Compatible quantum correlations: on extension problems for Werner and isotropic states

Transcription

1 Compatible quantum correlations: on extension problems for Werner an isotropic states Peter D. Johnson an Lorenza Viola Department of Physics an Astronomy, Dartmouth College, 6127 Wiler Laboratory, Hanover, NH 0755, USA (Date: September 12, 201) The limite sharability/joinability of entanglement was first quantifie in the seminal work by Coffman, Kunu, an Wootters, in terms of an exact (CKW) inequality obeye by the entanglement across the A-B, A- C an A-(BC) bipartitions, as measure by concurrence [4]. In a similar venue, several subsequent investigations attempte to etermine how ifferent entanglement measures can be use to iagnose failures of joinability, see e.g. [7, 15, 16]. More recently, significant progress has been mae in characterizing quantum correlations more general than entanglement [17, 18], in particular as capture by quantum iscor [19]. While it is now establishe that quantum iscor oes not obey a monogamy inequality [20], ifferent kin of limitations exist on the extent to which it can be freely share an/or communicate [21, 22]. Despite these important avances, a complete picture is far from being reache. What kin of limitations o strictly mark the quantum-classical correlation bounary? What ifferent quantum features are responsible for enforcing ifferent aspects of such limitaarxiv: v [quant-ph] 11 Sep 201 We investigate some basic scenarios in which a given set of bipartite quantum states may consistently arise as the set of reuce states of a global N-partite quantum state. Intuitively, we say that the multipartite state joins the unerlying correlations. Determining whether, for a given set of states an a given joining structure, a compatible N-partite quantum state exists is known as the quantum marginal problem. We restrict to bipartite reuce states that belong to the paraigmatic classes of Werner an isotropic states in imensions, an focus on two specific versions of the quantum marginal problem which we fin to be tractable. The first is Alice-Bob, Alice-Charlie joining, with both pairs being in a Werner or isotropic state. The secon is m-n sharability of a Werner state across N subsystems, which may be seen as a variant of the N-representability problem to the case where subsystems are partitione into two groupings of m an n parties, respectively. By exploiting the symmetry properties that each class of states enjoys, we etermine necessary an sufficient conitions for three-party joinability an 1-n sharability for arbitrary. Our results explicitly show that although entanglement is require for sharing limitations to emerge, correlations beyon entanglement generally suffice to restrict joinability, an not all unentangle states necessarily obey the same limitations. The relationship between joinability an quantum cloning as well as implications for the joinability of arbitrary bipartite states are iscusse. PACS numbers: 0.67.Mn, 0.65.U, 0.65.Ta I. INTRODUCTION Unerstaning the nature of quantum correlations in multiparty systems an the istinguishing features they exhibit relative to classical correlations is a central goal across quantum information processing (QIP) science [1], with implications ranging from conense-matter an statistical physics to quantum chemistry, an the quantum-to-classical transition. From a founational perspective, exploring what ifferent kins of correlations are, in principle, allowe by probabilistic theories more general than quantum mechanics further helps to ientify uner which set of physical constraints the stanar quantum framework may be uniquely recovere [2, ]. In this context, entanglement provies a istinctively quantum type of correlation, that has no analogue in classical statistical mechanics. A striking feature of entanglement is that it cannot be freely istribute among ifferent parties: if a bipartite system, say, A(lice) an B(ob), is in a maximally entangle pure state, then no other system, C(harlie), may be correlate with it. In other wors, the entanglement between A an B is monogamous an cannot be share [4 8]. This simple tripartite setting motivates two simple questions about bipartite quantum states: given a bipartite state, we ask whether it can arise as the reuce state of A-B an of A-C simultaneously; or, more generally, given two bipartite states, we ask if one can arise as the reuce state of A-B while the other arises as the reuce state of A-C. It shoul be emphasize that both of these are questions about the existence of tripartite states with given reuction properties. While formal (an more general) efinitions will be provie later in the paper, these examples serve to introuce the notions of sharing (1-2 sharing) an joining (1-2 joining), respectively. In its most general formulation, the joinability problem is also known as the quantum marginal problem (or local consistency problem), which has been heavily investigate both from a mathematical-physics [9 11] an a quantum-chemistry perspective [12, 1] an is known to be QMA-har [14]. Our choice of terminology, however, facilitates a uniform language for escribing the joinability/sharability scenarios. For instance, we say that the joinable correlations of A-B an A-C are joine by a joining state on A-B-C.

2 2 tions, an how oes this relate to the egree of resourcefulness that these correlations can have for QIP? While the above are some of the broa questions motivating this work, our specific focus here is to make progress on joinability an sharability properties in lowimensional multipartite settings. In this context, a recent paper [2] has obtaine a necessary conition for three-party joining in finite imension in terms of the subsystem entropies, an aitionally establishe a sufficient conition in terms of the trace-norm istances between the states in question an known joinable states. For the specific case of qubit Werner states [24], Werner himself establishe necessary an sufficient conitions for the 1-2 joining scenario [25]. With regars to sharability, necessary an sufficient conitions have been foun for 1-2 sharing of generic bipartite qubit states [26, 27], as well as for specific classes of quit states [28]. To the best of our knowlege, no conitions that are both necessary an sufficient for the joinability of generic states are available as yet. In this paper, we obtain necessary an sufficient conitions for both the three-party joinability an the 1-n sharability problems, in the case that the reuce bipartite states are either Werner or isotropic states on -imensional subsystems (quits). Though our results are restricte in scope of applicability, they provie key insights as to the sources of joinability limitations. Most importantly, we fin that stanar measures of quantum correlations, such as concurrence an quantum iscor, o not suffice to etermine the limitations in joining quantum correlations. Specifically, we fin that the joine states nee not be entangle or even iscorant in orer not to be joinable. Further to that, although separable states may have joinability limitations, they are, nonetheless, freely (arbitrarily) sharable. By introucing a one-parameter class of probability istributions, we provie a natural classical analogue to quit Werner an isotropic quantum states. This allows us to illustrate how classical joinability restrictions carry over to the quantum case an, more interestingly, to emonstrate that the quantum case emans limitations which are not present classically. Ultimately, this feature may be trace back to complementarity of observables, which clearly plays no role in the classical case. It is suggestive to note that the uncertainty principle was also shown to be instrumental in constraining the sharability of quantum iscor [21]. It is our hope that further pursuits of more general necessary an sufficient conitions may be aie by the methos an finings herein. The content is organize as follows. In Sec. II we present the relevant mathematical framework for efining the joinability an sharability notions an the extension problems of interest, along with some preliminary results contrasting the classical an quantum cases. Sec. III contains the core results of our analysis. In particular, after reviewing the efining properties of Werner an isotropic states on quits, in Sec. III A we motivate the appropriate choice of probability istributions that serve as a classical analogue, an etermine the resulting classical joinability limitations in Sec. III B. Necessary an sufficient conitions for three-party joinability of quantum Werner an isotropic states are establishe in Sec. III C, an contraste to the classical scenario. Sec. III D shows how the results on isotropic state joinability are in fact relate to known results on quantum cloning, whereas in Sec. III E we establish simple analytic expressions for the 1-n sharability of both Werner an isotropic states, along with iscussing constructive proceures to etermine m-n sharability properties for m > 1. In Sec. IV, we present aitional remarks on joinability an sharability scenarios beyon those of Sec. III. In particular, we outline generalizations of our analysis to N-party joinability, an show how bouns on the sharability of arbitrary bipartite states follow from the Werner an isotropic results. Concluing remarks an open questions are presente in Sec. V. For ease an clarity of presentation, the technical proofs of the results in Sec. III are presente in two separate Appenixes (A on joinability an B on sharability, respectively), together with the relevant group-representation tools. II. JOINING AND SHARING CLASSICAL VS. QUANTUM STATES Although our main focus will be to quantitatively characterize simple low-imensional settings, we introuce the relevant concepts with a higher egree of generality, in orer to better highlight the unerlying mathematical structure an to ease connections with existing relate notions in the literature. We are intereste in the correlations among the subsystems of a N-partite composite system S. In the quantum case, we thus require a Hilbert space with a tensor prouct structure: H (N) N i=1 H (1) i, im(h (1) i ) i, where H (1) i represents the iniviual single-particle state spaces an, for our purposes, each i is finite. In the classical scenario, to each subsystem we associate a sample space Ω i consisting of i possible outcomes, with the joint sample space being given by the Cartesian prouct: Ω (N) Ω 1... Ω N. Probability istributions on Ω (N) are the classical counterpart of quantum ensity operators on H (N). A. Joinability The input to a joinability problem is a set of subsystem states which, in full generality, may be specifie relative to a neighboorhoo structure on H (N) (or Ω (N) ) [29, 0]. That is, let neighborhoos {N j } be given as subsets of the set of inexes labeling iniviual subsystems, N k Z N. We can then give the following:

3 Definition II.1. [Quantum Joinability] Given a neighborhoo structure {N 1, N 2,..., N l } on H (N), a list of ensity operators (ρ 1,..., ρ l ) (D(H N1 ),..., D(H Nl )) is joinable if there exists an N-partite ensity operator w D(H (N) ), calle a joining state, that reuces accoring to the neighborhoo structure, that is, Tr ˆNk (w) = ρ k, k = 1,..., l, (1) where ˆN k Z N \ N k is the tensor complement of N k. The analagous efinition for classical joinability is obtaine by substituting corresponing terms, in particular, by replacing the partial trace over ˆNk with the corresponing marginal probability istribution. As remarke, the question of joinability has been extensively investigate in the context of the classical [1] an quantum [9, 2, 2, ] marginal problem. A joining state is equivalenty referre to as an extension or an element of the pre-image of the list uner the reuction map, while the members of a list of joinable states are also sai to be compatible or consistent. Clearly, a necessary conition for a list of states to be joinable is that they agree on any overlapping reuce states. That is, given any two states from the list whose neighborhoos are intersecting, the reuce states of the subsystems in the intersection must coincie. From this point of view, any failure of joinability ue to a isagreement of overlappping reuce states is a trivial case of non-compatible N-party correlations. We are intereste in cases where joinability fails espite the agreement on overlapping marginals. This consistency requirement will be satisfie by construction for the Werner an isotropic quantum states we shall consier in Sec. III. One important feature of joinability, which has recently been investigate in [4], is the convex structure that both joinable states lists an joining states enjoy. The set of lists of ensity operators satisfying a given joinability scenario is convex uner component-wise combination; this is because the same convex combination of their joining states is a vali joining state for the convex combine list of states. Similarly, the set of joining states for a given list of joine states is convex by the linearity of the partial trace. As mentione, one of our goals is to she light on limitations of quantum vs. classical joinability an the extent to which entanglement may play a role in that respect. That quantum states are subject to stricter joinability limitations than classical probability istributions o, can be immeiately appreciate by consiering two ensity operators ρ AB = Ψ B Ψ B = ρ AC, where Ψ B is any maximally entangle Bell pair on two qubits: no three-qubit joining state w ABC exists, espite the reuce state on A being manifestly consistent. In contrast, as shown in [2, 1], as long as two classical istributions have equal marginal istributions over A, p(a, B) an p(a, C) can always be joine. This is evience by the construction of the joining state: w(a, B, C) = p(a, B) p(a, C)/p(A). As pointe out in [2], although the above choice is not unique, it is the joining state with maximal entropy an represents an even mixture of all vali joining istributions. Although any two consistently-overlappe classical probability istributions may be joine, limitations on joining classical probability istributions o typically arise in more general joining scenarios. This follows from the fact that any classical probability assignments must be consistent with some convex combination of pure states. Consier, for example, a pairwise neighboorhoo structure, with an associate list of states p(a, B), p(b, C), an p(a, C), which have consistent single-subsystem marginals. Clearly, if each subsystem correspons to a bit, no convex combination of pure states gives rise to a probability istribution w(a, B, C) in which each pair is completely anticorrelate; in other wors, bits of three can t all isagree. In Sec. III C, we explicitly compare this particular classical joining scenario to analogous quantum scenarios. While all the classical joining limitations may be expresse by linear inequalities, the quantum joining limitations are significantly more complicate. The limitations arise from emaning that the joining operator be a vali ensity operator, namely, trace-one an nonnegative (which clearly implies Hermiticity). This fact is emonstrate by the following proposition, which may be reaily generalize to any joining scenario: Proposition II.2. For any two trace-one Hermitian operators Q AB an Q AC which obey the consistency conition Tr B (Q AB ) = Tr C (Q AC ), there exists a trace-one Hermitian joining operator Q ABC. Proof. Consier an orthogonal Hermitian prouct basis which inclues the ientity for each subsystem, that is, {A i B j C k }, where A 0 = B 0 = C 0 = I. Then we can construct the space of all vali joining operators Q ABC as follows. Let ABC be the imension of the composite system. The component along A 0 B 0 C 0 is fixe as 1/ ABC, satisfying the trace-one requirement. The components along the two-boy operators of the form A i B j I are fixe by the require reuction to Q AB, an similarly the components along the two-boy operators of the form A i I C k are etermine by Q AC. The components along the one-boy operators of the form A i I I, I B i I, an I I C i are etermine from the reuctions of Q AB an Q AC. This leaves the coefficients of all remaining basis operators unconstraine, since their corresponing basis operators are zero after a partial trace over systems B or C. Thus, requiring the joining operator to be Hermitian an normalize is not a limiting constraint with respect to joinability: any limitations are ue to the nonnegativity constraint. Unerstaning how non-negativity manifests is extremely ifficult in general an far beyon our scope here. We can nevertheless give an example in which the role of non-negativity is clear. Part of the job

4 4 of non-negativity is to enforce constraints that are also obeye by classical probability istributions. For example, in the case of a two-qubit state ρ, if X I ρ = 1 an I X ρ = 1, then X X ρ must equal 1. More generally, consier a set of mutually commuting observables {M i } k i=1 an any basis { m } in which all M i are iagonal. Any vali state must lea to a list of expectation values (Tr (ρm 1 ),..., Tr (ρm k )), whose values are element-wise convex combinations of the vertexes {( m M 1 m,..., m M k m ) m}. The interpretation of this constraint is that since commuting observables have simultaneously efinable values, just as classical observables o, probability istributions on them must obey the rules of classical probability istributions. We call on this fact when we compare the quantum joining limitations to the classical analogue ones in Sec. III C. Non-negativity constraints that o not arise from classical limitations on compatible observables may be labele as inherently quantum constraints, the most familiar being provie by uncertainty relations for conjugate observables [5, 6]. Although complementarity constraints are most evient for observables acting on the same system, complementarity can also give rise to a trae-off in the information about a subsystem observable vs. a joint observable. This fact is essentially what allows Bell s inequality to be violate. For our purposes, the complementarity that comes into play is that between overlapping joint observables (e.g., between S 1 S 2 an S 1 S for three qubits). We are thus generally intereste in unerstaning the interplay between purely classical an quantum joining limitations, an in the correlation trae-offs that may possibly emerge. Historically, as alreay mentione, a pioneering exploration of the extent to which quantum correlations can be share among three parties was carrie out in [4], yieling a characterization of the monogamy of entanglement in terms of the well-known CKW inequality: C 2 AB + C 2 AC (C 2 ) min A(BC), where C enotes the concurrence an the right han-sie is minimize over all pure-state ecompositions. Thus, with the entanglement across the bipartition A an (BC) hel fixe, an increase in the upper boun of the A-B entanglement can only come at the cost of a ecrease in the upper boun of the A-C entanglement. One may woner whether the CKW inequality may help in iagnosing joinability of reuce states. If a joining state w ABC is not a priori etermine (in fact, the existence of such a state is the entire question of joinability), the CKW inequality may be use to obtain a necessary conition for joinability, namely, if ρ AB an ρ AC are joinable, then C 2 AB + C 2 AC 1. (2) However, there exist pairs of bipartite states both unentangle (as the following Proposition shows) an nontrivially entangle (as we shall etermine in Sec. III.B, see in particular Fig. 2(a)) that obey the weak CKW inequality in Eq. (2), yet are not joinable. The key point is that while the limitations that the CKW captures are to be ascribe to entanglement, entanglement is not require to prevent two states from being joinable. In fact, weaker forms of quantum correlations, as quantifie by quantum iscor [19], are likewise not require for joinability limitations. Consier, specifically, so-calle classical-quantum bipartite states, of the form ρ = p i i i A σb, i p i = 1, i i where { i A } is some local orthogonal basis on A an σ i B is, for each i, an arbitrary state on B. Such states are known to have zero iscor [7]. Yet, the following hols: Proposition II.. Classical-quantum correlate states nee not be joinable. Proof. Consier the two quantum states ρ AB = ( X X X X + X X X X )/2, ρ AC = ( Z Z Z Z + Z Z Z Z )/2, on the pairs A-B an A-C, respectively. Both have a completely mixe reuce state over A an thus it is meaningful to consier their joinability. Let w ABC be a joining state. Then the outcome of Bob s X measurement woul correctly lea him to preict Alice to be in the state X or X, while at the same time the outcome of Charlie s Z measurement woul correctly lea him to preict Alice to be in the state Z or Z. Since this violates the uncertainty principle, w ABC cannot be a vali joining state. The existence of separable not joinable states has been inepenently reporte in [2]. While formally our example is subsume uner the more general one presente in Thm. 4.2 therein (strictly satisfying the necessary conition for joinability given by their Eq. (2.2)), it has the avantage of offering both a transparent physical interpretation of the unerlying correlation properties, an an intuitive proof of the joinability failure. B. Sharability As mentione, the secon joinability structure we analyze is motivate by the concept of sharability. In our context, we can think of sharability as a restricte joining scenario in which a bipartite state is joine with copies of itself. If H (2) H (1) 1 H (1) 2, consier a N-partite space that consists of m right copies of H (1) 1 an n left copies of H (1) 2, with each neighborhoo consisting of one right an one left subsystem, respectively (hence a total of mn neighborhoos). We then have the following: Definition II.4. [Quantum Sharability] A bipartite ensity operator ρ D(H L H R ) is m-n sharable if there

5 5 exists an N-partite ensity operator w D(H m L H n R ), calle a sharing state, that reuces left-right-pairwise to ρ, that is, Tr ˆLi ˆRj (w) = ρ, i = 1,..., m, j = 1,..., n, () where the partial trace is taken over the tensor complement of neighborhoo ij. Each m-n sharability scenario may be viewe as a specific joining structure with the aitional constraint that each of the joining states be equal to one another, the list being (ρ, ρ,..., ρ). In what follows, we shall take arbitrarily sharable to mean - sharable, whereas finitely sharable means that ρ is not m-n sharable for some m, n. Also, each property m-n sharable (sometimes also referre to as a m-n extenible ) is taken to efine a sharability criterion, which a state may or may not satisfy. It is worth noting the relationship between sharability an N-representability. The N-representability problem asks if, for a given (symmetric) p-partite ensity operator ρ on (H (1) 1 ) p, there exists an N-partite pre-image state for which ρ is the p-particle reuce state. N- representability has been extensively stuie for inistinguishable bosonic an fermionic subsystems [12, 1, 8] an is a very important problem in quantum chemistry [9]. We can view N-representability as a variant on the sharability problem, whereby the istinction between the left an right subsystems is lifte, an m+n = N. Given the p-partite state ρ as the share state, we ask if there exists a sharing N-partite state which shares ρ among all possible p-partite subsystems. In the setting of inistinguishable particles, the associate symmetry further constrains the space of the vali N-partite sharing states. Just as with 1-2 joinability, any classical probability istribution is arbitrarily sharable []. Likewise, similar to the joinability case, convexity properties play an important role towars characterizing sharability. If im(h (1) 1 ) = 1 L an im(h (1) 2 ) = 2 R, then it follows from the convexity of the set of joinable states lists that m-n sharable states form a convex set, for fixe subsystem imensions L an R. This implies that if ρ satisfies a particular sharability criterion, then any mixture of ρ with the completely mixe state also satisfies that criterion, since the completely mixe state is arbitrarily ( - ) sharable. Besies mixing with the ientity, the egree of sharability may be unchange uner more general transformations on the input state. Consier, specifically, completely-positive trace-preserving bipartite maps M(ρ) that can be written as a mixture of local unitary operations, that is, λ i U1 i V2 i ρu1 i V i 2, λ i = 1, (4) M(ρ) = i where U i 1 an V i 2 are arbitrary unitary transformations H L an H R, respectively. These (unital) maps form a proper subset of general Local Operations an Classical Communication (LOCC) [1]. We establish the following: i Theorem II.5. If ρ is m-n sharable, then M(ρ) is m- n sharable for any map M that is a convex mixture of unitaries. Proof. Let M(ρ) be expresse as in Eq. (4). By virtue of the convexity of the set of m-n sharable states (for fixe subsystem imensions), it suffices to show that each term, UV ρu V, in M(ρ) is m-n sharable. Let w be a sharing state for ρ, an efine w = ( U 1...U m V m+1...v m+n ) w ( U 1...U mv m+1...v m+n). Then, for any left-right pair of subsystems i an j, it follows that Tr i,j (w ) = U i V j Tr i,j (w) U i V j = U V ρu V = ρ UV. Hence, w is an m-n-sharing state for ρ UV, as esire. This result suggests a connection between the egree of sharability an the entanglement of a given state. In both cases, there exist classes of states for which these properties cannot be further egrae by locally acting maps (or any map for that matter). Obviously, LOCC cannot ecrease the entanglement of states with no entanglement, an convex unitary mixtures as above cannot increase the sharability of states with - sharability (because they are alreay as sharable as possible). These two classes of states can in fact be shown to coincie as a consequence of the fact that arbitrary sharability is equivalent to (bipartite) separability. This result has been appreciate in the literature [2,, 6, 40] an is creite to both [41] an [42]. We reprouce it here in view of its relevance to our work: Theorem II.6. A bipartite quantum state ρ on H L H R is unentangle (or separable) if an only if it is arbitrarily sharable. Proof. ( ) Let ρ be separable. Then for some set of ensity operators {ρ L i, ρr i }, it can be written as ρ = i λ iρ L i ρr i, with i λ i = 1. Let n an m be arbitrary, an let the N-partite state w, be efine as follows: w = i λ i (ρ L i ) m (ρ R i ) n, with N = m + n. By construction, the state of each L-R pair is ρ, since it follows straighforwarly that Eq. () is obeye for each i, j. Thus, w is a vali sharing state. ( ) Since ρ is arbitrarily sharable, there exists a sharing state w for arbitrary values of m, n. In particular, we nee only make use of a sharing state w for m = 1 an arbitrarily large n, whence we let n. Given w, let us construct another sharing state w, which is invariant uner permutations of the right subsystems, that is, let w = 1 S n π S n V π wv π, where S n {π} is the permutation group of n objects, acting on H n R via the natural n-fol representation,

6 6 V π ( i ψ i ) = i ψ π(i), i = 1,..., n. that w shares ρ: Tr ˆL, ˆR ( w) = 1 S n = 1 S n π S n Tr ˆL, ˆR ( V π wv π ) Tr ˆL,π( ˆRi) (w) = 1 S n π S n It then follows π S n ρ = ρ. Having establishe the existence of a symmetric sharing state w D(H L H R ), Fannes Theorem (see section 2 of [42]) implies the existence of a unique representation of w as a sum of prouct states, w = i λ iρ i L ρi R ρi R.... Reucing w to any L-R pair leaves a separable state. Thus, if ρ is 1-n sharable it must be separable. As we allue to before, a Corollary of this result is that in fact 1- sharability implies - sharability. In closing this section, we also briefly mention the concept of exchangeability [4, 44]. A ensity operator ρ on (H (1) 1 ) p is sai to be exchangeable if it is symmetric uner permutation of its p subsystems an if there exists a symmetric state w on (H (1) 1 ) (p+q) such that the reuce states of any subset of p subsystems is ρ for all q N. Similar to sharability, exchangeability implies separability. However, the converse only hols in general for sharability: clearly, there exist states which are separable but not exchangeable, because of the extra symmetry requirement. Thus, the notion of sharability is more irectly relate to entanglement than exchangeability is. III. JOINING AND SHARING WERNER AND ISOTROPIC STATES Even for the simplest case of two bipartite states with an overlapping marginal, a general characterization of joinability is extremely non-trivial. As remarke, no conitions yet exist which are both necessary an sufficient for two arbitrary ensity operators to be joinable; although, conitions that are separately necessary or sufficient have been recently erive [2]. In this Section, we present a complete characterization of the threeparty joining scenario an the 1-n sharability problem for Werner an isotropic states on arbitrary subsystem imension. We begin by introucing the relevant families of quantum an classical states to be consiere. A. Werner an isotropic quit states, an their classical analogues The usefulness of bipartite Werner an isotropic states is erive from their simple analytic properties an range of mixe state entanglement. For a given subsystem imension, Werner states are efine as the one-parameter family that is invariant uner collective unitary transformations [24] (see also [44]), that is, transformations of the form U U, for arbitrary U U(). The parameterization which we employ is given by ρ(ψ ) = 2 1 [ ( Ψ ) I 2 + (Ψ 1 ) V where V is the swap operator, efine by its action on any prouct ket, V ψφ φψ. This parameterization is chosen because Ψ is a Werner state s expectation value with respect to V, Ψ = Tr[V ρ(ψ )]. Non-negativity is ensure by 1 Ψ 1, an the completely mixe state correspons to Ψ = 1/. Furthermore, the concurrence of Werner states is simply given by [45] ], C(ρ(Ψ )) = Tr [ V ρ(ψ ) ] = Ψ, Ψ 0. (5) For Ψ > 0, the concurrence is efine to be zero, inicating separability. Werner states have been experimentally characterize for photonic qubits, see e.g. [46]. Interestingly, they can be issipatively prepare as the steay state of coherently riven atoms subject to collective spontaneous ecay [47]. Isotropic states are efine, similarly, as the oneparameter family that is invariant uner transformations of the form U U [48]. We parameterize these states as ρ(φ + ) = [ 2 ( Φ + ) I (Φ ) ] Φ + Φ +, where Φ + = 1/ i ii. The value of the parameter is given by the expectation value with respect [ to the partially transpose swap operator, Φ + = Tr V T A ] (AB) ρ(φ+ ), an is relate to the so-calle singlet fraction [49] by Φ + = F. Non-negativity is now ensure by 0 Φ +, whereas the concurrence is given by [50], C(ρ(Φ + 2 )) = ( 1) (Φ+ 1), Φ + 1, (6) an is efine to be zero for Φ + 1. Before introucing probability istributions that will serve as the analogue classical states, we present an alternative way to think of Werner states, which will prove useful later. First, the highest purity, attaine for the Ψ = 1 state, is 2/[( 1)], with the absolute maximum of 1 corresponing to the pure singlet state for = 2. Secon, collective projective measurements on a most-entangle Werner state return only isagreeing outcomes (e.g., corresponing to 1, but not 1 1 ). The following construction of bipartite Werner states emonstrates the origin of both of these essential features. For generic, the analogue to the singlet state is the following -partite fully anti-symmetric state: ψ = 1! π S sign(π)v π , (7)

7 7 where, as before, S {π} enotes the permutation group an { l } is an orthonormal basis on H (1) C. The above state has the property of being completely isagreeing, in the sense that a collective measurement returns outcomes that iffer on each quit with certainty. The most-entangle bipartite quit Werner state is nothing but the two-party reuce state of ψ. Thus, we can think of general bipartite quit Werner states as mixtures of the completely mixe state with the two-partyreuction of ψ. The inverse of 2/[( 1)] (the purity) is precisely the number of ways two its can isagree. Unerstaning bipartite Werner states to arise from reuce states of ψ will inform our construction of the classical analogue states, an also help us unerstan some of the results of Sec. III C an III E. For Werner states, increase entanglement correspons to increase isagreement for collective measurement outcomes. For isotropic states, increase entanglement correspons to increase agreement of collective measurements, but only with respect to the computational basis { i } relative to which such states are efine. It is this expression of agreement vs. isagreement of outcomes which carries over to the classical analogue states, which we are now reay to introuce. The relevant probability istributions are efine on the outcome space Ω Ω = {1,..., } {1,..., }. To resemble Werner an isotropic quantum states, these probability istributions shoul have completely mixe marginal istributions an range from maximal isagreement to maximal agreement. This is achieve by an interpolation between an even mixture of agreeing pure states, namely, (1, 1), (2, 2),..., (, ), an an even mixture of all possible isagreeing pure states, namely, (1, 2),..., (1, ), (2, 1),..., (, 1). That is: p(a = i, B = j) α = α δ i,j + 1 α ( 1) (1 δ i,j), (8) where α is the probability that the two outcomes agree. To make the analogy complete, it is esirable to relate α to both Ψ an Φ +. We efine α in the quantum cases to be the probability of obtaining k on system A, conitional to outcome k on system B for the projective measurement { ij ij }. For Werner states, this probability is relate to Ψ by p( k A k B ) W = Ψ α W, (9) an, similarly for isotropic states, we have p( k A k B ) I = Φ α I. (10) We may thus re-parameterize both the Werner an isotropic states in terms of their respective above-efine probabilities of agreement, namely: ρ(α W ) = ρ(α I ) = 1 1 [ (1 α W ) I (α 2 + W ) 1 ] V, (11) [ (1 α I ) I (α 2 + I 1 ) ] Φ + Φ +, (12) subject to the conitions 0 α W 2 + 1, α I 1. For Werner states, α W can rightly be consiere a probability of agreement because it is inepenent of the choice of local basis vectors in the projective measurement {U U ij ij U U }. For isotropic states, α I oes not have as irect an interpretation. We may nevertheless interpret α as a probability of basis-inepenent agreement if we pair local basis vectors on A with their complex conjugates on B. In other wors, α I can be thought of as the probability of agreement for local projective measurements of the form {U U ij ij U U } [51]. B. Classical joinability limitations In orer to etermine the joinability limitations in the classical case, we begin by noting that any (finiteimensional) classical probability istribution is a unique convex combination of the pure states of the system. In our case, there are five extremal three-party states, for which the two-party marginals are classical analogue states, as efine in Eq. (8). These are p(a, B, C agree) = 1 (i, i, i), i 1 p(a, B agree) = (i, i, j), ( 1) i j 1 p(a, C agree) = (i, j, i), ( 1) i j 1 p(b, C agree) = (j, i, i), ( 1) i j 1 p(all isagree) = ( 1)( 2) (i, j, k), i j k where (i, j, k) stans for the pure probability istribution p(a, B, C) = δ A,i δ B,j δ C,k. The first four of these states are vali for all 2 an each correspons to a vertex of a tetraheron, as epicte in Fig. 1(left). The fifth state is only vali for an correspons to the point (α AB, α AC, α BC ) = (0, 0, 0) in Fig. 1(right). Any vali three-party state for which the two-party marginals are classical analogue states must be a convex combination of the above states. Therefore, the joinable-unjoinable bounary is elimite by the bounary of their convex hull. For the = 2 case, the inequalities escribing these bounaries are explicitly given by the following: p(a, B, C agree) 0 α AB + α AC + α BC 1, p(c isagrees) 0 α AB + α AC + α BC 1, p(b isagrees) 0 α AB α AC + α BC 1, p(a isagrees) 0 α AB + α AC α BC 1,

8 8 where each inequality arises from requiring that the corresponing extremal state has a non-negative likelihoo. In the case, the inequality p(a, B, C agree) 0 is replace by α AB, α AC, α BC 0. C. Joinability of Werner an isotropic quit states We now present our results on the three-party joinability of Werner an isotropic states an then compare them to the classical limitations just foun in the previous section. While, as mentione, all the technical proofs are post-pone to Appenix A in orer to ease reaability, the basic iea is to exploit the high egree of symmetry that these classes of states enjoy. Consier Werner states first. Our starting point is to observe that if a tripartite state w ABC joins two reuce Werner states ρ AB an ρ AC, then the twirle state w ABC, given by w ABC = (U U U) w ABC (U U U) µ(u), (1) is also a vali joining state. In Eq. (1), µ enotes the invariant Haar measure on U(), an the twirling superoperator effects a projection into the subspace of operators with collective unitary invariance [52]. By invoking the Schur-Weyl uality [5], the guarantee existence of joining states with these symmetries allows one to narrow the search for vali joining states to the Hermitian subspace spanne by representations of subsystem permutations, that is, ensity operators of the form w = π S µ π V π, w W W, (14) where Hermiticity emans that µ π = µ π 1. Given w ABC which joins Werner states, each subsystem pair is characterize by the expectation value with the respective swap operator, Ψ ij = Tr[w ABC(V ij I ij )], where i, j {A, B, C} with i j. Hence, the task is to etermine for which (Ψ AB, Ψ BC, Ψ AC ) there exists a ensity operator w ABC consistent with the above expectations. Our main result is the following: Theorem III.1. Three Werner quit states with parameters Ψ AB, Ψ BC, Ψ AC are joinable if an only if (Ψ AB, Ψ BC, Ψ AC ) lies within the bi-cone escribe by 1 ± Ψ 2 Ψ BC + ωψ AC + ω2 Ψ AB, (15) for, or within the cone escribe by 1 Ψ 2 Ψ BC + ωψ AC + ω2 Ψ AB, Ψ 0, (16) for = 2, where Ψ = 1 (Ψ AB + Ψ BC + Ψ AC ), ω = ei 2π. (17) Similarly, if a tripartite state w ABC joins isotropic states ρ AB an ρ AC, then the isotropic-twirle state w ABC, given by w ABC = (U U U) w ABC (U U U) µ(u), (18) is also a vali joining state. A clarification is, however, in orer at this point: although we have been referring to the isotropic joinability scenario of interest as three-party isotropic state joining, this is somewhat of a misnomer because we effectively consier the pair B-C to be in a Werner state, as evient from Eq. (18). Compare to Eq. (14), the relevant search space is now partially transpose relative to subsystem A, that is, consisting of ensity operators of the form w = µ π Vπ TA, w W iso. (19) π S Our main result for three-party joinability of isotropic states is then containe in the following: Theorem III.2. Two isotropic quit states ρ AB an ρ AC an quit Werner state ρ BC with parameters Φ + AB, Φ+ AC, Ψ BC are joinable if an only if (Φ + AB, Φ+ AC, Ψ BC ) lies within the cone escribe by Φ + AB + Φ+ AC Ψ BC, (20) 1 + Φ + AB + Φ+ AC Ψ BC (21) 2 (Ψ BC 1) + 1 (eiθ Φ + AB + e iθ Φ + AC ), e ±iθ = ±i ( + 1)/(2) + ( 1)/(2), or, for, within the convex hull of the above cone an the point (0, 0, 1). The results of Theorems III.1 an III.2 as well as of Sec. III B are pictorially summarize in Fig. 1. We now compare these quantum joinability limitations to the joinability limitations in place for classical analogue states. As escribe in Sec. III B, the nonnegativity of p(a, B, C agree) an p(a isagree) is enforce by the two inequalities α AB + α AC + α BC 1 an α AB α AC + α BC 1, respectively. We expect the same requirement to be enforce by the analogue quantum-measurement statistics. For = 2, the bases of the Werner an isotropic joinability-limitation cones are etermine by Ψ AB + Ψ AC + Ψ BC 0 an Φ + AB + Φ+ AC Ψ BC 2, respectively. Writing own each of these parameters in terms of the appropriate probability of agreement α, as efine in Eqs. (9) an (10), we obtain α AB +α AC +α BC 1 an α AB α AC +α BC 1. Hence, for qubits, part of the quantum joining limitations are inee erive from the classical joining limitations. This is also illustrate in Fig. 1(left). Of course, one woul not expect the quantum scenario to exhibit violations of the classical joinability restrictions; still, it

9 9 FIG. 1. (Color online) Three-party quantum an classical joinability limitations for Werner an isotropic states, an their classical analogue, as parameterize by Eqs. (11), (12), (8), respectively. Left panel: Qubit case, = 2. The Werner state bounary is the surface of the arker cone with its vertex at (2/, 2/, 2/), whereas the isotropic state bounary is the surface of the lighter cone with its vertex at (1/, 1/, 2/). The classical bounary is the surface of the tetraheron. Right panel: Higher-imensional case, = 5. The Werner state bounary is the surface of the bi-cone with vertices at (0, 0, 0) an (1/, 1/, 1/), whereas the isotropic state bounary is the flattene cone with its vertex at (1/6, 1/6, 1/). The classical bounary is the surface of the two joine tetrahera. In both panels the grey line resting on top of the cones inicates the colinearity of the cone surfaces along this line segment. is interesting that states which exhibit manifestly nonclassical correlations may nonetheless saturate bouns obtaine from purely classical joining limitations. For, the only classical bounary which plays a role is the one which bouns the base of the isotropic joinability-limitation cone: Φ + AB + Φ+ AC Ψ BC. Again, in terms of the agreement parameters, this is (just as for qubits) α AB α AC + α BC 1. In the Werner case, the quantum joinability bounary is not clearly elineate by the classical joining limitations. We can nevertheless make the following observation. By the non-negativity of Werner states, the threeparty joinability region in Fig. 1(right) is require to lie within a cube of sie-length 2/( + 1) with one corner at (0, 0, 0). Consier the set of cubes obtaine by rotating from this initial cube about an axis through (0, 0, 0) an (2/(+1), 2/(+1), 2/(+1). It is a curious fact that the exact quantum Werner joinability region (the bi-cone) is precisely the intersection of all such cubes. Another interesting feature is that there exist trios of unentangle Werner states which are not joinable. For example, the point (Ψ AB, Ψ AC, Ψ BC ) = (1, 1, 0) correspons to three separable Werner states that are not joinable. This point is of particular interest because its classical analogue is joinable. Translating (1, 1, 0) into the agreement-probability coorinates, (α AB, α AC, α BC ) = (2/, 2/, 1/), we see that this point is actually on the classical joining limitation borer. Thus, these three separable, correlate states are not joinable for purely quantum mechanical reasons. Note that the point (α AB, α AC, α BC ) = (2/, 2/, 1/) oes correspon to a joinable trio of pairs in the isotropic three-party joining scenario: this point lies at the center of the face of the isotropic joinability cone, as seen in Fig. 1(left). The same fact hols for (2/, 1/, 2/) or (1/, 2/, 2/) when the Werner state pair in the isotropic joining scenario escribes A-C or A-B, respectively; in both cases, we woul have obtaine yet another cone in Fig. 1 that sits on a face of the classical tetraheron bounary. Having etermine the joinable trios of both Werner an isotropic states, we are now in a position to also answer the question of what pairs A-B an A-C of states are joinable with one another. In the Werner state case, this is obtaine by projecting the Werner joinability bicone own to the Ψ AB -Ψ AC plane, resulting in the following: Corollary III.. Two pairs of quit Werner states with parameters Ψ AB an Ψ AC are joinable if an only if Ψ AB, Ψ AC 1 2, or if the parameters satisfy (Ψ AB + Ψ AC )2 + 1 (Ψ AB Ψ AC )2 1, (22) or aitionally, in the case, if Ψ AB, Ψ AC 1 2. For isotropic states, we may similarly project the cone of Eq. (21) onto the Φ + AB -Φ+ AC plane to obtain the 1-2 joining bounary. This yiels the following: Corollary III.4. Two pairs of quit isotropic states with parameters Φ + AB an Φ+ AC are joinable if an only if they lie within the convex hull of the ellipse (Φ + AB / + Φ+ AC / 1)2 (1/ 2 + (Φ+ AB / Φ+ AC /)2 ) ( 2 1)/ 2 = 1, an the point (Φ + AB, Φ+ AC ) = (0, 0). (2)

10 10 these quarants show that there is also a trae-off between the amount of classical correlation in one pair an the amount of entanglement in the other pair. In fact, the smoothness of the bounary curve as it crosses from one of the pairs being entangle to unentangle suggests that, at least in this case, entanglement is not the correct figure of merit in iagnosing joinability limitations. (a) (b) D. Isotropic joinability results from quantum cloning FIG. 2. (Color online) Two-party joinability limitations for Werner an isotropic quit states. (a) Werner states. The shae region correspons to joinable Werner pairs, with the lighter region being vali only for. The roune bounary is the ellipse etermine by Eq. (22). This explicitly shows the existence of pairs of entangle Werner states that are within the circular bounary etermine by the weak CKW inequality, Eq. (2), yet are not joinable. (b) Isotropic states. The three regions correspon to the joinable pairs of isotropic states for = 2, = an = This shows how, in the limit of large, the trae-off in isotropic state parameters becomes linear, consistent with known results on -imensional quantum cloning [54]. Lastly, by a similar projection of the isotropic cone given by Eqs. (20)-(21), we may explicitly characterize the Werner-isotropic hybri 1-2 joining bounary: Corollary III.5. An isotropic state with parameter Φ + AB an a Werner state with parameter Ψ BC are joinable if an only if they lie within the convex hull of the ellipse (Φ + AB / + Φ+ AC / 1)2 (1/ 2 + (Φ+ AB / Φ+ AC /)2 ) ( 2 1)/ 2 = 1, (24) an the point (Φ + AB, Ψ BC ) = (0, 1), an, for, within the aitional convex hull introuce by the point (Φ + AB, Ψ BC ) = (0, 1). The above results give the exact quantum-mechanical rules for the two-pair joinability of Werner an isotropic states, as pictorially summarize in see Figs. 2(a) an 2(b). A number of interesting features are worth noticing. First, by restricting to the line where Ψ AB = Ψ AC, we can conclue that qubit Werner states are 1-2 sharable if an only if Ψ 1/2, whereas for, all quit Werner states are 1-2 sharable. As we shall see, this agrees with the more general analysis of Sec. III E. Secon, some insight into the role of entanglement in limiting joinability may be gaine. In the first quarant of Fig. 2(a), where neither pair is entangle, it is no surprise that no joinability restrictions apply. Likewise, it is not surprising to see that, in the thir quarant where both pairs are entangle, there is a trae-off between the amount of entanglement allowe between one pair an that of the other. But, in the secon an fourth quarants we observe a more interesting behavior. Namely, Interestingly, the above results for 1-2 joinability of isotropic states can also be obtaine by rawing upon existing results for asymmetric quantum cloning, see e.g. [54, 55] for 1-2 an 1- asymmetric cloning an [56 58] for 1-n asymmetric cloning. One approach to obtaining the optimal asymmetric cloning machine is to exploit the Choi isomorphism [59] to translate the construction of the optimal cloning map to the construction of an optimal operator (or a telemapping state ). This connection is mae fairly clear in [56, 58]; in particular, singlet monogamy refers to the trae-off in fielities of the optimal 1-n asymmetric cloning machine or, equivalently, to the trae-off in singlet fractions for a (1 + n) quit state. We escribe how the approach to solving the optimal 1-n asymmetric cloning problem may be rephrase to solve the 1-n joinability problem for isotropic states. The state Ψ escribe in Eq. (4) of [56] is a 1-n joining state for n isotropic states characterize by singlet fractions F 0,j (relate to the isotropic state parameter by F 0,j = Φ + 0,j /, as note). The bouns on the singlet fractions are etermine by the normalization conition of Ψ, together with the requirement that Ψ be an eigenstate of a certain operator R efine in Eq. () of [56]. That Ψ is an isotropic joining state is reaily seen from its construction, an that it may optimize the singlet fractions (hence elineate the bounary in the {F 0,j } space) is proven in [58]. Our contribution here is the observation that this result provies the solution to the 1-n joinability of isotropic states. The equivalence is establishe by the fact that optimality is preserve in either irection by the Choi isomorphism. Quantitatively, the bounary for 1-n optimal asymmetric cloning, is given by Eq. (6) in [56] in terms of singlet fractions. Specializing to the 1-2 joining case an rewriting in terms of Φ +, we have Φ + AB + Φ+ AC ( 1) ( Φ +AB n Φ AC) + As one may verify, this is equivalent to the result of Corollary III.4. In light of this connection, the fact that, as increases, the isotropic-joinability cone of Fig. 1(right) becomes flattene own to the α AB -α AC plane is irectly relate to the linear trae-off in the isotropic state parameters for the semi-classical limit, as iscusse in [54]. Within our three-party joining picture, we can give a partial explanation of this fact: namely, it is a

11 11 consequence of the classical joining bounary in tanem with the upper limit on the agreement parameter α BC for the Werner state on B-C: α BC 2/( + 1). In the limit of, these two bounaries conspire to limit the (A-B)-(A-C) isotropic state joining bounary to a triangle, as explicitly seen in Fig. 2(b)(right). For the general 1-n isotropic joining scenario, the quantum-cloning results aitionally imply the following: Theorem III.6. A list of n isotropic states characterize by parameters Φ + 0,1,..., Φ+ 0,j is 1-n joinable if an only if the (positive-value) parameters satisfy n j=1 Φ + 0,j ( 1) + 1 n + 1 ( n j=1 Φ + 0,j) 2. (25) Interestingly, similar to our iscussion surrouning Eq. (2), the authors of [56] argue how the singlet monogamy boun can lea to stricter preictions (e.g., on grounstate energies in many-boy spin systems) than the stanar monogamy of entanglement bouns base on CKW inequalities [4, 7]. E. Sharability of Werner an isotropic quit states We next turn to sharability of Werner an isotropic states in imension, beginning from the important case of 1-n sharing. For Werner states, a proof base on a representation-theoretic approach is given in Appenix B. Although we expect a similar proof to exist for isotropic states, we obtain the esire 1-n sharability result by builing on the relationship with quantum cloning problems highlighte above. We then outline a constructive proceure for etermining the more general m-n sharability of Werner states. Our main results are containe in the following: of Werner state sharing, Eq. (26) implies that a finite parameter range exists where the corresponing Werner states are not sharable. In contrast, for, every Werner state is at least 1-2 sharable. This simply reflects the fact that ψ (recall Eq. (7)) provies a 1-( 1) sharing state for a most-entangle quit Werner state. With isotropic state sharing, for all there is, again, a finite range of isotropic states which are not sharable. The simplicity of the results in Eqs. (26)-(27) is intriguing an begs for intuitive interpretations. Consier a central quit surroune by n outer quits. If the central quit is in the same Werner or isotropic state with each outer quit, then Theorems III.7 an III.8 can be reinterprete as proviing a boun on the sums of concurrences. For Werner states, we have that the sum of all the central-to-outer concurrences cannot excee the number of moes by which the systems may isagree (i.e., 1). In the isotropic state case, the sum of the n pairwise concurrences cannot excee the maximal concurrence value given by C max, = 2( 1)/. These rules o not hol in more general joining scenarios, as we alreay know from Sec. III C. There, we foun that the trae-off between A-B concurrence an A-C concurrence is not a linear one, as such a simple sum rule woul preict; instea, it traces out an ellipse (recall Fig. 2(a)). Starting from the proof of Thm. III.7 foun in Appenix B, in conjunction with similar representationtheoretic tools, it is possible to evise a constructive algorithm for etermining the m-n sharability of Werner states. The basic observation is to realize that the most-entangle m-n sharable Werner state correspons to the largest eigenvalue of a certain Hamiltonian operator H m,n, which is in turn expressible in terms of Casimir operators. Calculation of these eigenvalues may be obtaine using Young iagrams. Although we lack a general close-form expression for max(h m,n ), the require calculation can nevertheless be performe numerically. Representative results for n-m sharability of low-imensional Werner states are shown in Table I. Theorem III.7. A quit Werner state with parameter Ψ is 1-n sharable if an only if Ψ 1 n. (26) Theorem III.8. A quit isotropic state with parameter Φ + is 1-n sharable if an only if Φ n. (27) Proof. Specializing Eq. (25) to the case of equal parameters for all n isotropic states, the above result immeiately follows. As state in [56], this is consistent with the well known result for optimal 1-n symmetric cloning. A pictorial representation of the above sharability results, specialize to qubits, is given in Fig.. In the case FIG.. Pictorial summary of sharability properties of qubit Werner an isotropic states, accoring to Eqs. (26) an (27). The arrow-heae lines epict the parameter range for which states satisfy each of the sharability properties isplaye to the right an left, respectively. The vertical ticks between en points of these ranges inicate the points at which subsequent 1-n sharability properties begin to be satisfie.