. Sanders, Graph states for quantum se- 46, 062103 (2005). cret sharing, Phys. Rev. A 78, 042309 (2008). [59] J. Watrous, Bipartite subspaces having no bases distinguishable by local operations and classical communication, Phys. Rev. 7

APPENDIX

A. Here we give the local protocol which shows that the set of states [given by (2)]

|00i ± |11i |02i |20i |22i |01i ± |10i |12i |21i is locally reducible, with only one subset (the Bell basis) being locally indistinguishable. The protocol goes like this: Both Alice and Bob perform the measurement that distinguishes the subspaces spanned by {|0i , |1i} and {|2i}. The following table summarizes the outcomes and inferences.

Alice → subspace subspace Bob {|0i , |1i} {|2i} ↓ subspace Bell Basis |20i / |21i {|0i , |1i} locally locally (9) indistinguishable distinguishable subspace |02i / |12i |22i {|2i} locally locally distinguishable distinguishable

B. Proof of irreducibility of the Bell basis (Proposition 2): Let πα = Mα† Mα, where Mα is the Krauss operator. Since the measurement is orthogonality-preserving, for every α, the following states must be pairwise orthogonal to each other.

(I ⊗ Mα)|ψ1i = (I ⊗ Mα)(|00i + |11i) (I ⊗ M )|ψ2i = (I ⊗ M )(|00i − |11i) α α (10) (I ⊗ Mα)|ψ3i = (I ⊗ Mα)(|01i + |10i)

(I ⊗ Mα)|ψ4i = (I ⊗ Mα)(|01i − |10i)

Setting the inner products to zero, we solve for the matrix elements:

hψ1|I ⊗ πα|ψ2i =0 a00 − a11 =0 a00 = a11

hψ1|I ⊗ πα|ψ3i =0 a01 + a10 =0 a01 = a10 =0 (11)

hψ1|I ⊗ πα|ψ4i =0 a01 − a10 =0

As we can see πα must be proportional to the identity.

C. Proof of irreducibility of the GHZ basis: The states in the GHZ basis are given by:

|G1i = |0i|0i|0i + |1i|1i|1i, |G2i = |0i|0i|1i + |1i|1i|0i, |G3i = |0i|1i|0i + |1i|0i|1i, |G4i = |0i|1i|1i + |1i|0i|0i, (12) |G5i = |0i|0i|0i − |1i|1i|1i, |G6i = |0i|0i|1i − |1i|1i|0i, |G7i = |0i|1i|0i − |1i|0i|1i, |G8i = |0i|1i|1i − |1i|0i|0i. 8

The proof works exactly the same way as the previous one. Let’s first consider Charlie. A measurement by Charlie can be I defined by a set of POVM elements {πi}, i πi = . In the {|0i, |1i} basis the matrix form of πi = Mi†Mi is given by: P a00 a01 (13) a10 a11!

For the outcome i, the post-measurement states are,

(I ⊗ I ⊗ Mi)|G1i = (I ⊗ I ⊗ Mi)(|0i|0i|0i + |1i|1i|1i),

(I ⊗ I ⊗ Mi)|G2i = (I ⊗ I ⊗ Mi)(|0i|0i|1i + |1i|1i|0i),

(I ⊗ I ⊗ Mi)|G3i = (I ⊗ I ⊗ Mi)(|0i|1i|0i + |1i|0i|1i), (I ⊗ I ⊗ M )|G4i = (I ⊗ I ⊗ M )(|0i|1i|1i + |1i|0i|0i), i i (14) (I ⊗ I ⊗ Mi)|G5i = (I ⊗ I ⊗ Mi)(|0i|0i|0i − |1i|1i|1i),

(I ⊗ I ⊗ Mi)|G6i = (I ⊗ I ⊗ Mi)(|0i|0i|1i − |1i|1i|0i),

(I ⊗ I ⊗ Mi)|G7i = (I ⊗ I ⊗ Mi)(|0i|1i|0i − |1i|0i|1i),

(I ⊗ I ⊗ Mi)|G8i = (I ⊗ I ⊗ Mi)(|0i|1i|1i − |1i|0i|0i).

As the measurement is orthogonality-preserving, the above states must be mutually orthogonal. Setting the inner products to zero we can easily solve for the matrix elements:

hG1|I ⊗ I ⊗ πi|G5i =0 a00 − a11 =0 a00 = a11

hG1|I ⊗ I ⊗ πi|G2i =0 a01 + a10 =0 a01 = a10 =0 (15)

hG1|I ⊗ I ⊗ πi|G6i =0 a01 − a10 =0

Thus πi is proportional to a 2 × 2 identity matrix. Since the argument applies to all possible outcomes, Charlie cannot go first and from the symmetry of the states, neither can Alice or Bob.

D. Consider the following GHZ states (unnormalized) on C2 ⊗ C2 ⊗ C2

|0i |0i |0i ± |1i |1i |1i A B C A B C (16) |0iA |1iB |1iC ± |1iA |0iB |0iC

The above states cannot be distinguished across A|BC (locally equivalent to the Bell basis) but are distinguishable across the other two bipartitions B|CA and C|AB.

E. Proof of Lemma 1: The twelve product states are given by:

|ψ1i = |1i|2i|1+2i, |ψ2i = |1i|2i|1 − 2i,

|ψ3i = |1i|3i|1+3i, |ψ4i = |1i|3i|1 − 3i,

|ψ5i = |2i|1+2i|1i, |ψ6i = |2i|1 − 2i|1i, (17) |ψ7i = |3i|1+3i|1i, |ψ8i = |3i|1 − 3i|1i,

|ψ9i = |1+2i|1i|2i, |ψ10i = |1 − 2i|1i|2i,

|ψ11i = |1+3i|1i|3i, |ψ12i = |1 − 3i|1i|3i.

I Suppose that Alice goes first. Her measurement is defined by a set of POVM elements {πl}, l πl = 3 3. In matrix form, πl × = M †M can be written as (in {|1i, |2i, |3i} basis) l l P 9

e11 e12 e13 πl = Ml†Ml = e21 e22 e23 . (18) e31 e32 e33     As the measurement is orthogonality preserving then after any given outcome, say l, the post measurement states (Ml ⊗ I ⊗ I)|ψki, k = 1,..., 12 remain pairwise orthogonal to each other. Setting the inner products of the post measurement states equal to zero, we solve for the matrix elements:

hψ5|πl ⊗ I ⊗ I|ψ7i =0 e23 =0

hψ7|πl ⊗ I ⊗ I|ψ5i =0 e32 =0

hψ1|πl ⊗ I ⊗ I|ψ5i =0 e12 =0 hψ5|π ⊗ I ⊗ I|ψ1i =0 e21 =0 l (19) hψ3|πl ⊗ I ⊗ I|ψ7i =0 e13 =0

hψ7|πl ⊗ I ⊗ I|ψ3i =0 e31 =0

hψ9|πl ⊗ I ⊗ I|ψ10i =0 e11 = e22

hψ11|πl ⊗ I ⊗ I|ψ12i =0 e11 = e33

Thus we see that πl is proportional to the 3 × 3 identity matrix. Since the argument holds for all outcomes, Alice cannot go first and from the symmetry neither can Bob or Charlie.

F. Tables I and II

Table I. Off-diagonal elements Sl. No. States Elements Sl. No. States Elements

(1) |1+2i|2i, |1i|1i a21 = a12 =0 (2) |1i|1i, |1+3i|3i a31 = a13 =0 (3) |1i|1i, |1+2i|6i a61 = a16 =0 (4) |1i|1i, |1+3i|8i a81 = a18 =0 (5) |2i|5i, |1+2i|2i a52 = a25 =0 (6) |2i|5i, |1+2i|6i a65 = a56 =0 (7) |3i|9i, |1+3i|3i a93 = a39 =0 (8) |3i|9i, |1+3i|8i a98 = a89 =0 (9) |1i|4 + 5i, |1+2i|2i a42 = a24 =0 (10) |1i|4 + 5i, |1+2i|6i a46 = a64 =0 (11) |1+2i|2i, |1+3i|3i a32 = a23 =0 (12) |1+2i|2i, |1+2i|6i a62 = a26 =0 (13) |1+2i|2i, |1+3i|8i a82 = a28 =0 (14) |1+2i|2i, |2i|7 + 8i a72 = a27 =0 (15) |1i|7 + 9i, |1+3i|3i a73 = a37 =0 (16) |1i|7 + 9i, |1+3i|8i a87 = a78 =0 (17) |1+2i|2i, |1i|7 + 9i a92 = a29 =0 (18) |1+3i|3i, |3i|2 + 5i a53 = a35 =0 (19) |1+3i|3i, |1+2i|6i a63 = a36 =0 (20) |1+3i|3i, |1+3i|8i a83 = a38 =0 (21) |1+3i|3i, |3i|4 + 6i a43 = a34 =0 (22) |1+2i|6i, |1+3i|8i a86 = a68 =0 (23) |1+2i|6i, |2i|7 + 8i a76 = a67 =0 (24) |1+2i|6i, |1i|7 + 9i a96 = a69 =0 (25) |1+3i|8i, |3i|2 + 5i a85 = a58 =0 (26) |1+3i|8i, |1i|4 + 5i a84 = a48 =0 (27) |3i|2 + 5i, |3i|4 + 6i a54 = a45 =0 (28) |2i|5i, |2i|1 + 4i a51 = a15 =0 (29) |2i|5i, |2i|3 + 9i a95 = a59 =0 (30) |3i|1 + 7i, |3i|2 + 5i a75 = a57 =0 (31) |2i|3 + 9i, |2i|7 + 8i a97 = a79 =0 (32) |3i|1 + 7i, |3i|9i a91 = a19 =0 (33) |3i|4 + 6i, |3i|9i a94 = a49 =0 (34) |1i|4 + 5i, |1i|7 + 9i a74 = a47 =0 (35) |3i|1 + 7i, |3i|4 + 6i a41 = a14 =0 (36) |1i|7 + 9i, |1i|1i a71 = a17 =0

Note: The analysis was done according to the serial numbers, and that’s how the table should be read/followed. In some cases, the inner product conditions give us the values of aij s, i 6= j right away. For example, consider entry (1): Here, for the pair of states {|1+2i |2i , |1i |1i}, the conditions h1+2|1i h2 |Πα| 1i =0 and h1|1+2i h1 |Πα| 2i =0 yield a21 = a12 =0. 10

In some other cases, e.g. entry (9) the inner-product conditions give the equations: (a) a42 + a52 = 0 and (b) a24 + a25 = 0 which can’t be solved directly. But in entry (5) we have already obtained the values for a25 and a52, both of which are zero. Therefore, a42 = a24 =0.

Table II. Diagonal elements states elements states elements

|1i|4 + 5i, |1i|4 − 5i a44 = a55 |2i|1 + 4i, |2i|1 − 4i a11 = a44 |1i|7 + 9i, |1i|7 − 9i a77 = a99 |3i|1 + 7i, |3i|1 − 7i a11 = a77 |3i|2 + 5i, |3i|2 − 5i a22 = a55 |2i|3 + 9i, |2i|3 − 9i a33 = a99 |2i|7 + 8i, |2i|7 − 8i a77 = a88 |3i|4 + 6i, |3i|4 − 6i a44 = a66

G. Strongly nonlocal OPB on C4 ⊗ C4 ⊗ C4

First, we obtain a small set of states similar to (5); there are eighteen such states:

|1i|2i|1 ± 2i |2i|1 ± 2i|1i |1 ± 2i|1i|2i S1 = |1i|3i|1 ± 3i |3i|1 ± 3i|1i |1 ± 3i|1i|3i (20) |1i|4i|1 ± 4i |4i|1 ± 4i|1i |1 ± 4i|1i|4i

As the above states are locally distinguishable in bipartitions, we add the twisted states given below:

|2i|3i|1 ± 2i |3i|1 ± 2i|2i |1 ± 2i|2i|3i |2i|4i|1 ± 2i |4i|1 ± 2i|2i |1 ± 2i|2i|4i |3i|4i|1 ± 3i |4i|1 ± 3i|3i |1 ± 3i|3i|4i S2 = (21) |4i|3i|1 ± 4i |3i|1 ± 4i|4i |1 ± 4i|4i|3i |4i|2i|1 ± 4i |2i|1 ± 4i|4i |1 ± 4i|4i|2i |3i|2i|1 ± 3i |2i|1 ± 3i|3i |1 ± 3i|3i|2i

Note that, in both blocksthe second and third columns are obtained by cyclic permutation of the first column [similar property holds for (4)]. We can now complete the above set by adding the "simple product" states–in this case there are ten of them:

|2i|3i|4i, |3i|4i|2i, |4i|2i|3i, |2i|4i|3i, |4i|3i|2i, S3 = (22) ( |3i|2i|4i, |1i|1i|1i, |2i|2i|2i, |3i|3i|3i, |4i|4i|4i. )

4 4 4 The union of the above three sets S = S1 ∪ S2 ∪ S3 form the desired OPB on C ⊗ C ⊗ C . In what follows, we first show that S is locally irreducible and then we will prove that S is strongly nonlocal.

First we show that S is locally irreducible. To show this, we first prove that that S1 is locally irreducible. We write the states (20 ) as: 11

|ψ1i = |1i|2i|1+2i, |ψ2i = |1i|2i|1 − 2i,

|ψ3i = |1i|3i|1+3i, |ψ4i = |1i|3i|1 − 3i,

|ψ5i = |1i|4i|1+4i, |ψ6i = |1i|4i|1 − 4i,

|ψ7i = |2i|1+2i|1i, |ψ8i = |2i|1 − 2i|1i,

|ψ9i = |3i|1+3i|1i, |ψ10i = |3i|1 − 3i|1i, (23)

|ψ11i = |4i|1+4i|1i, |ψ12i = |4i|1 − 4i|1i,

|ψ13i = |1+2i|1i|2i, |ψ14i = |1 − 2i|1i|2i,

|ψ15i = |1+3i|1i|3i, |ψ16i = |1 − 3i|1i|3i.

|ψ17i = |1+4i|1i|4i, |ψ18i = |1 − 4i|1i|4i.

I Suppose Alice goes first. Alice’s measurement is defined by a set of POVM elements {πl}, l πl = 4 4. Each element πl = × M †M is given by a 4 × 4 matrix written in {|1i, |2i, |3i, |4i} basis: l l P

e11 e12 e13 e14 e21 e22 e23 e24 πl = Ml†Ml = . (24) e31 e32 e33 e34   e41 e42 e43 e44     We assume that this measurement is orthogonality-preserving. Therefore, the states {(Ml ⊗ I ⊗ I) |ψki}, k =1,..., 18 must be orthogonal to each other. Setting the inner products of the post measurement states equal to zero, we solve for the matrix elements eij as shown in the table below:

hψ1|πl ⊗ I ⊗ I|ψ7i =0 e12 =0

hψ7|πl ⊗ I ⊗ I|ψ1i =0 e21 =0

hψ3|πl ⊗ I ⊗ I|ψ9i =0 e13 =0

hψ9|πl ⊗ I ⊗ I|ψ3i =0 e31 =0

hψ7|πl ⊗ I ⊗ I|ψ9i =0 e23 =0

hψ9|πl ⊗ I ⊗ I|ψ7i =0 e32 =0

hψ5|πl ⊗ I ⊗ I|ψ11i =0 e14 =0

hψ11|πl ⊗ I ⊗ I|ψ5i =0 e41 =0 (25)

hψ7|πl ⊗ I ⊗ I|ψ11i =0 e24 =0

hψ11|πl ⊗ I ⊗ I|ψ7i =0 e42 =0

hψ9|πl ⊗ I ⊗ Iψ11i =0 e34 =0

hψ11|πl ⊗ I ⊗ I|ψ9i =0 e43 =0

hψ13|πl ⊗ I ⊗ I|ψ14i =0 e11 = e22

hψ15|πl ⊗ I ⊗ I|ψ16i =0 e11 = e33

hψ17|πl ⊗ I ⊗ I|ψ18i =0 e11 = e44

We see that πl must be proportional to the identity. As the argument applies to all outcomes, all the POVM elements must be proportional to the identity, and therefore Alice cannot go first, and from the symmetry of the states, neither can Bob, nor Charlie. Thus S1 is locally irreducible. Since S1 is a subset of S the argument holds for S as well, and therefore, S is locally irreducible. We now prove that S is strongly nonlocal following the steps in the proof involving the states (4). Consider the bipartition A|BC. We have already shown that Alice cannot go first when all of them are separated (as S is irreducible). But for Alice, 12 the situation doesn’t change because she is still separated from BC. Therefore, even in the configuration A|BC Alice cannot begin a local protocol by performing some local NOPM on her subsystem.

We now come to the subsystem BC. First we rewrite the states in Si, i = 1, 2, 3 to reflect this fact. For clarity, we denote computational basis states of HBC in the following way: |1ii → |ii ; |2ii → |i + 4i ; |3ii → |i + 8i ; |4ii → |i + 12i and rewrite the states:

|1i|5 ± 6i, |2i|1 ± 5i, |1 ± 2i|2i, S1 (A|BC)= |1i|9 ± 11i, |3i|1 ± 9i, |1 ± 3i|3i, (26) |1i|13 ± 16i, |4i|1 ± 13i, |1 ± 4i|4i.

|2i|9 ± 10i, |3i|2 ± 6i, |1 ± 2i|7i, |2i|13 ± 14i, |4i|2 ± 6i, |1 ± 2i|8i, |3i|13 ± 15i, |4i|3 ± 11i, |1 ± 3i|12i, S2 (A|BC)= (27) |4i|9 ± 12i, |3i|4 ± 16i, |1 ± 4i|15i, |4i|5 ± 8i, |2i|4 ± 16i, |1 ± 4i|14i, |3i|5 ± 7i, |2i|3 ± 11i, |1 ± 3i|10i.

|2i|12i, |3i|14i, |4i|7i, |2i|15i, |4i|10i, S3 (A|BC)= (28) ( |3i|8i, |1i|1i, |2i|6i, |3i|11i, |4i|16i. )

For any general measurement on BC described by a POVM {πα}, each element πα can be represented by a 16 × 16 matrix with the elements denoted by ai,j , i, j = 1,..., 16. We proceed exactly the same way as in the previous proofs. The table below shows that all the diagonal elements are equal.

states elements states elements

|2i|1 + 5i, |2i|1 − 5i a1,1 = a5,5 |3i|1 + 9i, |3i|1 − 9i a1,1 = a9,9

|4i|1 + 13i, |4i|1 − 13i a1,1 = a13,13 |3i|2 + 6i, |3i|2 − 6i a2,2 = a6,6

|4i|3 + 11i, |4i|3 − 11i a3,3 = a11,11 |3i|4 + 16i, |3i|4 − 16i a4,4 = a16,16

|1i|5 + 6i, |1i|5 − 6i a5,5 = a6,6 |1i|9 + 11i, |1i|9 − 11i a9,9 = a11,11 (29)

|1i|13 + 16i, |1i|13 − 16i a13,13 = a16,16 |4i|5 + 8i, |4i|5 − 8i a5,5 = a8,8

|3i|5 + 7i, |3i|5 − 7i a5,5 = a7,7 |2i|9 + 10i, |2i|9 − 10i a9,9 = a10,10

|4i|9 + 12i, |4i|9 − 12i a9,9 = a12,12 |2i|13 + 14i, |2i|13 − 14i a13,13 = a14,14

|3i|13 + 15i, |3i|13 − 15i a13,13 = a15,15 |2i|3 + 11i, |2i|3 − 11i a3,3 = a11,11 Tables III and IV show that all the off-diagonal elements are zero. The tables should be read according to the serial numbers (that’s how the matrix elements were evaluated using the orthogonality-preserving conditions).

Therefore we have shown that πα must be proportional to the identity. As this argument holds for all possible outcomes, all POVM elements must also be proportional to the identity and hence, trivial. Therefore, Bob and Charlie cannot go first. This shows that the set is locally irreducible in the bipartition A|BC and from the symmetry of the states it must also be locally irreducible in the other two bipartitions. Hence, S is strongly nonlocal. 13

Table III. Off-diagonal terms sl. no. states elements sl. no. states elements

(1) |1+2i|2i, |1+3i|3i a2,3 = a3,2 =0 (2) |1+2i|2i, |1+4i|4i a2,4 = a4,2 =0 (3) |1+2i|2i, |1+2i|7i a2,7 = a7,2 =0 (4) |1+2i|2i, |1+2i|8i a2,8 = a8,2 =0 (5) |1+2i|2i, |1+3i|12i a2,12 = a12,2 =0 (6) |1+2i|2i, |1+4i|15i a2,15 = a15,2 =0 (7) |1+2i|2i, |1+4i|14i a2,14 = a14,2 =0 (8) |1+2i|2i, |1+3i|10i a2,10 = a10,2 =0 (9) |1+2i|2i, |1i|1i a2,1 = a1,2 =0 (10) |1+3i|3i, |1+4i|4i a3,4 = a4,3 =0

(11) |1+3i|3i, |1+2i|7i a3,7 = a7,3 =0 (12) |1+3i|3i, |1+2i|8i a3,8 = a8,3 =0 (13) |1+3i|3i, |1+3i|12i a3,12 = a12,3 =0 (14) |1+3i|3i, |1+4i|15i a3,15 = a15,3 =0 (15) |1+3i|3i, |1+4i|14i a3,14 = a14,3 =0 (16) |1+3i|3i, |1+3i|10i a3,10 = a10,3 =0 (17) |1+3i|3i, |1i|1i a3,1 = a1,3 =0 (18) |1+4i|4i, |1+2i|7i a4,7 = a7,4 =0 (19) |1+4i|4i, |1+2i|8i a4,8 = a8,4 =0 (20) |1+4i|4i, |1+3i|12i a4,12 = a12,4 =0 (21) |1+4i|4i, |1+4i|15i a4,15 = a15,4 =0 (22) |1+4i|4i, |1+4i|14i a4,14 = a14,4 =0 (23) |1+4i|4i, |1+3i|10i a4,10 = a10,4 =0 (24) |1+4i|4i, |1i|1i a4,1 = a1,4 =0

(25) |1+2i|7i, |1+2i|8i a7,8 = a8,7 =0 (26) |1+2i|7i, |1+3i|12i a7,12 = a12,7 =0 (27) |1+2i|7i, |1+4i|15i a7,15 = a15,7 =0 (28) |1+2i|7i, |1+4i|14i a7,14 = a14,7 =0 (29) |1+2i|7i, |1+3i|10i a7,10 = a10,7 =0 (30) |1+2i|7i, |1i|1i a7,1 = a1,7 =0 (31) |1+2i|8i, |1+3i|12i a8,12 = a12,8 =0 (32) |1+2i|8i, |1+4i|15i a8,15 = a15,8 =0 (33) |1+2i|8i, |1+4i|14i a8,14 = a14,8 =0 (34) |1+2i|8i, |1+3i|10i a8,10 = a10,8 =0 (35) |1+2i|8i, |1i|1i a8,1 = a1,8 =0 (36) |1+3i|12i, |1+4i|15i a12,15 = a15,12 =0 (37) |1+3i|12i, |1+4i|14i a12,14 = a14,12 =0 (38) |1+3i|12i, |1+3i|10i a12,10 = a10,12 =0

(39) |1+3i|12i, |1i|1i a12,1 = a1,12 =0 (40) |1+4i|15i, |1+4i|14i a15,14 = a14,15 =0 (41) |1+4i|15i, |1+3i|10i a15,10 = a10,15 =0 (42) |1+4i|15i, |1i|1i a15,1 = a1,15 =0 (43) |1+4i|14i, |1+3i|10i a14,10 = a10,14 =0 (44) |1+4i|14i, |1i|1i a14,1 = a1,14 =0 (45) |1+3i|10i, |1i|1i a10,1 = a1,10 =0 (46) |1+2i|2i, |2i|3 + 11i a2,11 = a11,2 =0 (47) |1+2i|2i, |2i|4 + 16i a2,16 = a16,2 =0 (48) |1+2i|2i, |2i|9 + 10i a2,9 = a9,2 =0 (49) |1+2i|2i, |2i|6i a2,6 = a6,2 =0 (50) |1+2i|2i, |2i|13 + 14i a2,13 = a13,2 =0 (51) |1+2i|2i, |1i|5 + 6i a2,5 = a5,2 =0 (52) |1+2i|7i, |2i|3 + 11i a7,11 = a11,7 =0

(53) |1+2i|7i, |2i|4 + 16i a7,16 = a16,7 =0 (54) |1+2i|7i, |2i|9 + 10i a7,9 = a9,7 =0 (55) |1+2i|7i, |2i|6i a7,6 = a6,7 =0 (56) |1+2i|7i, |2i|13 + 14i a7,13 = a13,7 =0 (57) |1+2i|7i, |1i|5 + 6i a7,5 = a5,7 =0 (58) |1+2i|8i, |2i|3 + 11i a8,11 = a11,8 =0 (59) |1+2i|8i, |2i|4 + 16i a8,16 = a16,8 =0 (60) |1+2i|8i, |2i|9 + 10i a8,9 = a9,8 =0 14

Table IV. Off-diagonal terms sl. no. states elements sl. no. states elements

(61) |1+2i|8i, |2i|6i a8,6 = a6,8 =0 (62) |1+2i|8i, |2i|13 + 14i a8,13 = a13,8 =0 (63) |1+2i|8i, |1i|5 + 6i a8,5 = a5,8 =0 (64) |1+3i|3i, |3i|11i a3,11 = a11,3 =0 (65) |1+3i|3i, |3i|4 + 16i a3,16 = a16,3 =0 (66) |1+3i|3i, |3i|1 + 9i a3,9 = a9,3 =0 (67) |1+3i|3i, |3i|2 + 6i a3,6 = a6,3 =0 (68) |1+3i|3i, |3i|13 + 15i a3,13 = a13,3 =0 (69) |1+3i|3i, |3i|5 + 7i a3,5 = a5,3 =0 (70) |1+3i|12i, |3i|11i a12,11 = a11,12 =0

(71) |1+3i|12i, |3i|4 + 16i a12,16 = a16,12 =0 (72) |1+3i|12i, |3i|1 + 9i a12,9 = a9,12 =0 (73) |1+3i|12i, |3i|2 + 6i a12,6 = a6,12 =0 (74) |1+3i|12i, |3i|13 + 15i a12,13 = a13,12 =0 (75) |1+3i|12i, |3i|5 + 7i a12,5 = a5,12 =0 (76) |1+3i|10i, |3i|11i a10,11 = a11,10 =0 (77) |1+3i|10i, |3i|4 + 16i a10,16 = a16,10 =0 (78) |1+3i|10i, |3i|1 + 9i a10,9 = a9,10 =0 (79) |1+3i|10i, |3i|2 + 6i a10,6 = a6,10 =0 (80) |1+3i|10i, |3i|13 + 15i a10,13 = a13,10 =0 (81) |1+3i|10i, |3i|5 + 7i a10,5 = a5,10 =0 (82) |1+4i|4i, |4i|3 + 11i a4,11 = a11,4 =0 (83) |1+4i|4i, |4i|16i a4,16 = a16,4 =0 (84) |1+4i|4i, |4i|9 + 12i a4,9 = a9,4 =0

(85) |1+4i|4i, |4i|5 + 8i a4,5 = a5,4 =0 (86) |1+4i|4i, |4i|2 + 6i a4,6 = a6,4 =0 (87) |1+4i|4i, |4i|1 + 13i a4,13 = a13,4 =0 (88) |1+4i|15i, |4i|3 + 11i a15,11 = a11,15 =0 (89) |1+4i|15i, |4i|16i a15,16 = a16,15 =0 (90) |1+4i|15i, |4i|9 + 12i a15,9 = a9,15 =0 (91) |1+4i|15i, |4i|5 + 8i a15,5 = a5,15 =0 (92) |1+4i|15i, |4i|2 + 6i a15,6 = a6,15 =0 (93) |1+4i|15i, |4i|1 + 13i a15,13 = a13,15 =0 (94) |1+4i|14i, |4i|3 + 11i a14,11 = a11,14 =0 (95) |1+4i|14i, |4i|16i a14,16 = a16,14 =0 (96) |1+4i|14i, |4i|9 + 12i a14,9 = a9,14 =0 (97) |1+4i|14i, |4i|5 + 8i a14,5 = a5,14 =0 (98) |1+4i|14i, |4i|2 + 6i a14,6 = a6,14 =0

(99) |1+4i|14i, |4i|1 + 13i a14,13 = a13,14 =0 (100) |3i|5 + 7i, |3i|11i a5,11 = a11,5 =0 (101) |3i|5 + 7i, |3i|4 + 16i a5,16 = a16,5 =0 (102) |3i|5 + 7i, |3i|2 + 6i a5,6 = a6,5 =0 (103) |3i|5 + 7i, |3i|13 + 15i a5,13 = a13,5 =0 (104) |4i|5 + 8i, |4i|9 + 12i a5,9 = a9,5 =0 (105) |3i|2 + 6i, |3i|4 + 16i a6,16 = a16,6 =0 (106) |3i|2 + 6i, |3i|11i a6,11 = a11,6 =0 (107) |3i|2 + 6i, |3i|13 + 15i a6,13 = a13,6 =0 (108) |4i|2 + 6i, |4i|9 + 12i a6,9 = a9,6 =0 (109) |4i|9 + 12i, |4i|3 + 11i a9,11 = a11,9 =0 (110) |4i|9 + 12i, |4i|16i a9,16 = a16,9 =0 (111) |2i|9 + 10i, |2i|13 + 14i a9,13 = a13,9 =0 (112) |4i|3 + 11i, |4i|16i a11,16 = a16,11 =0

(113) |2i|3 + 11i, |2i|13 + 14i a11,13 = a13,11 =0 (114) |2i|13 + 14i, |2i|4 + 16i a13,16 = a16,13 =0 (115) |2i|1 + 5i, |2i|6i a1,6 = a6,1 =0 (116) |2i|1 + 5i, |2i|13 + 14i a1,13 = a13,1 =0 (117) |2i|1 + 5i, |2i|3 + 11i a1,11 = a11,1 =0 (118) |2i|1 + 5i, |2i|4 + 16i a1,16 = a16,1 =0 (119) |2i|1 + 5i, |2i|9 + 10i a1,9 = a9,1 =0 (120) |3i|1 + 9i, |3i|5 + 7i a1,5 = a5,1 =0