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Generalizations of entanglement based on coherent states and convex sets

Howard Barnum,∗ Emanuel Knill,† Gerardo Ortiz,‡ and Lorenza Viola§

Los Alamos National Laboratory, MS B256, Los Alamos, NM 87545(Dated: October 25, 2018)

Unentangled pure states on a bipartite system are exactly the coherent states with respectto the group of local transformations. What aspects of the study of entanglement are appli-cable to generalized coherent states? Conversely, what canbe learned about entanglementfrom the well-studied theory of coherent states? With thesequestions in mind, we charac-terize unentangled pure states as extremal states when considered as linear functionals onthe local Lie algebra. As a result, a relativized notion of purity emerges, showing that thereis a close relationship between purity, coherence and (non-)entanglement. To a large extent,these concepts can be defined and studied in the even more general setting of convex cones ofstates. Based on the idea that entanglement is relative, we suggest considering these notionsin the context of partially ordered families of Lie algebrasor convex cones, such as those thatarise naturally for multipartite systems. The study of entanglement includes notions of lo-cal operations and, for information-theoretic purposes, entanglement measures and ways ofscaling systems to enable asymptotic developments. We propose ways in which these maybe generalized to the Lie-algebraic setting, and to a lesserextent to the convex-cones set-ting. One of our motivations for this program is to understand the role of entanglement-likeconcepts in condensed matter. We discuss how our work provides tools for analyzing thecorrelations involved in quantum phase transitions and other aspects of condensed-mattersystems.

∗barnum@lanl.gov†knill@lanl.gov‡g ortiz@lanl.gov§lviola@lanl.gov

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Contents

I. Introduction 3

II. Entanglement for Bipartite Quantum Systems 5A. Characterizing Product States 5B. Local Quantum Maps 7C. Communication Complexity 13D. Resource Scaling 13E. Measures of Entanglement 14

III. The Lie-algebraic Setting 16A. Purity, Coherence and Entanglement 17B. Local Quantum Maps 19C. Communication Complexity 21D. Resource Scaling 22E. Measures of Relative Entanglement 22F. Other Measures 23

IV. The Convex Cones Setting 24A. Convex Cones 24B. Local Maps 25C. Measures of Relative Entanglement 27D. Monotonicity for Explicit Liftable Maps 27

V. Discussion 29A. Further Examples and Extensions 29B. Relevance to Condensed Matter Physics 30C. Conclusion 32

A. Comparison of the Settings for Generalized Entanglement 32

Acknowledgments 35

References 35

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I. INTRODUCTION

Entangled states are joint states of two or more distinguishable quantum systems that cannotbe expressed as a mixture of products of states of each system. Entangled states can exhibit quan-tum correlations between the two systems that have no local classical interpretation. One of themost important developments in the study of quantum mechanics was the characterization of thesecorrelations by Bell [1, 2], whose many experimental verifications [3, 4] (see also [5] and thereferences therein) have given further support to the validity of quantum mechanics. Entangledstates are now widely considered to be the defining resource of quantum communication, enablingprotocols such as quantum teleportation [6] and leading to great improvements in the communi-cation efficiency of certain multi-party tasks [7, 8]. As a result, entanglement is being activelyinvestigated both from a physical and from an information-theoretic perspective.

So far, nearly all studies of entanglement involved two or more distinguishable quantum sub-systems. As a result, investigations of entanglement have focused on understanding how quantumsystems are made up from subsystems and how this differs fromclassical systems. However,there are a number of signs that the assumption of distinguishable quantum subsystems is toonarrow to capture all properties of states that one might like to ascribe to entanglement. Severalauthors [9, 10, 11, 12, 13, 14, 15, 16, 17] have considered entanglement-related notions for bosonsand fermions. For example, consider the state of one photon in two coupled cavities. Being thestate of one particle, there is a tendency to expect that there is no entanglement, because one par-ticle cannot be entangled. On the other hand, each cavity is aquantum system. From the pointof view of these two quantum systems, the state where the photon is in an equal superposition ofbeing in either cavity can be represented as(|10〉 + |01〉)/

√2 and is clearly entangled. Another

example involving photons is provided by optical “cat states” [18, 19]. In this case, cat states arequantum superpositions of sufficiently distinct coherent states in a mode. As the name suggests,such states are thought to involve entanglement. They certainly have distinctive non-classical be-havior, but since they exist in a single system (the mode) thestrict interpretation of entanglementbased on subsystems would indicate that no entanglement is present. A third example is that con-sisting of a number of fermions in a lattice. The “simple” states for such a system are describedby the so-called Slater determinants (see, for example, [20], p. 7), which describe the wavefunc-tion of noninteracting fermions. Because the fermions in such a wavefunction are independent,one expects that no entanglement is present in such a state. However, from the point of view ofthe lattice modes, most Slater determinants exhibit entanglement [21]. The three examples makeit clear that the presence or absence of entanglement depends on the physically relevant point ofview. Here we propose that this point of view depends on the relationships between different Liealgebras of observables that determine the dynamics and ourability to control the system of inter-est. In particular, the extent to which entanglement is present depends on the observables used tomeasure a system and describe its states.

One of our goals is to show that the relationships between product states, separable states andentangled states are at least Lie-algebraic in nature, and to some extent even more general. Thismakes it possible to study the salient features of entanglement without reference to subsystems,using instead whatever Lie algebras are physically relevant. For the case of bipartite quantumsystems, the relevant Lie algebrah consists of the unilocal operators (operators of the formA ⊗I or I ⊗ B). To show that the ideas of entanglement, separability and product states do notcritically depend on the two subsystems, we provide severalways in which product states canbe characterized in terms ofh alone. All of these ways lead to the same concept for generalsemisimple Lie algebras, namely that of generalized coherent states [22, 23, 24]. It is therefore

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natural to consider product states to be special kinds of coherent states. From this perspective,separable states are mixtures of coherent states, and pure entangled states are incoherent purestates. Another way to think about these structures is to realize that the coherent states are exactlythose states which are relatively pure, that is, extremal with respect to the set of expectationsof observables in the Lie algebra. Thus, pure states are entangled if they appear to be mixed withrespect to the Lie algebra’s expectations. In the case of bipartite quantum systems, this is an aspectof entanglement that has long been considered a key nonclassical property of quantum mechanics:Pure entangled states have mixed reduced density operatorswhereas, for example, in classicalprobability no pure state can have a mixed marginal. See, forexample, [25], p. 298, [26], p. 116and [27], p. 306.

The recognition that incoherence naturally generalizes entanglement makes explicit the depen-dence of the notion of entanglement on the relevant Lie algebra and makes available the tools ofthe theory of generalized coherent states [23, 24] for investigating aspects of entanglement. Toextend the power of this perspective to the information-theoretic applications of entanglement re-quires introducing measures of entanglement, generalizing the ways in which entanglement canbe manipulated and providing a means for using states as a resource. In bipartite systems, there isan abundance of measures of entanglement, many of which generalize naturally. Further measuresarise naturally in the general context and specialize to potentially interesting measures for mul-tipartite systems that have not yet been considered. In bipartite systems, a key role is played byLOCC (local quantum operations and classical communication) maps. We propose several classesof maps for general semisimple Lie algebras that, in the caseof bipartite systems, are related toLOCC. A desirable property of entanglement measures is thatthey are monotone non-increasingunder LOCC. We can show monotonicity properties for some classes of maps in the general set-ting. To introduce the notion of states as a resource and enable asymptotic analysis, we considerschemes for associating Lie algebras with tensor products of systems defined by a given represen-tation of a semisimple Lie algebra.

For the purpose of determining what are the essential properties of states needed to study entan-glement, we introduce a setting even more general than Lie algebras. Since the states when viewedas linear functionals on observables form a convex cone, we generalize the definitions to the set-ting where we have two or more convex cones related by positive maps. The cones represent thefamily of states as linear functionals on the Lie algebras. In the case of bipartite systems, these arethe local Lie algebra and the Lie algebra of all operators. The map relating the two state spaces isthe restriction map of linear functionals. The definitions relating to separability and entanglementonly require this structure. Entanglement measures can also be defined based only on convexity,and so can various notions of local maps.

In taking seriously the idea that entanglement is a relativenotion, one finds that in many cases,there are many more than two relevant Lie algebras. In the bipartite case, we can consider thehierarchy of algebras consisting of the trivial Lie algebra, the algebra of operators acting on thefirst system, that acting on the second system, the sum of these, and the algebra of all operators.When there are more than two systems, the number of differentways of combining local Liealgebras multiplies. For photons, there is the Lie algebra of passive linear operations, of activelinear operations, and that of all linear and nonlinear operations. To these one might add the Liealgebras acting locally on the modes, etc. It is in the increasing amount of information that isavailable about states as more operators are added that crucial quantum properties emerge. Webelieve that in studying a given system, it is beneficial to consider coherence and entanglementproperties at multiple levels.

Independently of the work reported here, Klyachko [28] has recently proposed a generalization

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of entanglement for representations of semisimple Lie groups. His starting point is an extremalityproperty that we use as one of the equivalent characterizations of product (in general, coherent)states. Klyachko’s work is focused on the geometric invariant theory approach for investigatingstates with respect to one Lie group of operators. This approach leads to useful classifications ofthe orbits of states under the Lie group’s action. In this context, he discusses how the notions ofclassical realism that lead to Bell’s inequalities [1] generalize to the Lie algebraic setting. He alsointroduces notions of maximal entanglement and another interesting entanglement measure.

In Section II, we introduce the basic notions required for generalizing separability and entan-glement by reviewing the example of bipartite systems from the point of view of Lie algebrasand coherence. The generalization to semisimple Lie-algebras is explained in Section III, and theextent to which the generalization depends only on the relationships between convex cones is dis-cussed in Section IV. For reference, the different settingsfor studying entanglement are comparedin Table A. The paper concludes with a discussion of other relevant examples and the potentialapplications to condensed matter. We assume familiarity with the basic concepts of quantum infor-mation and entanglement. A good reference for quantum information theory is [29]. For reviewsof entanglement, see [30, 31, 32]. We also use results from the basic theory of Lie algebras. De-tails can be found in books such as [33, 34, 35, 36]. For physically motivated treatments of Liealgebras, see [37, 38, 39, 40, 41]. References for convexityand convex cones include [42, 43].

II. ENTANGLEMENT FOR BIPARTITE QUANTUM SYSTEMS

The standard setting for studying entanglement involves two (or more) distinguishable quantumsubsystems forming abipartite system. The properties of entanglement are most salient if thequantum subsystems are spatially well separated, with communication between the sites restrictedto classical signals subject to speed-of-light limitations. Let the state space of two such quantumsubsystems be given by the Hilbert spacesHa andHb of dimensionNa andNb, respectively. Thejoint state space of the bipartite system isHab = Ha ⊗Hb. All state spaces and operator algebrasare assumed to be finite dimensional. See Section V A for a brief discussion of the need andpossibilities for extensions to infinite dimensional systems. Product states are pure states ofHab

of the form|ψ〉 ⊗ |φ〉. Entangled pure states are states ofHab that are not expressible as a productstate. It is necessary to generalize the state space to mixtures of pure states, that is probabilitydistributions over pure states. For this purpose, one uses density matrices to represent states. Adensity matrixρ is pure ifρ = |ψ〉〈ψ| for some|ψ〉. Equivalently, it is pure if tr(ρ2) = 1, or ifρ is extremal in the set of density matrices (see below). A separable state is a mixture of productstates. Its density matrix is therefore aconvex combinationof product states, which is a sum ofthe form

∑k pk|ψk〉〈ψk| ⊗ |φk〉〈φk|, where(pk)k is a probability distribution [44]. We will use

the expressions “convex combination” and “mixture” interchangeably. A non-separable state issaid to be entangled. It is worth recalling that separable states can have non-classical features. Forexample, see [45, 46].

A. Characterizing Product States

In our approach, the key distinction between entangled and separable states is the differencebetween the way things look locally and globally. The local observables are operators of theformA ⊗ I andI ⊗ B. For our purposes, it is convenient to allow arbitrary operators as observ-ables, not only hermitian ones. Since non-hermitian operators can be expressed as complex linear

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combinations of hermitian operators, expectations of suchoperators are readily computed fromexpectations of hermitian operators.

If a pure state of the two systems in unentangled, then it is completely determined by the expec-tation values of the local observables. To specify a pure entangled state requires knowledge of thecorrelations, which are expectations of operators of the formA⊗B. Note that this method for dis-tinguishing between unentangled and entangled states doesnot extend to mixtures. A generic sep-arable state can contain non-trivial correlations. An example is(|0〉〈0|⊗|0〉〈0|+|1〉〈1|⊗|1〉〈1|)/2.Here the two subsystems are classically correlated. Nevertheless, it is possible to characterize sep-arability by investigating the structure of states in termsof their expectations of local versus globalobservables.

There are four non-trivial Lie algebras of observables thatdetermine the structure of the bipar-tite system. Letha (hb) be the Lie algebra of operators of the fromA⊗I (I⊗B) acting on systema(b). We call these theunilocalalgebras, because they consist of operators acting on one subsystemonly. Thelocal Lie algebra is given byhl = ha + hb. Let g be the Lie algebra of all operators onHab. As defined, these four Lie algebras are complex. However, asfamilies of operators they are†-closed, that is, closed under hermitian conjugation. LetRe(h) be the set of hermitian operatorsin h. For a hermitian-closed space of operatorsh, h = Re(h) + iRe(h), wherei =

√−1. Using

complex Lie algebras simplifies the representation theory and is useful for defining generalizationsof local quantum maps (see Section II B). Although exponentialseA for non-skew-hermitian op-erators are not unitary, they can be interpreted as Lie-algebraically definable operators associatedwith postselected outcomes in an implementation of a quantum map.

A simple way of characterizing product states without referring to the underlying partition intotwo subsystems can be based on unique ground states. Aunique ground stateof a hermitian oper-ator is a unique minimum-eigenvalue eigenstate. Operatorswith degenerate minimum-eigenvalueeigenspaces do not have a unique ground state. In general, wecall the the minimum-eigenvalueeigenspace of an operator theground space.

Theorem 1 |ψ〉 ∈ Hab is a product state iff it is the unique ground state of an operator in Re(hl).

Proof: Suppose that|ψ〉 is the unique ground state ofH = A ⊗ I + I ⊗ B ∈ Re(hl). Theground space ofH is the intersection of the ground spaces ofA⊗ I andI ⊗B, which are productsubspaces. Thus, a unique ground state is a product state. Conversely, let|ψ〉 = |φa〉 ⊗ |φb〉.Choose an operatorA (B) onHa (Hb) such that|φa〉 (|φb〉) is the unique ground state ofA (B).Then|ψ〉 is the unique ground state ofA⊗ I + I ⊗B ∈ Re(hl).

We can use Theorem 1 to define a generalization of a product state for any hermitian-closedLie algebra of operators. As we will see in Section III, this generalization agrees with the notionof generalized coherent states.

The distinction between product and entangled states can also be viewed in terms of puritywith respect to the relevant algebra of operators. It can be seen that product states are exactly thestates whose reduced density matrices on each of the two subsystems are pure. The two reduceddensity matrices for a state completely determine the expectations of the observables in the localLie algebra. To prepare for generalizing these observations, consider states as linear functionalson the Lie algebras in question. We define anh-stateto be a linear functionalλ on the operators ofh inducedby a density matrixρ according toλ(C) = tr(ρC). The set ofh-states is denoted byh+.In the present setting, states are completely determined bythe linear functional on the Lie algebraof all operatorsg induced by their density matrix. Ag-stateλ can be restricted to each of the Liealgebrasha, hb andhl. For example, the restrictionλha of λ to ha determines the expectations of

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observables on the first subsystem, and therefore the reduced density matrix associated with thestate.

Consider the seth+l of hl-states. This set is closed underconvex(or probabilistic) combination.That is, if theλk are hl-states, then so is

∑k pkλk for any probability distribution(pk)k. By

compactness, all states inh+l can be obtained as convex combinations ofextremalstates (orextremepointsof h+l ). Extremal states are states not expressible as a convex combination of other states.If the only information available about a state are the expectations of observables inhl, then statesthat induce extremal expectations, that is, extremal elements ofh+l , are those about which there isthe least uncertainty. It therefore makes sense to call suchstatespure, or hl-pure, to be specific.

Theorem 2 Anhl-state is pure iff it is induced by a pure product state.

Proof: Consider a density matrixρ inducing thehl-stateλ. The stateλ is determined by the re-duced density matrices ofρ. It is possible to find a probabilistic combination of pure product stateswith the same reduced density matrices, which therefore also inducesλ. This implies that everyhl-state is expressible as a probabilistic combination ofhl states induced by pure product states.Consequently, the purehl-states are among those induced by pure product states. Conversely, ifλ is not pure, thenλ can be nontrivially expressed in the formpλ1 + (1 − p)λ2 where theλk arehl-states. It follows that the two reduced density matrices that can be deduced fromλ are not bothpure: They are mixtures of the reduced density matrices deduced fromλk, and sinceλ1 6= λ2, atleast one of these mixtures is nontrivial.

The previous theorem shows that the difference between pureunentangled states and pure en-tangled states is that as expectations ofhl, the latter are not extremal. If the only informationthat is available are expectations ofC ∈ hl, it is not possible to distinguish between entangledstates and unentangled mixed (that is, separable) states. To distinguish, we need expectations ofother operators. It is worth noting what it means to have access only to expectations of sets ofobservables. Given only a single instance of a quantum system, the expectations cannot be in-ferred. On the other hand, with sufficiently powerful control, it is possible to realize a projectivemeasurement of the eigenvalues of observables, a process that gives information not just about theexpectation of an observable, but also about the expectations of its powers. One situation whereaccess to expectations only is realistic is when the quantumsystem must be accessed collectivelyin large ensembles involving mostly identical states. In anappropriate weak interaction and largeensemble limit, the effect on other large systems reveals the expectations of observables involvedin the interaction, whereas the effect on the systems in the ensemble tends to a unitary evolutionwith the observable as a Hamiltonian. The weak interaction therefore naturally limits the availablecontrol to Lie algebras generated by a small number of observables. An example where this situa-tion occurs for systems that are best modeled as being quantum is nuclear magnetic resonance ofmolecules in the liquid state [47].

B. Local Quantum Maps

One can compare states in the context of information processing resources by considering fam-ilies of “local” quantum maps that can be used to convert states. For bipartite systems, as well asfor multipartite systems in general, the most important such family, LOCC, consists of maps thatcan be implemented with local quantum maps with access to ancillas and classical communica-tion (see [29], Sect. 12.5). A larger family, the separable quantum maps, have an operator-sumrepresentation consisting of operators of the formA ⊗ B. Separable quantum maps are readily

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generalized to the Lie algebraic setting, whereas we have not yet found an equally convincinggeneralization of LOCC.

A quantum mapis a trace-preserving completely-positive linear transformation of density op-erators. Rather than define these terms, we use the fact that every quantum map can be written inthe operator-sum representation asρ →

∑k CkρCk

†, with∑

k Ck†Ck = I. We will also consider

completely-positive maps, which have the same form, but don’t require the constraint on theCk.To define LOCC, we make the sequenceC = (Ck)k explicit and defineC(ρ) =

∑k CkρCk

†. Notethat the sequenceC is not uniquely determined by the map. We callC anexplicit map. See [29], p.372 for how to determine when two explicit maps act the same. To avoid trivial degeneracies, weassume that the operators that define an explicit map are always non-zero. If(Dk)k is a sequenceof explicit quantum maps, then the conditional compositionof C and(Dk)k is the quantum mapwith operator sequence(DklCk)kl and actionρ →

∑klDklCkρCk

†Dkl†. A unilocal quantum map

is a map of the form(Ak ⊗ I)k or (I ⊗ Bk)k. LOCC is the set of quantum maps obtained asconditional compositions of unilocal maps. The length of the composition is associated with thenumber of rounds of classical communication. Aseparable mapis a completely positive map withan explicit form given by(Ak ⊗ Bk)k. Note that all LOCC maps are necessarily separable. Theset of separable maps has been called SLOCC [48, 49, 50] and can be viewed as maps that can beimplemented with LOCC and postselection based on the communication record.

Quantum maps as defined here are often called “quantum operations” [29], though the latterterm is sometimes extended to include non-trace-preserving completely positive maps. In thismanuscript, we use the word “map” to refer to linear functions of spaces other than the Hilbertspace of the quantum system under consideration. We use the word “operator” to refer to lin-ear functions from the Hilbert space to itself. An importantrole in defining various notions oflocal maps is played by explicit maps, which in the bipartiteand in the Lie algebraic setting arecompletely-positive by definition. There is the potential for confusion in referring to explicit maps.For example, an explicit map can be separable without the operators in the explicit representationhaving the necessary product form. To simplify the terminology, we position the adjective “ex-plicit” such that it applies to all modifiers between it and the word “map”. For example, an explicitseparable mapC = (Ck)k satisfies that eachCk is a product operator, whereas this is not requiredof separable explicit maps.

Separable maps can be defined fromhl without reference to the two component subsystems.

Theorem 3 A completely positive map is separable iff it has an explicitrepresentation(Ck)k withCk ∈ ehl .

By definition,ehl is the topological closure of the set of all exponentials of operators inhl. Thenotion of closure may be based on the norm induced by the matrix inner product tr(A†B).

Proof: ehl consists of all non-zero determinant operators of the formA⊗B. Thusehl containsall invertible product operators, which are dense in the setof product operators. The set of productoperators is closed.

There are separable quantum maps that are not LOCC [45]. The goal is to define or construct,with minimal reference to the two subsystems, quantum maps that respect locality better thanseparable ones. For example, in order to construct the family of LOCC maps, it is sufficient to beable to determine when an operator inhl is unilocal, and when a family of unilocal operators allact on the same side. With this ability, one can construct LOCC as was done above, by conditionalcomposition. If the ability does not depend on the bipartitenature of the system, there is hope thatLOCC has a non-trivial generalization.

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We have two approaches to obtaining families of separable quantum maps with stronger localityproperties. The first approach is based on the observation that unilocal maps induce well-definedtransformations ofha-, hb- andhl-states. To formally define what this means, letC be an explicitmap. ThenC acts on the set of linear functionalsg∗ of g according toC(λ)(X) = λ(

∑k Ck

†XCk).It will be clear from context whether we are applyingC to operators or to linear functionals. ThemapC, but not its explicit form, is determined by the action ong-states. Note also thatg-stateslinearly span all linear functionals ong, and similarly forh states withh one ofha, hb or hl. C

induces a well-defined transformation ofh-states if we can complete the following commutativediagram with a mapC′ of h∗:

g∗C−→ g∗

restrict ↓ ↓ restrict

h∗C

′

−→ h∗

(1)

whereh∗ is the set of linear functionals onh. Equivalently, wheneverλ1 andλ2 areg-states thatagree onh, that is, for whichλ1h = λ2h, it is the case thatC(λ1)h = C(λ2)h. Equivalently, ifλ is a linear functional ong such thatλh = 0, thenC(λ)h = 0. The last statement is equivalentto the statement thatC preserves the nullspace of the restriction map. If any of theabove propertieshold, we say thatC can belifted to h. Its lifting is the mapC′ induced onh-states.

In the present setting, the notion of liftability can be simplified by using the canonical (viathe trace inner product) isomorphismµ betweenh∗ andh. Because the trace inner product isnon-degenerate when restricted to the†-closed set of operatorsh, the isomorphismµ is uniquelydetermined by the identityλ(C) = tr(µ(λ)†C) for all C ∈ h. In particular, for the algebrag of alloperators onHab, if λ ∈ g∗ is induced by the operatorX, thenµ(λ) = X. In general, we say thatthe linear functionalλ is inducedby µ(λ). Let tra (trb) denote the partial trace mapping operatorsonHab to operators onHb (Ha, respectively). We have the following identities:

µ(λha) = trb(µ(λ))⊗ I/Nb,

µ(λhb) = I/Na ⊗ tra(µ(λ)),

µ(λhl) = trb(µ(λ))⊗ I/Nb + I/Na ⊗ tra(µ(λ))− tr(µ(λ))(I ⊗ I)/(NaNb).

These identities witness the fact that the reduced density matrices of a state determine the inducedlinear functionals on the local Lie algebras. In the range ofµ, the nullspaces of the restrictionmaps toha, hb, andhl correspond to the spaces spanned byA ⊗ B with B,A and bothA andB,respectively, traceless. Using the fact that product operators are a basis of all operators onHa⊗Hb,it can be seen that the explicit mapC lifts to ha iff tr b

∑k Ck(A⊗ B)Ck

† = C′(A)tr(B) for somemapC′. Equivalently, it lifts iff whenever tr(B) = 0, then trb

∑k Ck(A ⊗ B)Ck

† = 0. Similarstatements can be made abouthb. C lifts to hl iff whenever both tr(A) = 0 and tr(B) = 0, thentrb∑

k Ck(A⊗ B)Ck† = 0 and tra

∑k Ck(A⊗B)Ck

† = 0.Most completely positive maps, even LOCC ones, cannot be lifted. An example for two qubits

is the “conditional reset” map that first measures qubita, and if the measurement outcome is|1〉,it resets qubitb to |0〉. However, the unilocal maps are liftable. In fact, they are liftable to bothha andhb, as are (unconditional) compositions of unilocal maps. This is the case because suchmaps are determined by their actions on the reduced density matrices. This suggests that liftableexplicit quantum maps could be used as a generating set for quantum maps with more locality thenseparable quantum maps. We next discuss some of the properties of liftable separable maps andtheir relationship to LOCC.

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Theorem 4 Let C = (C1 ⊗ C2) be a one-operator, explicit separable map liftable tohl. ThenC = αU ⊗ V withU andV unitary.

Proof: Liftability implies that ifA andB are traceless, thenC1AC1† andC2BC2

† are traceless.This implies that the mapf : A → C1AC1

† satisfies that tr(f(A)) = α1tr(A) for someα1.Thusf/α1 is trace preserving, from which it follows thatC1

†C1 = α1I. For the same reason,C2

†C2 = α2I. The conclusion of the theorem now follows, withα =√

|α1α2|.

Theorem 5 Let C be an explicit separable map that lifts to the identity map onhb. ThenC isunilocal, acting on systema only.

Proof: WriteC = (Dk ⊗Ek)k, whereDk ⊗Ek 6= 0 for all k. By assumption and applying themap toI ⊗B, ∑

k

tr(DkDk†)EkBEk

† = NaB. (2)

If for somek, Ek 6∝ I, we can find|ψ〉〈ψ| such thatEk|ψ〉〈ψ|Ek†’s one-dimensional range does

not contain|ψ〉. Because for alll, tr(DlDl†) > 0, the left side of Equation 2 also has this property,

contradicting the identity. HenceEk = αkI for eachk and the result follows.

Theorem 5 characterizes unilocal maps but has the disadvantage that we have to refer explicitlyto the unilocal Lie algebras, thus requiring more information about locality than that provided byhl alone. This suggests the following problem:

Problem 6 Are separable quantum maps that lift tohl LOCC? Are theyLOCC if they lift to bothha andhb?

If the answer to this problem is “no”, then we are interested in the question of whether theexplicit separable quantum maps that are liftable tohl generate all separable quantum maps byconditional composition.

In order to be able to conditionally compose explicit separable quantum maps that are LOCCwithout departing from LOCC, we need the explicit representations to have the additional propertythat they can be LOCC implemented in such a way that the communication record reveals which ofthe operators in the sequence occurred. Following our convention for using the adjective “explicit”,we call an explicit quantum map with this property anexplicit LOCC map.

Problem 7 Are there explicit separable quantum maps that areLOCC but not explicitLOCC?

If the answers to this problem and to Problem 6 are “yes”, thenone has to consider the strength-ening of the questions in Problem 6 where “separable” is replaced by “explict separable” and“LOCC” by “explicit LOCC”. This is required so that conditional composition can be used with-out leaving LOCC. Here is one case where we can prove that a family of quantum maps is explicitLOCC.

Theorem 8 Let C = (Dk ⊗ Ek) be an explicit separable quantum map that lifts tohl with theadditional property that(Dk

†Dk)k is linearly independent. ThenEk = γkUk with Uk unitary. Inparticular,C is an explicitLOCCmap.

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Proof: Using the identification of linear functionals with operators, consider linear functionalsλ of g induced byA⊗B with tr(B) = 0. The restriction ofλ to hl is induced by tr(A)I ⊗B ∈ hl.The restriction has only scalar dependence onA. Restricting after applyingC gives the linearfunctional induced by

∑

k

tr(DkADk†)I/Na ⊗ EkBEk

† +∑

k

DkADk† ⊗ tr(EkBEk

†)I/Nb (3)

Note that because tr(B) = 0, and the assumption that the map is trace preserving, the contributionto I⊗ I vanishes. Because of liftability, the same scalar dependence applies to this expression. Bycyclicity of the trace, tr(DkADk

†) = tr(ADk†Dk). Because theDk

†Dk are independent, we canchooseAl such that tr(AlDk

†Dk) = δlk. Hence the following are all scalar multiples of the sameoperator, where the scalar is independent ofB:

Ol = I/Na ⊗ ElBEl† +∑

k

DkAlDk† ⊗ tr(EkBEk

†)I/Nb. (4)

Computing the partial trace over the first system, we get

tra(Ol) = ElBEl† +∑

k tr(AlDk†Dk)tr(BEk

†Ek)/Nb by cyclicity of trace,= ElBEl

† +∑

k tr((Al ⊗ B)(Dk†Dk ⊗ Ek

†Ek))/Nb because tr is multiplicative for⊗,= ElBEl

† + tr(Al ⊗ B)/Nb becauseC is a quantum map,= ElBEl

† because tr(B) = 0.

Consequently, the operatorsElBEl† are all proportional with constant of proportionality indepen-

dent ofB. ConsiderE = Er. We have

ElBEl† = αlEBE

† (5)

for all tracelessB, whereαltr(Ar) = tr(Al). Reformulating, we get that for all tracelessB,tr(BEl

†El) = αltr(BE†E). HenceEl†El = αlE

†E + βlI for someβl. The trace-preservingcondition requires that

I ⊗ I =∑

k

Dk†Dk ⊗ Ek

†Ek (6)

=∑

k

Dk†Dk ⊗

(αkE

†E + βkI)

=

(∑

k

αkDk†Dk

)⊗E†E +

(∑

k

βkDk†Dk

)⊗ I (7)

Suppose that the traceless part ofE†E is not zero, Then∑

k αkDk†Dk = 0, which is possible

only if αk = 0 for all k (by independence). But by constructionαr = 1, soE†E is a multipleof the identity, henceE = Er is a multiple of a unitary operator, sayEr = γrUr. Returning tothe trace-preserving condition (Equation 6) and using the fact thatr was arbitrary, we find that∑

k γkDk†Dkγk = I. This makesD = (γkDk ⊗ I)k a unilocal quantum map. TheUk can be

implemented conditionally on whichDk occurs in a unilocal implementation ofD, henceC isLOCC.

12

Corollary 9 LetC = (D1 ⊗ E1, D2 ⊗ E2) be an explicit separable quantum map that lifts tohl.ThenC is explicitLOCC.

Proof: The result follows by Theorem 8 unlessD2†D2 = α′D1

†D1 andE2†E2 = β ′E1

†E1 forsomeα andβ. In this case, using the trace-preserving condition,D1

†D1 ⊗E1†E1 ∝ I ⊗ I making

all operators proportional to unitaries. Such an map can be realized explicitly with LOCC by firstcreating a shared random variable, then implementing localunitaries conditional on the randomvariable.

Every explicit unilocal quantum map can be obtained as a composition of binary unilocal quan-tum maps, where abinary quantum map is an explicit quantum map consisting of two operators.The modifier “explicit” is assumed when using the modifier “binary”. We can therefore use thecorollary to characterize LOCC as the quantum maps obtainedby conditional composition of bi-nary separable quantum maps that lift tohl.

Instead of using liftability as the basis for generalizing LOCC and other classes of local maps,one can use the spectral properties of the constituent operators of an explicit quantum map. Thisidea is motivated by the following result:

Theorem 10 An operator inRe(hl) that has a maximal ground space is unilocal.

Maximal means maximal among ground spaces different fromH of operators inRe(hl).Proof: An operator inRe(hl) is of the formA ⊗ I + I ⊗ B. By subtracting a multiple of the

identity, we can assume thatA andB traceless, not both zero. If they are both non-zero, then theoperator’s ground space is strictly contained in that ofA⊗ I, hence not maximal.

For future reference, an operator whose traceless part is zero or satisfies the condition of The-orem 10 is said to bemaximally unilocal. Note that except forNa = Nb = 2, not all unilocaloperators inRe(hl) are maximally unilocal. However, two maximally unilocal operatorsC1 andC2 with ground spacesH1 andH2 such thatH2 = eDH1 for someD ∈ hl act on the same side.Also, if C1 is maximally unilocal andC2 = eDC1e

−D with D ∈ hl, thenC2 is unilocal and actson the same side. We call a family of operators contained in the span ofeDCe−D |D ∈ hl withC maximally unilocal anm-compatibleunilocal family. With this definition, we have:

Theorem 11 An explicit unilocal quantum map consists of an m-compatible unilocal family ofoperators.

Proof: Every unilocal one-dimensional projector is maximally unilocal, and the span of theconjugates underehl of one such projector consists of all operators acting on thesame side.

Using this theorem, we can characterize LOCC as the set of quantum maps obtained by condi-tional composition of explicit m-compatible quantum maps.However, this characterization is notdirectly related to the definition of separable maps. To do sorequires introducing explicit quantummaps whose operators are exponentials of members of an m-compatible family. Also note that inaddition to using linear closure in the definition of m-compatibility, we could have used closureunder commutators (Lie bracket). In the bipartite setting,this makes no difference. Alternatively,we could have left out linear closure and just used conjugation underehl. We do not know whetherconditional composition of the resulting quantum maps yields LOCC. See the discussion of thistopic in Section III B.

13

C. Communication Complexity

In the study of multiparty protocols, an important issue is the communication complexity ofconverting one state to another using LOCC maps. The communication complexity is defined asthe number of classical bits that need to be communicated. The communication complexity of aparticular LOCC map to a given state can be determined from a representation as a conditionalcomposition. This can be done by adding the resources used ineach round. The contribution froma round depends on the previous map in the sequence of conditional compositions, as we nowexplain. Suppose that the initial state’s density matrix isgiven byρ, the total explicit quantummap before the round under consideration isC, and this is then conditionally composed with thefamily of unilocal explicit quantum mapsBk. In general, given an explicit quantum mapD appliedto density matrixρ, the average number of bits needed to communicate the outcomes is given byH(D, ρ) = −

∑k pk log pk, wherepk = tr(ρDk

†Dk) is the probability of outcomeDk. This isof course an asymptotic expression assuming knowledge ofρ. In other cases one might prefer tojust uselog |D| as the number of bits required. In any case, the contributionto the communicationcomplexity of the current round is the average communication complexity for transmitting theinformation in the outcomes of the conditionally applied maps. This quantity is given by

∑

k

tr(ρCk†Ck)H(Bk, CkρCk

†/tr(ρCk†Ck)). (8)

The contributions from each round are added up to obtain the communication complexity of thesequence of conditional compositions. Depending on the application, the contribution of the lastround can be omitted as its outcomes need not be communicatedto implemented the quantummap. Note that if the detailed outcomes in one round are not required for conditioning in the nextrounds, then the explicit maps can be modified to defer these outcomes until the last round, whichis one reason to omit the contribution of the last round.

In general, the goal is to implement a given communication task with (near) minimum commu-nication complexity. By determining the complexity according to Expression 8, we can generalizecommunication complexity to any scheme for defining a familyof quantum maps by conditionalcomposition, including the generalized local maps to be introduced for the Lie algebraic setting inSection III.

D. Resource Scaling

An important aspect of information theory involves asymptotic characterizations of the rela-tionships between information resources and of the complexity of tasks. To asymptotically scaleup a problem, one usually creates tensor copies of the bipartite states involved and then investi-gates their relationships in the context of the now much larger bipartite system. The relationshipbetween the local Lie algebras of the individual bipartite subsystems and the one obtained afterforming the tensor products requires a construction other than the usual products. We did not findan obvious way of implementing such a construction that doesnot rely on knowledge of additionalstructure. It may be the case that one must have knowledge of how the representation ofhl wasconstructed. Nevertheless, there are a few things we can saythat may help in better understandinghow resources can be scaled and how to implement asymptotic analyses.

We construct the spaceH = Hab ⊗ . . .⊗Hab as ann-fold tensor product of copies ofHab. Lethl,k be the local Lie algebra acting on thek’th factor. LethL be the local Lie algebra forH, where

14

H is bipartitioned intoHa ⊗ . . .⊗Ha andHb ⊗ . . .⊗Hb. Defineha,k, hA, hb,k andhB likewise.The group of permutations onn elements acts onH by permuting the tensor factors. The goal isto establish howhL relates to thehl,k. It suffices to consider the casen = 2, because we can viewhL as the smallest Lie algebra that contains the appropriate Lie algebras obtained for each pair offactors.

LetG2(hx) (x ∈ a, b, l) be the set of operatorsC onHab ⊗Hab such that for all operatorsXonHab, tr1(C(X† ⊗ I)) ∈ hx,2 and tr2(C(I ⊗X†)) ∈ hx,1. Here, tri is tracing out thei’th factorwith respect to the tensor productHab ⊗Hab. In words,G2(hx) is the set of operators which looklocally like operators inhx.

Theorem 12 G2(ha) = hA,G2(hb) = hB, butG2(hl) strictly containshL.

Proof: The definition ensures thathA ⊆ G2(ha). Let C ∈ G2(ha). We can writeC =∑klrs αklrs(Ak ⊗Bl)⊗ (Ar ⊗Bs) with (Ak)k and(Bl)l orthonormal bases of operators including

the identity. The ordering of the tensor product is according to (Ha ⊗Hb)⊗ (Ha ⊗Hb). Supposethatαkl0rs is non-zero for somel0 with Bl0 6= I. Then usingX = Ar ⊗ Bs in the definition ofG2

and tracing out we get∑

kl αklrsAk ⊗ Bl, which is not inha due to the termBl0 . By symmetry,this establishes the first two identities. The third statement follows from the observation that anyoperator of the form(A⊗ I)⊗ (I ⊗B) is inG2(hl). If A andB are traceless, this operator is notin hL.

The above theorem provides ways of constructinghA andhB but nothL. However, one canconstructhL as the Lie algebra generated byhA andhB. This depends on the bipartition onlythrough its emergence from having the two unilocal Lie algebras.

Another way in which one can attempt to constructhX involves using a group of unitary op-erators that extends the permutations groupSn acting on the factors.Sn by itself is insufficient,in the sense that the Lie algebra generated bygCg† for g a permutation operator andC ∈ hx,k isjust

⊕k hx,k. A sufficiently large extension suffices. An example is the groupU ⊗ V , with U and

V acting on the tensor products of theHa andHb factors, respectively, which generateshX fromhx,1 by conjugation. The problem is whether such an extension canbe chosen naturally. An ideathat does not work but might have some independent interest is to consider the Lie algebrah′Xgenerated bygCg† with C ∈ hx,1 andg a unitary operator in the group algebra generated by thepermutation operators. To see that this does not yield the desired Lie algebras, lets be the swapoperator. Theng = (I + is)/

√2 is unitary, butg((A⊗ I)⊗ (I ⊗ I))g† is not inhA.

E. Measures of Entanglement

For pure states|ψ〉 of a bipartite system the generally accepted and information-theoreticallymeaningful measure of entanglement is given by the von Neumann entropy of either one of thereduced density matrices for|ψ〉 [51]. Thus, the entanglement of|ψ〉 can be computed as theShannon entropy of the spectrum of the reduced density matrix on the first (or, equivalently, thesecond) system. Forhl-states, the underlying Hilbert space is not directly accessible. However,there are natural complexity measures associated with the convex structure of these states. Todefine such measures, letS be a Schur-concave function of probability distributions.By definition,Schur-concave functions are permutation invariant and concave (see for example [52], pp. 40).That is, ifp andq are two probability distributions of the same length where the probabilities ofq are a permutation of those ofp, thenS(p) = S(q); and if p = rp1 + (1 − r)p2 for r ≥ 0,thenS(p) ≥ rS(p1) + (1 − r)S(p2). An example of a Schur-concave function is the Shannon

15

entropy. For a pure state|ψ〉 defineS(|ψ〉) to beS evaluated on the spectrum of the reduceddensity matrices. For anhl-stateλ, define

S(λ) = infS(p) |λ =∑

k

pkλk with λk hl-pure. (9)

We will routinely overload the functionS. Which definition is intended is communicated throughthe argument. So far, the argument type can be a probability distribution, a state inHab or anhl-state.

Theorem 13 If thehl-stateλ is induced by a pure state|ψ〉 on the bipartite system, thenS(λ) =S(|ψ〉).

Proof: Using the Schmidt decomposition, we can write|ψ〉 =∑

k

√pk|φk〉⊗|ϕk〉 with (|φk〉)k

and(|ϕk〉)k orthonormal bases andS(|ψ〉) = S(p). If the λk are the purehl-states induced by|φk〉 ⊗ |ϕk〉, thenλ =

∑k pkλk. It follows thatS(λ) ≤ S(|ψ〉).

To prove thatS(λ) ≥ S(|ψ〉), write λ =∑

k pkλk, with λk hl-pure andS(p) arbitrarily closeto S(λ). To be specific,S(p) ≤ S(λ) + ǫ. By Theorem 2, theλk are pure product states. Letλkbe induced by|φk〉 ⊗ |ϕk〉. Defineρ =

∑k pk|φk〉〈φk| ⊗ |ϕk〉〈ϕk|. Then trb(ρ) =

∑k pk|φk〉〈φk|

and is equal to the corresponding reduced density matrix for|ψ〉. It therefore suffices to prove thatS(p) is at leastS evaluated on the spectrum ofρa. One way to see this it is to writeρa = APA†,whereA consists of unit-length columns (the|φk〉) andP is the diagonal matrix with thepk’s onthe diagonal. The eigenvalues ofρa are the same as those ofP 1/2A†AP 1/2. This matrix has thepkon the diagonal. The result now follows from the fact thatp is a transformation of the spectrumby a doubly stochastic matrix (see, for example, [29], page 513), doubly stochastic matrices areconvex combinations of permutation matrices (see, for example, [53], page 36; [29], page 574),and concavity ofS.

Theorem 13 makes it possible to introduce entanglement measures without reference to the un-derlying pair of systems while being faithful to the known measures for such systems. We extendthe entanglement measureS to mixed states by a second minimization over convex representationsas pure states [54]. To do so, consider ag-stateλ induced by the density matrixρ. With respectto the convex set ofg-states,λ is pure iff ρ is pure. The distinction between separability and en-tanglement can be seen to be one associated with the purity ofa state from the points of view ofgandhl. Thus, we define

S(λ; hl) = inf∑

k

pkS(λkhl) |λ =∑

k

pkλk with λk g-pure (10)

Because of the isomorphism between density matricesρ andg-states, this expression defines anentanglement measure for arbitrary bipartite density matrices. In anticipation of the generaliza-tions to come, we explicitly introduced the Lie algebrahl as a parameter.

Suppose thatS(p) = 0 iff p is pure, that is,pk = δjk for somej. We call such anS proper.Then ag-stateλ satisfiesS(λ; hl) = 0 iff it is a mixture of product states, which justifies thinking ofS as an entanglement measure. Several properties are desirable of an entanglement measure [51].For example, the measure should be convex and it should be non-increasing under LOCC maps.Both of these properties are satisfied byS as defined above [48].

Entanglement measures can be based on asymptotic convertibility of states with respect to afamily of local maps. For example, one can defineR(ρ, σ) as the asymptotic supremum ofr/s,

16

wherer is the number of asymptotically good copies ofρ that can be constructed froms copiesof σ given any number of additional product states and using separable quantum maps. For moreprecise definitions of this sort, see [55]. If there is a reasonable choiceσ of a maximally entangledstate, thenR(ρ, σ) can be considered to be an entropy of formation andR(σ, ρ) an entropy ofdistillation. By varying the constraints on the quantum maps different measures are obtained.

III. THE LIE-ALGEBRAIC SETTING

To generalize the notions introduced in the previous section requires not much more than re-moving the connection between the local Lie algebra and the bipartite system. As a consequencewe will learn that product states are generalized coherent states.

We fix a finite dimensional Hilbert spaceH (Hab in the bipartite setting) and consider statesfrom the point of view of various†-closed, complex Lie algebras of operators acting onH. Ul-timately, we consider families of Lie algebras(hx)x acting onH and ordered by inclusion. Butfirst we consider one†-closed Lie algebrah. By default we assume thatI is a member of ouroperator Lie algebras. The set of traceless operators ofh is denoted byh. The abstract Lie algebrafaithfully represented byh is denoted byh. The assumption thath is †-closed implies thath isreductive (see, for example, [56], Sect. 1.7). AreductiveLie algebrar is one that consists of thedirect product of an abeliana and a semisimple Lie algebras (see, for example, [56], Sect. 1.7,or [33], p. 102). Thedirect productis in the category of Lie algebras and homomorphisms ofLie algebras and corresponds, after exponentiation, to thedirect product of groups. In this caseit means that as vector spaces,r = a ⊕ s, wherea commutes withs. For Lie algebras,x andycommuteiff [x, y] = 0. A semisimpleLie algebra is one which is a direct product of simple Liealgebras, where asimpleLie algebra is one that is not abelian and has no proper ideals. Reduc-tiveness of our Lie algebras is useful because the finite-dimensional semisimple Lie algebras andtheir representations have been completely classified (see, for example, [33]). Ifh is irreducible asa set of operators, then the abelian part consists only of multiples of the identity operator, and thesemisimple part consists of the traceless operators.

The two examples forh to keep in mind arehl in the bipartite setting and the set of generatorsof the spatial rotations of a spin-1 particle. In the second example, the Hilbert space is threedimensional with basis|−1〉, |0〉 and|1〉 corresponding to the three states with definite spin alongz. The Lie algebrah is spanned by the identity together with the spin operatorsJz, Jx andJy. Thecorresponding abstract Lie algebra is1 × sl2C, where1 is the one-dimensional Lie algebra. Aslinear spaces, this is the same as1⊕ sl2C, the operator× emphasizes the fact that the constructionis a direct product, so that the two Lie algebras commute. Forthis example we takeg to consist ofall operators.

Before proceeding, we recall the basic properties of semisimple Lie algebras that are neededto define generalized coherent states and relate them our characterizations of product states in thebipartite setting.

A Cartan subalgebra ofh is a maximal abelian subalgebra whose elements are diagonalizable(that is,semisimple). According to a fundamental result for Lie algebras, Cartan subalgebras existand are conjugate (hence isomorphic) with respect to an operator in eh ([33], pp. 81-87; [35],Thm. D.22, p. 492; [34], Thm. 4.1.2, p. 263). Every diagonalizable operator inh is contained insome Cartan subalgebra. If the operator is hermitian, the Cartan subalgebra can be chosen to be†-closed. Letc be a Cartan subalgebra ofh, thenH can be decomposed into the joint eigenspacesfor c, H =

⊕αHα, where theα are distinct linear functionals onc such that for|ψ〉 ∈ Hα and

A ∈ c, A|ψ〉 = α(A)|ψ〉. ([33], p. 107; [35], p. 199 eq. (14.4)). TheHα are called theweight

17

spacesfor c and theα are called theweights. In general, a weight for a Cartan subalgebra isa linear functional for which there exists a finite dimensional representation with a non-emptycorresponding weight space. The abstract Lie algebrah can be represented on itself by the Liebracket. This is called theadjoint representation ofh. The weights for this representation arecalled roots. It turns out that the geometrical properties of the roots determine the Lie algebra.The roots are in effect also linear functionals onc. There are special sets of roots calledsimpleroot systems(or bases) that span the linear functionals onc and have the property that every rootis either a positive, or a negative integral combination of simple roots. The former are calledpositiveroots. The definition depends on the choice of simple roots, but not in a crucial way,because all simple root systems are isomorphic via a specialkind of isomorphism (a member ofthe so-called Weyl group, [33], p. 51; [35], Prop. D.29, p. 494). The weights can be partiallyordered by definingα ≤ α′ if α′ −α is a positive integral sum of simple roots. With this ordering,in an irreducible representation, there is a unique minimumweight, whose weight space is one-dimensional ([33], pp. 108–109; [35], Prop. 14.13, pp. 202–203). The minimum weight statedepends on the choice of Cartan subalgebra and simple roots.However,eiRe(h) acts transitivelyon the set of minimum weight vectors. Furthermore, every minimum weight vector can be obtainedby means of a†-closed Cartan subalgebra ofh. The minimum weight space has the property thatit is annihilated by operators inh which are in root spaces associated with negative roots. In fact,this is another characterization of the minimum weight space (see the definition and theorem in[33], p. 108). Usually, treatments of semisimple Lie algebras focus on the maximum weights of arepresentation. Here we choose to use the equivalent minimum weights because of the relationshipto ground states of Hamiltonians. The basic properties of Cartan subalgebras and the notions ofroots and weights extend from semisimple to reductive Lie algebras by adjoining the abelian part.

A family of generalized coherent states consists of an orbitof a dynamical group acting on astate space [23, 24]. According to this definition, every state is in a family of generalized coherentstates. As a result, an important part of the theory of generalized coherent states is to choose thoseorbits that best generalize the properties of the coherent states familiar in optics. In our case, thedynamical groups are Lie groups generated by semisimple Liealgebras. If the goal is to choosestates that are in a sense the most classical, then there are strong arguments for choosing the mini-mum weight states of a representation of the Lie group. Theorem 14 below provides some of thesearguments. We therefore use the termgeneralized coherentstate, or simplycoherentstate, to referspecifically to minimum weight states of a Lie algebra. Because we only consider finite dimen-sional representations, our treatment does not directly apply to the conventional coherent states ofoptics, for example. In this case, the relevant Lie algebra is the Heisenberg algebra, which is notsemisimple (or reductive). The standard†-closed representation is therefore necessarily infinite.The theory of coherent states suggests that extensions to such Lie algebras and representations arepossible [23].

A. Purity, Coherence and Entanglement

For a†-closed Lie algebra of operatorsh onH, defineh-states as before as linear functionalsonh induced by a state’s density matrixρ according toλ(A) = tr(ρA). Observe again that the seth+ of h-states is convex closed.Pureh-states are extreme points ofh+. Suppose that theh-stateλis induced by the density matrixρ. We can projectρ ontoh with respect to the trace inner product.Denote the projection map ontoh byPh. Becauseh is †-closed, the projectionPh(ρ) is a hermitianoperator inh. Furthermore,λ is also induced byPh(ρ), that is,λ(A) = tr(Ph(ρ)A) = tr(ρA) forA ∈ h. Note that in general,Ph(ρ) is not positive. For example, letρ be the density matrix for|1〉

18

in the spin-1 example. Another important observation is thatPh(ρ) depends only onλ. That is, ifρ andρ′ both induceλ, thenPh(ρ) = Ph(ρ

′).We now assume thath acts irreducibly onH. If it does not act irreducibly, decomposeH into

irreducible invariant subspaces forh and consider each of these subspaces separately. Define theh-purity of λ as tr(Ph(ρ)

2), whereλ is induced by the density matrixρ. This is of course thelength ofPh(ρ) according to the trace-inner-product norm. Theh-purity is bounded above by theconventional purity tr(ρ2), which is theg-purity with g the algebra of all operators onH. Thisgeneralization of purity is useful because according to Theorem 14 below, the pureh-states areexactly the states with maximumh-purity.

The goal of the remainder of this subsection is to give a number of useful characterizations ofpureh-states. In particular, we show that they are exactly the coherent states forh. We first statethe characterization theorem and then discuss the equivalent characterizations before proving thetheorem.

Theorem 14 The following are equivalent for a density matrixρ inducing theh-stateλ:

(1) λ is a pureh-state.

(2) ρ = |ψ〉〈ψ| with |ψ〉 the unique ground state of someH in Re(h).

(3) ρ = |ψ〉〈ψ| with |ψ〉 a minimum-weight vector (for some simple root system of someCartansubalgebra) ofh.

(4) λ has maximumh-purity.

(5) ρ is a one-dimensional projector ineh.

This theorem is a synthesis of various largely known resultsin the representation theory ofsemisimple Lie algebras and coherent states. Statements (1) and (2) are motivated by Theorems 2and 1, respectively. (2) also provides an interpretation of many meanfield ground states as coherentstates. This is because meanfield Hamiltonians are often expressed as operators in a small Liealgebra, in particular, operators quadratic in the creation and annihilation operators.

Statement (3) is one of the definitions of generalized coherent states. For other characterizationsof generalized coherent states, see [23, 24].

Statement (4) is a version of the minimum variance principle for coherentstates [57, 58]. Thevariance of a state|ψ〉 with respect toRe(h) is computed as the expectation of an “invariantuncertainty operator”. For a state|ψ〉, this expectation is given by

∑

i

xixi −∑

i

〈ψ|xi|ψ〉〈ψ|xi|ψ〉, (11)

where(xi)i is a basis ofRe(h), and(xi)i is the dual basis with respect to the trace inner product.This is a linear function of theh-purity because the second sum is the negative of the purity up toa constant due to our inclusion of the identity operator.

Statement (5) is motivated by the results concerning the classical simulatability of fermioniclinear optics [59, 60]. Simulatability depends crucially on the fact that the initial state preparationsand the measurements outcomes can be expressed in terms of projectors ineh.

19

Proof of Theorem 14. (2) ⇒ (3): Let c be a†-closed Cartan subalgebra ofh containingH.We can perturbH slightly without affecting the ground space by adding a generic element ofc tomake sure thatH is generic, that is, so that the commutant ofH is c. ThecommutantofH is the setof elements ofh that commute withH. It therefore suffices to show that ground states of genericelements ofc are minimum weight for an ordering of the roots. Note that forno non-zero rootαis α(H) = 0, because otherwiseH is not generic. Thus we can call a root positive ifα(H) > 0,and there is some simple root system for which this coincideswith the definition of positive roots.A ground state is annihilated by the root spaces ofh that correspond to the negative roots. Thisimplies that it is a minimum weight state.

(3) ⇒ (2): Every minimum-weight vector|ψ〉 has minimum weight for some†-closed Cartansubalgebrac with root basisα1, . . . , αd. There is a hermitian memberH of c for whichαk(H) > 0for eachk. |ψ〉 is the ground state ofH.

(2) ⇒ (5): Letλ be the eigenvalue of|ψ〉 forH. Then the desired projector islimt→∞ e(−H+λ)t.((5) & ((2) ⇒ (3))) ⇒ (3): Let |ψ〉〈ψ| = limk e

−Hk , with Hk ∈ h. Then |ψ〉〈ψ| =

limk e−Hke−Hk

†

. The operators in the limit are now hermitian, which impliesthat they can bewritten in the forme−hk , with hk hermitian inh. For sufficiently largek, the minimum eigenvalueof hk must be unique. This eigenvalue must go to zero and the eigenvalue gapδk of hk goes toinfinity. Thus for sufficiently largek, the ground state|ψk〉 of hk is projectively well-defined.Because of ((2) ⇒ (3))), |ψk〉 is a minimum weight state. Minimum weight states form an orbitof eRe(h), a compact set. Thus there is a cluster point|ψ0〉 of the |ψk〉. It must be the case that|ψ0〉 ∝ |ψ〉. Hence|ψ〉 is minimum weight.

(4) ⇒ (1): By convexity of purity.(1) ⇒ (3): Let c be the†-closed Cartan subalgebra containing the projection ofρ into h. We

call this asupportingCartan subalgebra ofρ. Let Hα be the weight spaces with respect to thisCartan subalgebra. Thenλ is zero on the non-zero root spaces with respect toc. Sinceρ is amixture of normalized superpositions of weight vectors|vα〉 ∈ Hα, it follows thatλc is a convexcombination of weights. But the weights are all in the convexclosure of the set of minimumweights with respect to different orderings of the roots. Extremality therefore requires thatλc isgiven by a minimum weight. Let|ψ〉 be the corresponding minimum weight state. By choice ofc,λ is also induced by|ψ〉〈ψ|. The density matrixρ cannot have a contribution to the mixture withdifferent weight spaces, as otherwise,λc is in the strict interior of the convex closure of the set ofminimum weights. Thatρ = |ψ〉〈ψ| now follows from the fact that due to irreducibility ofh, theminimum weight spaces are one-dimensional.

Note that supporting Cartan subalgebras’ weight spaces generalize the Schmidt basis used todiagonalize reduced density matrices in the bipartite setting. See Theorem 23.

((1) & (3)) ⇒ (4): Because all minimum weight states are in the same orbit ofeiRe(h), everyminimum weight state has the same purity. By extremality andconvexity of purity, minimumweight states have maximum purity.

B. Local Quantum Maps

We can use Theorem 3 to generalize separable maps. Thus we define h-separablequantummaps to be those with an explicit form(Ak)k with Ak ∈ eh. To generalize LOCC maps, onecan always return to the multipartite setting by using the fact that by semisimplicity,h can beuniquely represented as a product of simple Lie algebrash = ×khk (see, for example, [33], p.

20

23). The state space then factors as⊗kHk, with hk acting onHk only. We defineh-LOCC mapsby conditional composition of explicit(hk+CI)-separable quantum maps. This definition is moregeneral than the usual notion of LOCC maps for multipartite systems becausehk can be differentfrom slnC or its representationhk need not be the first fundamental representation.

In the bipartite setting, we discussed two other ways in which LOCC maps can be characterized.One way used liftability to well defined maps ofh-states. The other used restrictions on theoperators based on their eigenspaces. We consider how theseideas can lead to other interestingfamilies of quantum maps.

A subfamily of the explicith-separable quantum maps is obtained by requiring that each op-erator lifts toh. Such quantum maps are calledexplicit h-liftable quantum maps. (Recall ourconvention for using the word “explicit”.) In the bipartitesetting, Theorem 4 implies that all suchquantum maps are mixtures of unitaries, a small subfamily ofthe LOCC maps. The conclusion ofTheorem 4 does not hold in general. For example, trivially, if h consists of all operators onH, thenall quantum maps are in this family. One nice property of the family of explicith-liftable quantummaps is that there is a straightforward proof of monotonicity for a large family of entanglementmeasures, see Theorem 29.

A family of quantum maps that includes the explicith-liftable ones consists of theh-separablequantum maps that are liftable toh. In the bipartite setting, this family may be larger than thefamily of LOCC maps, see Problem 6. In the general setting, wepose the following problem:

Problem 15 Is the family of quantum maps obtained by conditional composition of explicit h-separable quantum maps that are liftable toh strictly smaller than the family ofh-separable quan-tum maps?

Based on Theorem 8 and its corollary, one might want to consider the family of maps consistingof binaryh-separable quantum maps. Unfortunately, this family can betrivial in the sense that inmany cases it consist of mixed unitary quantum maps only. Forexample, consider the spin-1 Liealgebra and suppose that(A,B) is an explicit separable quantum map. We haveA,B ∈ eh andA†A + B†B = I. The operatorsA†A andB†B are ineh and can be written in the formeHA andeHB with HA andHB in Re(h). ThusHA = αI + ~x · ~J . With a suitable rotation, we can assumethatHA = α + βJz. This ensures thateHA is diagonal in the basis|−1〉, |0〉, |1〉 and has diagonalentrieseα−β , eα, eα+β. It follows thateHB is diagonal also, and hence of the same form withα′

andβ ′. Their sum isI, and it can be checked that the solutions satisfyβ = β ′ = 0. HenceA andB are proportional to unitaries.

One idea for avoiding the possible triviality of binaryh-separable quantum maps is to usek-aryquantum maps. That is, consider extremalk-ary h-separable quantum maps. A quantum map isextremalif its action on density matrices is not a convex combinationof other quantum maps.Because mixed unitary quantum maps are not extremal unless they are unitary, the spin-1 exampleshows that there may be no such extremal quantum maps fork = 2. Let kmin be the minimumk > 1 for which such quantum maps exist. Let the family ofminimally generatedseparablequantum maps consist of explicit quantum maps obtained by conditional composition of unary orextremalkmin-ary h-separable quantum maps. Because of Corollary 9, this family is the family ofLOCC maps in the bipartite setting.

Problem 16 What is the relationship between the family of minimally generatedh-separablequantum maps,h-LOCCand andh-separable quantum maps?

Another family of quantum maps that might be interesting is obtained by adding the liftabilitycondition to the generators of the family in the above problem.

21

We now move on to considering families ofh-separable quantum maps that are characterizedby generators with large ground spaces. Based on Theorem 10,we can define amaximallyh-unilocal operator to be an operator inRe(h) whose ground space is maximal. These operatorshave a Lie algebraic characterization.

Theorem 17 Maximallyh-unilocal operators are the ones that are proportional to a an operatorof the dual basis to a simple root system of a†-closed Cartan subalgebra ofh.

The dual basis of a simple root system corresponds to the fundamental weights via the isomor-phism induced by the Killing form. The Killing form is the symmetric bilinear form associatedwith the trace in the adjoint representation. Thek’th fundamental weightλk for a simple rootsystem consisting of the rootsαl has the property that ifhl = [xl, yl] with xl andyl members of theroot space forαl and for−αl, respectively, thenλk(h) = 0 except forl = k. It also has minimumlength among weights satisfying this property. Fundamental weights are important because all therepresentations of a Lie algebra can be built from ones whoseminimum weight is fundamental.

Proof of Theorem 17. LetH ∈ Re(h) and choose a†-closed Cartan subalgebrac containingH and an ordering of the roots such that for positive rootsα, α(H) ≥ 0. Let (αk)k be the simpleroot system for this ordering. LetH0 be the ground space ofH. ThenH0 is a union of weightspaces ofc. By definition of the ground space, ifX is in the root space for a negative root,thenXH0 ⊆ H0. In particular,H0 contains the weight space for the minimum weightλ0 of thechosen ordering of the roots. Furthermore,H0 consists exactly of the weightsλ such thatλ− λ0is a positive integral combination of positive rootsα with α(H0) = 0. ThusH0 is non-triviallymaximal iffαk(H0) = 0 for all but onek = k0. Givenk0, the set of operators with this property isnecessarily one-dimensional and contains one that contributes to the dual basis of the simple rootsystem. This follows from the fact that the simple roots are abasis of the dual space ofc.

The maximallyh-unilocal operators fall into different classes dependingon the associated fun-damental weight. However, it is likely that ifh is simple, then the linear span of theeh conjugatesof a given maximallyh-unilocal operator is all ofh. We do not know whether this holds in gen-eral, but it is certainly the case forha andhb andg. This implies that if we define m-compatibilityas in the bipartite setting and close under conditional composition, we might get allh-LOCCmaps. So define ah-compatiblefamily of operators as a family consisting of theeh conjugates ofa maximallyh-unilocal operator.

Problem 18 Does conditional composition of explicit separable quantum maps with operatorsfrom anh-compatible family generate the family ofh-LOCC maps?

For now, the properties of the various families of quantum maps are largely unknown and offera fruitful area of further investigation.

C. Communication Complexity

Communication complexity can be defined exactly as in the bipartite setting for any of thefamilies of explicit quantum maps defined by conditional composition in the previous section.

22

D. Resource Scaling

The goal is to determine what might be reasonable choices of “scaled” Lie algebrash⊛n actingon H⊗n extending the action ofh on each factor so as to be consistent with the correspondingpicture for bipartite systems. It makes sense to require that h⊛n be contained inGn(h), the set ofoperatorsX with the property that ifY is an operator acting as the identity on thek’th factor ofH⊗n, then the partial trace ofXY onto thek’th factor is inh acting on this factor. In the bipartitecase, it was possible to obtain the desiredh⊛n by appealing to the two unilocal Lie algebras con-tained inh. We can similarly use any generating Lie subalgebras. That is, leth be generated byLie subalgebrashk. With respect to these Lie subalgebras, we can defineh⊛n as the Lie algebragenerated byGn(hk). In this case, it makes sense to definehk

⊛n = Gn(hk). At this point, we donot know to what extent this scheme is useful in analyzing theasymptotic relationships betweenstates from the point of view ofh. As a potentially interesting alternative, the scheme based onextensions of the permutation group discussed in the last paragraph of of Section II D can of coursebe applied to any Lie algebra of operators.

E. Measures of Relative Entanglement

From the point of view ofh, incoherent pure states ofH look like a mixture of coherent states.This is because theh-state induced by an incoherent state is a proper convex combinations of pureh-states. However, incoherent pure states can exhibit generalized entanglement provided that itis possible to refer to operators outside ofh. We therefore need access to observables in a largerLie algebra. Letg ⊃ h be a Lie algebra of operators onH. Theorem 14 applies tog as well, andin general, not all pureg-states are pure when restricted toh. Note that ag-state that restricts toa pureh-state state is necessarily pure. So it makes sense to call a pure g-stateh-coherentif itrestricts to a pureh-state.

The goal of this section is to find ways to quantify therelative entanglementof g-states withrespect toh. The idea is thath-coherentg-states are not entangled, while any other pureg-state isdefinitely entangled, but the extent of entanglement depends in some way on how far the state isfrom being pure when restricted toh. Once the entanglement of pureg-states has been quantified,this can be extended to arbitraryg-states.

Let S be a Schur-concave function of probability distributions.Then we can defineS(λ) forh-statesλ andS(λ′; h) for g-statesλ′ as we did in Section II E. In the bipartite setting,S(λ) isconcave as a function ofh-statesλ.

Problem 19 For whichh is S a concave function ofh-states?

ThatS(λ; h) is a convex function ofg-statesλ will be shown in the more general setting ofconvex cones, where we will also discuss the issue of monotonicity of S under the various notionsof generalized local quantum maps.

Another measure that can be used for quantifying generalized entanglement is based on purity.Let p(λ′) denote theh-purity of anh-stateλ′. We can define, for ag-stateλ,

p(λ; h) = sup∑

k

pkp(λkh) |λ =∑

k

pkλk with λk pure forg (12)

Thenp(λ; h) achieves its maximum exactly at the states that are mixturesof h-coherent states, andp(λ; h) is convex inλ. Mixtures ofh-coherent states are generalized separable states.

23

Observe that for bipartite pure states, the purity is a linear function of the Renyi entropy givenby −∑k p

2k where thepk are the eigenvalues of the reduced density matrices. In thiscase, the

Renyi entropy can be derived from the Schur-concave function S((pk)k) = −∑k p2k.

It is possible to define resource-based measures of relativeentanglement as discussed at the endof Section II E, with the caution that asymptotic versions ofsuch measures depend on whether auseful notion of scaling for resources can been found.

One advantage of relativizing measures of entanglement by using pairsh ⊆ g, is that one canbetter investigate properties of states on systems with a hierarchy of meaningful choices for Liealgebras. Multipartite systems are examples where this situation arises. For every subsets ofthe subsystems, there is the algebrahs of operators acting only on the subsystems ins, and thehs can be summed over a partition of the subsystems to obtain generalizations ofhl. These Liealgebras are ordered by inclusion. Given a state, one can, for every pairk ⊆ l, determine thestate’s generalized entanglement and use these quantitiesto characterize different types of statesand localize the extent to which they are entangled. Other examples with multiple, physicallymotivated Lie algebras are discussed in Section V A.

F. Other Measures

We mention two other types of relative entanglement measures for states that may generalizethe bipartite setting. One is based on the amplitudes in a representation of a state as a superpositionof coherent states, the other uses supporting Cartan subalgebras as a generalization of the Schmidtbasis. Since both of them can be extended to mixedg-states using the construction repeatedly usedabove (see Equation 10), we discuss them only for pureg-states. Since these are induced by purestates ofH and the relativization comes in through the extension, we define the measures for allpure states|ψ〉 ∈ H.

LetS be a Schur-concave function and|ψ〉 a state that induces a pureg-state. We can define anentanglement measure by minimizing theS-complexity of|ψ〉’s renormalized square amplitudesin writing |ψ〉 as a superposition of coherent states. Formally:

Sa(|ψ〉) = infS(p) | pk = |αk|2/∑

k

|αk|2 where|ψ〉 =∑

k

αk|ψk〉 with h-coherent|ψk〉.

(13)Note that by irreducibility ofh, every state is in the span of the coherent states forh.

Problem 20 Is Sa(|ψ〉) = S(|ψ〉) in the bipartite setting?

S(|ψ〉) is defined for the bipartite setting before Theorem 13.A limiting case of this definition is theh-rank of |ψ〉 defined as the minimum number of states

needed to represent|ψ〉 as a superposition of coherent states. Theh-rank is obtained as the limit ofthe Schur-concave functionsSr : p →

∑k p

1/rk asr → ∞. A special case of theh-rank has a long

history in quantum chemistry (see, for example [61], p. 69) and has been proposed in the contextof entanglement for fermions in [12], and for bosons in [13, 17].

Problem 21 What is the relationship between the amplitude-based (Sa(|ψ〉)) and the convexity-based (S(λ)) measures of entanglement for pureg-states?

Sa satisfies that for properS, Sa(|ψ〉) = 0 iff |ψ〉 is coherent forh. The measureSC(|ψ〉) basedon supporting Cartan subalgebras does not satisfy this. To defineSC(|ψ〉), let c be a supporting

24

Cartan subalgebra ofh for |ψ〉〈ψ|. Let Pα be the projectors onto the weight spaces ofc. We candefine

SC(|ψ〉) = inf S((|Pα|ψ〉|2)α), (14)

where the minimization is over supporting Cartan subalgebras. In the generic case, there is onlyone supporting Cartan subalgebra. Nevertheless it would benice if the minimization was redun-dant.

Problem 22 Is S((|Pα|ψ〉|2)α) as introduced above independent of the choice of supportingCar-tan subalgebra?

Note thatSC(|ψ〉) is zero for any|ψ〉 contained in a weight space for some Cartan subalgebraof h and that in general, such states are not coherent forh. Furthermore, these weight spaces areusually not one-dimensional. Nevertheless, this measure generalizes the bipartite setting.

Theorem 23 Assume the bipartite setting withh = hl. The weight spaces of a supporting Cartansubalgebra for|ψ〉 are the one-dimensional spaces associated with tensor products of Schmidtbasis elements for each side for some choice of Schmidt basis. HenceSC(|ψ〉) = S(|ψ〉).

This implies that for the bipartite setting, the answer to Problem 22 is “yes”.Proof: The projection of|ψ〉〈ψ| into hl is given by = ρa⊗ I/Nb+ I/Na⊗ ρb − I/Na⊗ I/Nb

whereρa andρb are the respective reduced density matrices. The supporting Cartan subalgebrasare the Cartan subalgebras that commute with. These are necessarily of the formca ⊗ I + I ⊗cb, whereca andcb are†-closed Cartan subalgebras ofha andhb that commute withρa andρb,respectively. Therefore,ca (cb) is generated by the projectors onto an orthogonal basisBa (Bb) ofeigenstates ofρa (ρb, respectively). The associated weight spaces are one-dimensional, spannedby tensor products of members ofBa andBb. Because the members ofBa andBb can be pairedto form a Schmidt basis for|ψ〉, the result follows.

IV. THE CONVEX CONES SETTING

Many of the notions introduced for†-closed operator Lie algebras can be generalized evenfurther. For example, we can work with any linear space of operators and study properties ofthe convex set of linear functionals induced by states. In fact, as pointed out in Section V A,there are physically interesting cases where this may be necessary. In this section we focus on theconvexity properties of the state space and investigate theextent to which local maps and measuresof generalized entanglement can still be defined and retain their features.

A. Convex Cones

A convex coneC is a subset of a real linear spaceU closed under positive linear combinations.That is, ifx, y ∈ C andp, q ≥ 0, thenpx+qy ∈ C. To avoid degeneracies, we assume thatU is thespan ofC. Let C consist of the non-zero elements ofC. The coneC is pointedif there is a linearfunctional tr (thetrace) on U such tr(C) > 0. Equivalently,C is pointed ifC ∩ (−C) = 0.We assume thatU is finite dimensional and thatC is closed in the usual topology forU . Forthe remainder of this paper, aconeis a closed, pointed, convex cone equipped with the positivelinear functional tr. For our purposes cones represent spaces of unnormalized pure and mixed

25

states. In the Lie-algebraic setting, the cone is given by the set of linear functionalsλ ∈ h∗ thatare nonnegative multiples ofh-states. The trace is given by evaluation ofλ at the identityI ∈ h.If λ is induced by the matrixρ, evaluation at the identity gives the usual trace, tr(ρ). We refer tomembersx ∈ C with tr(x) = 1 asstates. Thepure states ofC are extremal states ofC. Ourassumptions onC imply that every state ofC is a convex combination of pure states.

In the Lie-algebraic setting, we explicitly introduced a second Lie algebrag when discussingmeasures of relative entanglement. Before we introduced such measures,g was implicitly present,but was trivially associated with the set of all operators. This is because the fact thath-statesare induced by density operators plays a crucial role. In theconvex cones setting, there is noequally obvious way in which states are induced, so we explicitly introduce anouterconeD ⊆ V ,whose states induce the states onC via a linear mapπ : V → U satisfyingπ(D) = C, andx ∈ D, tr(x) = 1 implies tr(π(x)) = 1, that is,π is trace preserving. In the Lie algebraic setting,π is simply the restriction map: Ifλ is ag-state, thenπ(λ) = λh ∈ h+. We refer toC as theinnercone. Ifx is a pure state ofC thenπ−1(x) is convex closed and its extremal states are pure statesinD. Note that in the Lie-algebraic setting,π−1(x) for a pureh-statex is a pureg-state. We defineseparablestates ofD to be states in the convex closure ofπ−1(x) |x is pure inC. We denotethe cone generated by the separable states ofD asDsep (this depends onC). A pure statex of Dsep

satisfies thatπ(x) is pure inC.As we discuss the extent to which we can define suitable generalizations of various notions

to the convex cones setting, it is worth keeping in mind what the two cones correspond to in thebipartite setting. In this setting,D is isomorphic to the cone of positive operators onHa⊗Hb, withtr the usual trace functional. The trace one operators are the density matrices.C is determinedby the reduced density matrices. Formally,C is isomorphic to the cone of operators of the formA ⊗ I/Nb + I/Na ⊗ B + αI/Na ⊗ I/Nb with A, B traceless andA + αI/Na andB + αI/Nb

positive. The connection tohl-states is discussed in Section II B. The map fromD toC takesρ totrb(ρ)⊗ I/Nb + I/Na ⊗ tra(ρ)− tr(ρ)(I/Na ⊗ I/Nb).

B. Local Maps

A positivemap ofD is a linear mapA : V → V such thatA(D) ⊆ D. The mapA is tracepreservingif tr(x) = tr(A(x)) for all x. This definition corresponds to positive, but not necessarilycompletely positive maps in the Lie algebraic setting. Without the algebraic structure available forstates, it is not possible to define a unique “tensor product”of cones, as would be required todistinguish between positive and completely positive maps[62, 63] (cited in [64]). Because of theabsence of a suitable tensor product construction, we also do not have any suggestions for how toaddress asymptotic questions by resource scaling.

The family of positive maps ofD is closed under positive combinations and hence form a cone(without a trace). In the Lie-algebraic, or even the bipartite setting, the extreme points of this coneare not easy to characterize (see, for example, [64], p. 1927, [65]). However, the extreme points ofthe cone of completely positive maps are certainly extremality preserving in the following sense:A positive mapA of D is extremality preservingif for all extremalx ∈ D, A(x) is extremal.There are extremality preserving positive, not completelypositive, maps. An example is partialtransposition for density operators of qubits. We call a positive map that is a mixture of extremalitypreserving mapsq-positive. It is possible to recapture the idea of complete positivityby explicitlyintroducing a cone representing the “tensor product” extension ofD. This will be discussed afterdefining liftability. In the bipartite setting, the family of q-positive maps ofD is between thefamily of positive maps and the family of completely positive maps acting on density matrices on

26

Hab.The next step is to define a family of maps that generalizes theseparable maps. Call a positive

mapA of D C-separable if it is a mixture of extremality-preserving positive mapsAk that arealso extremality-preserving and positive forDsep. In the bipartite setting, this definition includesmaps such as the swap, which exchanges the two subsystems andis not separable, in addition tosome non-completely positive operations. Note that if the Lie-algebraic definition of separabilityis used, operations like the swap are excluded because they are not in the Lie group generated byhl: The swap induces an exterior automorphism ofhl. From the point of view of entanglement,including the swap can make sense because it obviously does not increase entanglement.

One tool used to narrow the family of separable quantum maps was based on liftability. Thedefinition of liftability immediately generalizes to our cones. We say that a positive mapA onDcan be lifted toC if A preserves the nullspace ofπ, or, equivalently, if there exists a positive mapA′ onC such thatπ(A(x)) = A′(π(x)). In this case, we say thatA′ is the lifting ofA toC.

Using liftability, we can add more cones to try to capture theidea of complete positivity or toexclude maps like the swap. For complete positivity, introduce one more coneE and positive trace-preserving mapσ : E → D (onto). In the setting where states are defined by density matrices on aHilbert spaceH of dimensiond, E represents the cone generated by density matrices onH⊗H′,with H′ of dimension at leastd2. With this cone in hand, we can try to get the completely positivemaps by considering only maps that are a mixture of extremality preserving mapsAk obtainedas liftings of extremality preserving positive mapsBk onE. Whether this works depends on theanswer to the following problem:

Problem 24 LetA be a positive map on operators ofH⊗H′ with dim(H′) ≥ dim(H)2. SupposethatA preserves the set of rank one operators and that it lifts to a mapA′ of operators onH. IsA′ completely positive?

To exclude the swap, it suffices to introduce cones included inC to represent density matricesonHa andHb and require liftability to both of these cones.

The other tool used to restrict separable maps involves operators with maximal ground spaces.It is not clear how to apply this tool to the convex cone setting since the distinction betweenpositive and negative eigenvalues is not easily recovered in the actionρ→ AρA†.

To be able to generate families of maps by a kind of locality preserving composition requiresthe idea of conditional composition based on explicit maps.An explicit positive mapA onD isgiven byA = (Ak) withAk extremality preserving positive maps. For explicit separability, theAk

are required to beC-separable. In addition, we can impose the liftability condition on eachAk. Wecall the latterexplicitC-liftable separable maps.The idea of Section III B to restrict the separablemaps by using certain minimal explicit separable maps can beapplied in the convex cones setting.However, without the strong symmetry present in the Lie-algebraic setting, the definition ofkmin

(Section III B) is unlikely to be as natural. However, one could investigate the families of mapsobtained by replacingkmin by 2, 3, . . ..

Conditional composition can be used to generate a family of maps as before. One can thenreadily generalize communication complexity to the resulting conditionally composed maps.

27

C. Measures of Relative Entanglement

The entanglement measures defined on the basis of a Schur-concave functionS are intrinsicallydefined using only convexity. Thus, for statesx ∈ C,

S(x) = infS(p) |x =∑

k

pkxk with xk pure, (15)

and for statesx ∈ D,

S(x;C) = inf∑

k

pkS(π(xk)) |x =∑

k

pkxk with xk pure. (16)

In general,S(x) is not concave, though this is the case in the bipartite setting and if the set ofstates is a simplex. In the latter case, the expression of a point as a convex combination of extremepoints is unique.

Problem 25 For which convex sets isS(x) concave for all Schur-concaveS?

Theorem 26 S(x;C) is convex inx.

Proof: Let y = px1 + (1 − p)x2 be a convex combination of statesx1, x2 ∈ D. We showthatS(y;C) ≤ pS(x1;C) + (1 − p)S(x2;C), from which the theorem follows. For every way ofexpressingxk =

∑l pklxkl as a convex combination of pure states ofD we havey =

∑l(pp1lx1l+

(1− p)p2lx2l). Thus

S(y;C) ≤∑

l(pp1lS(π(x1l)) + (1− p)p2lS(π(x2l))) by definition,= p

∑l p1lS(π(x1l)) + (1− p)

∑l p2lS(π(x2l)).

The last two sums can be chosen to be arbitrarily close toS(x1;C) andS(x2;C).

Purity as defined in the Lie algebraic setting does not generalize to the setting of convex conesunlessC has a well-defined center and satisfies that all its pure states are equidistant from thecenter in a natural metric.

D. Monotonicity for Explicit Liftable Maps

A desirable property for measures of entanglement is that they are nonincreasing under thefamily of maps that are considered to be local.

Problem 27 For which of the families of maps that we have introduced isS(x;C) (or, more specif-ically, S(x; h)) nonincreasing?

In the bipartite setting, it has been shown thatS(x; hl) is nonincreasing under LOCC maps [48].Here we show that this is the case in the convex cones setting for the family of trace-preservingexplicit liftableC-separable maps of cones. With the cones that arise in the bipartite setting, thisfamily of maps includes the explicit liftable separable quantum maps. (See also Problem 24.) Themonotonicity result is easy to see for the later family because in this case, the family of mapsconsists of mixtures of product unitaries.

28

For x 6= 0 in a cone, definex = (x) = tr(x)−1x to be the unique state proportional tox. Ifx = 0, definex = 0. We say that the functionΥ : D → R is explicitly nonincreasingfor thetrace-preserving explicit positive mapA = (Ak)k if for extremal statesx ∈ D,

Υ(x) ≥∑

k

pkΥ(Ak(x)), (17)

wherepk = tr(Ak(x)). The property of being explicitly nonincreasing is useful as a sufficientcondition for being nonincreasing.

Lemma 28 Suppose thatS(x;C) is explicitly nonincreasing for the trace-preserving explicit pos-itive mapA. ThenS(x;C) is nonincreasing forA.

The Lemma holds for anyΥ defined from its values on pure states according toΥ(x) =inf∑k pkΥ(xk) |x =

∑k pkxk with xk pure.

Proof: Let A = (Ak)k with Ak positive and writepk = tr(Ak(x)) To prove the lemma, firstconsider an extremalx. Then

S(A(x);C) = S(∑

k Ak(x);C)

≤∑

k pkS(Ak(x);C) by convexity,≤ S(x;C) by being explicitly nonincreasing.

For a nonextremalx, writex =∑

l qlxl with xl pure and∑

l qlS(xl;C) arbitrarily close toS(x;C).Note that for purey, S(y;C) = S(π(y)). Then

S(A(x);C) = S(∑

l qlA(xl);C) by linearity,≤ ∑

l qlS(A(xl);C) by convexity and trace preservation,≤∑

l qlS(xl;C) by extremality ofxl.

The result now follows because the the right hand side is arbitrarily close toS(x, C).

Theorem 29 If A is a trace-preserving explicit liftableC-separable map ofD, thenS(x;C) isexplicitly nonincreasing underA.

Proof: Let A = (Ak)k with eachAk liftable to C andC-separable. Writepk = tr(Ak(x)).Because of Lemma 28, it is sufficient to prove Inequality 17. Let x be a pure state ofD. Letπ(x) =

∑l qlyl be a convex representation ofπ(x) in terms of pure states ofC such thatS(q) is

arbitrarily close toS(x;C) = S(π(x)). We can find pure stateszl ∈ D such thatπ(zl) = yl. Thusx =

∑l zl + z for somez with π(z) = 0. With the appropriate interpretation ofAk(x)/pk when

pk = 0,

π(Ak(x)) = π(Ak(x)/pk)= π(Ak(

∑l qlzl + z)/pk)

=∑

l(ql/pk)π(Ak(zl)) sinceAk preserves the nullspace ofπ,

=∑

l(rlkql/pk)π(Ak(zl)) with rlk = tr(Ak(zl)).

29

SinceAk is C-separable andzl is pure inDsep, so isAk(zl). Thus, by definition,S(Ak(x);C) =

S(π(Ak(x))) ≤ S((rlkql/pk)l). To prove the desired inequality, bound as follows:

∑k pkS(Ak(x);C) ≤

∑k pkS((rlkql/pk)l)

≤ S(∑

k pk(rlkql/pk)l) by Schur concavity,= S((

∑k rlkql)l)

= S((ql)l) = S(q) becauseA is trace preserving,

which is arbitrarily close toS(x;C).

Conditional composition of trace-preserving explicit liftableC-separable maps preserves ex-plicit liftability and C-separability. Nevertheless it is useful to know circumstances that guaranteethat conditional composition preserves monotonicity ofS(x;C).

Theorem 30 Suppose thatS(x;C) is explicitly nonincreasing under the trace-preserving explicitextremality-preserving mapsA = (Ak)k andBk. Then it is explicitly nonincreasing under theconditional compositionE of A followed by theBk. E is also an explicit extremality-preservingmap.

Proof: Letx be a pure state ofD. ThatE is also an explicit extremality-preserving map is clear.Write pk = tr(Ak(x)) andqkl = tr(Bkl(Ak(x)))/pk. If pk = 0, setqkl = 0. To prove Inequality 17,compute∑

kl qklpkS( (Bkl(Ak(x)));C)

=∑

kl qklpkS( (Bkl(Ak(x)));C)

≤ ∑k pkS(Ak(x);C) because theBk are explicitly non-

increasing and theAk(x) are extremal.≤ S(x;C) becauseA is explicitly nonincreasing.

V. DISCUSSION

A. Further Examples and Extensions

The traditional setting for studies of entanglement is thatof bipartite systems. Our investigationshows that the more general theory based on Lie algebras exhibits most of the features associatedwith bipartite entanglement, and a significant number of these features can even be found in theconvex cones setting. As a result, we hope that the general theory provides new insights intobipartite entanglement and its generalizations to multipartite systems. Relativizing the idea ofentanglement has the advantage of being able to immediatelyuse the entire hierarchy of local Liealgebras and associated entanglement measures in the multipartite setting.

There are other settings where multiple, physically motivated Lie algebras occur. We givefour examples of such settings. The first example involves spectrum generating algebras (SGAs).SGAs are used to determine the spectrum (eigenvalues and eigenspaces) of quantum systems.SGAs provide the starting point for one or more chains of Lie subalgebras that are used for ob-taining algebraic bases of states and for expanding the Hamiltonian as a linear combination of

30

invariant (Casimir) operators belonging to the chains. When such an expansion contains only in-variant operators of a single algebraic chain, the system exhibits a dynamical symmetry, and thecorresponding spectrum can be calculated exactly using therepresentation theory of Lie subalge-bras. In the generic case where operators from multiple chains occur (that is, distinct dynamicalsymmetries coexist), the SGA approach may still make it possible to accurately represent theHamiltonian in terms of a small number of algebraic operators. Since they were introduced innuclear physics [66], SGA methods have been successfully applied to a variety of problems inmolecular, atomic, and condensed matter physics [67]. Using the approach developed here, onecould investigate the states’ relationships to the families of coherent states associated with the Liesubalgebras and quantify their relative entanglement.

An example we have already mentioned as motivation for our work involves fermions inNmodes. In this case, in addition to the algebra of all relevant operators, there is the Lie algebrahp of number-preserving operators quadratic in the creation and annihilation operators. Theseoperators can be expressed in the forma†Ma whereM is anN×N matrix. Thehp-coherent statesare the Slater determinants (see, for example, [20], p. 7) and represent independent fermions. Ifthe Lie algebra is enlarged toha consisting of all operators that are homogeneous quadraticinthe creation and annihilation operators, coherent states include BCS states [68], which can bethought of as describing independent fermion-pairs. Therefore, from this perspective, BCS statesare unentangled. On the other hand, they have entanglement with respect to the pairhp ⊂ ha ofLie algebras.

The example of fermions generalizes to anyons. Anyons as defined in quantum field theoryinclude particles with fractional exchange statistics [69]. To apply our theory to anyons requiresusing features of the convex cones setting. This is because the various sets of operators quadraticin the creation and annihilation operators are Lie algebrasonly for fermions and bosons [70].This was one of our motivations for extending the formalism.The convex cones can be definedas the set of linear functionals induced by states on sets of operators as before and investigatedusing essentially the same basic tools. Further investigation is required to determine whetherspecial properties not available in the convex cones setting still apply to quadratic anyonic operatorfamilies.

For bosons inN modes, four algebras frequently play an important role. Thesmallest,hplconsists of the operators of the forma†Ma, whereM is anN × N matrix anda is the vector ofannihilation operators of theN modes. This algebra generates the passive linear optics operators.A second Lie algebra,hs is the one that generates shifts in the canonical variables associated withthe modes and consists of operators at most linear in the creation and annihilation operators. TheLie algebrahal ⊇ hpl + hs consisting of all operators that are at most quadratic in theannihilationand creation operators is the algebra that generates all linear optics operators. Finally, there is thealgebra of all relevant operators. The usual coherent states of optics and harmonic oscillators arethehs-coherent states.

Although much of our proposal can be applied to the example ofbosons, caution is requiredin generalizing the finite dimensional theory to the infinitedimensional state spaces of bosonicmodes. In addition, algebras likehs are not semisimple or reductive, requiring an extension of thetheory, as can be done for the theory of coherent states [23, 24].

B. Relevance to Condensed Matter Physics

Entanglement, and our generalizations of it, may be important in the understanding of physicalphenomena. For example, the concept of “quantum phase transitions” [71] involves a qualitative

31

change in the behavior of correlation functions at zero temperature, i.e. in a pure ground state, asparameters in a system’s Hamiltonian are varied. In some cases an order parameter is associatedto the transition, in others a topological order. Since classical pure states cannot exhibit correla-tions, this is an essentially quantum phenomenon. Moreover, the presence of correlations betweensubsystems in a pure state can serve as a definition of entanglement, so quantum phase transitionsmight be considered to be due to a qualitative change in the nature of entanglement. Therefore,quantifying and classifying entanglement may help characterizing a quantum phase transition.Can measures of entanglement distinguish between a broken symmetry and a topological phasetransition? Can one classify quantum critical points? It isessential in this regard to have a notionof entanglement that need not make reference to locality or subsystems. Whether the correlationfunctions that best characterize a given phase transition are those of distinguishable subsystems(say, lattice sites) or some other kind of correlations (say, two-particle correlation functions forsystems of indistinguishable particles) may determine whether standard entanglement, or insteadsome generalization of it, provides appropriate concepts.Even standard entanglement is relativeto a distinguished factorization of a total Hilbert space into “local” ones, though this is usuallyunproblematic in quantum information settings. In other settings, such as many-body condensed-matter systems, different factorizations may occur on a more equal footing as “global” transfor-mations typically play a natural role. Thus a system of interacting bosons or fermions on a latticemay be viewed in terms of a factorization of the state space into distinguishable lattice sites, butthe Fourier transformation from position modes to momentummodes may provide an alternativefactorization; and it may also be that for some problems, correlations between particles, ratherthan modes, are relevant, taking us beyond the distinguishable-subsystems framework of standardentanglement theory.

The introduction of “quasiparticles”, or transformationssuch as the Jordan-Wigner transforma-tion [72, 73], may further alter the algebraic language we use to analyze the system; our motivationfor such transformations may be mathematical (easier solvability in one algebraic language than inanother) or physical (one algebra better exhibits the physical structure of the system’s dynamics, orof our interactions with it). In either case, the coherent states formalism is often known to be use-ful, and tools and concepts from quantum information theory, such as generalized entanglementmeasures, generalized LOCC and asymptotics may help as well. Initial work in the direction ofconnecting the information theoretic approach to entanglement to condensed matter can be foundin [74, 75, 76].

To give a more explicit example, Landau quasiparticles refer to thosedressedparticles of theoriginal interacting system that weakly interact as a result of transferringmost of the real interac-tions into the properties of the quasiparticles themselves. As a result, these quasiparticles may bequalitatively different from the original particles, an example of which is provided by the compos-ite fermions in the quantum Hall setup [77]. But how do we construct those quasiparticles? Weakinteractions can be related to weak correlations and, therefore,weak generalized entanglement. Ifone can re-express the original problem in a language such that the Hamiltonian operator belongsto the quadratic expressions in the language’s generating operators (for example, creation and an-nihilation operators) then we know that the quasiparticlesare non-interacting. Otherwise, we needto quantify the degree of “entanglement” (in the ground state, say) to determine whether thepar-ticlesgenerated by the language interact sufficiently weakly to behave as true quasiparticles. Theuse of hierarchical languages may help to address this issue[78].

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C. Conclusion

We have outlined a program whose goal is to tie together the theory of entanglement and thetheory of coherent states. We implemented the first few stepsof this program starting with theobservation that fundamental concepts of the theory of coherent states coincide with conceptsfrom the theory of entanglement. We extended this observation by providing general definitions ofthe key information-theoretic notions in entanglement theory. In particular, we introduced severalclasses of quantum maps to the Lie algebraic setting appropriate for coherent state theory thatgeneralize the idea of separable maps for multipartite systems and approach LOCC. The numerousopen problems attest to the richness of this program.

After noting that many of the notions that we generalized can, to some extent, be stated evenmore generally in the context of convex cones, we made this explicit by investigating appropriatedefinitions for convex cones. Except for the convex cones arising as spaces of linear functionals onoperator families induced by states, most such convex conesare not physically relevant. Neverthe-less, they help us appreciate what aspects of the various models are required in order to investigatedifferent properties of generalized entanglement and their information-theoretic implications.

The main conclusion of our program so far is that conventional entanglement is a special caseof a much more general theory with many of the same features. Furthermore, it is clear that entan-glement is a relative property of states, requiring that states that are mixed from one perspectivecan be pure from other, more powerful perspectives. Once this relativity is recognized, it is pos-sible to investigate relative entanglement of states when many physically motivated perspectivescoexist. Examples include multipartite systems, condensed matter systems, and systems whosedynamics is described by the chain of Lie algebras associated with a dynamical symmetry or aspectrum generating algebra.

APPENDIX A: COMPARISON OF THE SETTINGS FOR GENERALIZED ENTANGLEMENT

The following table shows the three settings as generalizations of the bipartite setting.

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Bipartite systems Lie algebras Cones

Structure: Ha ⊗Hb, a tensorproduct of two Hilbertspaces.

I ⊆ h ⊆ g, †-closed Lie algebrasof operators on a Hilbert spaceH.

Closed, convex conesC ⊆ D withtraces, andπ : D → C a linear,trace-preserving, map ontoC.

States: Full or reduced densitymatrices.

Linear functionals onh or g inducedby density matrices.

Trace-one elements ofC orD.

Specializationto bipartitesystems:

h = A⊗ I + I ⊗B, g is the set ofall operators onHa ⊗Hb.

C ≃ (A,B) | A (B) positive onHa (Hb), D ≃ C | C positive onH, π(C) = (tra(C), trb(C)).

Specializationto Lie algebras:

C (D) consist of the linearfunctionals induced onh (g) bypositiveρ onH asx → tr(ρx). π isthe restriction map.

Distinguishedpure states:

Product pure states. Coherent (or, equivalently, pure)h-states.

Statesx ∈ D such thatπ(x) is purein C.

Distinguishedmixed states:

Separable states. Convex combinations ofg-states thatrestrict to coherenth-states.

The coneDsep of separable states inD consisting of convex combinationsof statesx ∈ D such thatπ(x) is purein C.

Pure stateentanglementmeasures:

Von-Neumann entropyfor pure states.Unilateral purity.

S Schur concave,λ anh-state:S(λ)= infS(p) | λ =

∑k pkλk with λk

h-coherent,pk ≥ 0.h-purity.Measures based on amplitudes(Sa(λ)) and supporting Cartansubalgebras (SC(λ)).

Forx a pure state inC, S(x) =infS(p) | x =

∑k pkxk with xk

pure,pk ≥ 0.

Mixed stateentanglementmeasures:

Given pure stateentanglement measureS: S(ρ) =inf

∑k pkS(ρk) |∑

k pkρk = ρ, ρk is apure product state,pk ≥ 0.

Given anh-state measureS and ag-stateλ, S(λ) = inf∑k pkS(λk) |∑

k pkλk = λ, λkh is coherent,pk ≥ 0.

Given aC-measureS, x a state inD.S(x) = inf∑k pkS(π(xk)) |∑

k pkxk = x, π(xk) is pure,pk ≥ 0.

Properties ofentanglementmeasures:

Convex. Monotoneunder LOCC.

Convex. Monotone under explicitliftable separable quantum maps.

Convex. Monotone undertrace-preserving explicit liftableC-separable maps ofD.

Maximallyentangledstates:

Bell states. See [28]. Undefined.

Non-classicalityof entangledstates:

Bell inequalities. See [28]. Undefined.

Hierarchies: Add the unilateralalgebras.

Arbitrary family of operator Liealgebras ordered by inclusion.

Arbitrary family of cones, partiallyordered by trace-preserving ontomaps.

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Bipartite systems Lie algebras Cones

Local unitaryoperators:

Product unitaryoperators.

eiRe(h). Positive linear isomorphismf : D → D such thatπf = fπ forsome isomorphismf : C → C.Caution: Defs of local maps do notalways specialize to thecorresponding defs for Lie algebras.

Local operators:Product operators. eh. C-product maps:Extremality-preserving positive mapsf : D → D that preserve extremalityin Dsep also.

Separable maps:ρ →∑k AkρA

†k,

where theAk areproduct operators.

ρ →∑k AkρA

†k, whereAk ∈ eh.

Caution: Defs of local maps do notalways specialize to thecorresponding defs for bipartitesystems.

C-separable maps:x →∑k Ak(x),

where theAk areC-product maps

Unilocaloperators:

A⊗ I, I ⊗A. Operators ofh with maximal groundspaces? Operators whose action liftsto h-states?

C-product maps ofD that lift toC?

Compatiblefamilies ofone-sided localoperators:

Operators acting onthe same subsystem.

Operators conjugate undereh to onewith maximal ground spaces?

Undefined.

LOCC: Monoid generated byconditionalcomposition ofexplicit unilocalquantum maps.

Monoid generated by conditionalcomposition of explicit quantummaps consisting of compatiblefamilies?Monoid generated by conditionalcomposition of explicit liftableseparable quantum maps?. . .

Monoid generated by conditionalcomposition of trace-preservingexplicit liftableC-separable maps?

Communicationcomplexity:

Defined in terms of outcome probabilities in each step of a conditional composition.

Knownmonotonicity ofentanglementresults:

Under LOCC maps. Under explicitly liftable separablequantum maps.

Under trace-preserving explicitliftable C-separable maps.

Resourcescaling:

By tensor product,preserving orientationof the bipartition.

Grow Lie algebras over tensorproducts ofH using partial traces.May require additional structure?

Undefined.

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Acknowledgments

We thank Cristian Batista for the original motivation for investigating entanglement in asubsystem-independent way by pointing out that Slater determinants should be considered unen-tangled. We also thank the other “coherent tangles,” in particular Jim Gubernatis, Leonid Gurvits,Rolando Somma and Wojciech Zurek for stimulating discussions. We acknowledge support fromthe DOE (contract W-7405-ENG-36).

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