Exact and Asymptotic Measures of Multipartite Pure State Entanglement

Charles H. Bennett1, Sandu Popescu2, Daniel Rohrlich3, John A. Smolin1, and Ashish V. Thapliyal4 1IBM Research Division, Yorktown Heights, NY 10598, USA — bennetc, [email protected] 2Isaac Newton Institute, Cambridge University, Cambridge, UK and BRIMS, Hewlett-Packard Labs., Stoke Giﬀord, Bristol BS12 6QZ, UK — [email protected] 3School of Physics and Astronomy,Tel Aviv University, Tel Aviv, Israel — [email protected] 4 Dept. of Physics, Univ. of California, Santa Barbara, CA 93106, USA, [email protected] (December 28, 1999)

negligible (o(n)) amount of quantum communication. In an effort to simplify the classification of pure entangled states of multi (m)-partite quantum systems, we study exactly and asymptotically (in n) reversible transformations among nth tensor powers of such states (ie n copies of the state shared among the same m parties) under local quantum operations and classical communication (LOCC). For ex- I. INTRODUCTION act transformations, we show that two states whose marginal one-party entropies agree are either locally-unitarily (LU) Entanglement, first noted by Einstein-Podolsky-Rosen equivalent or else LOCC-incomparable. In particular we (EPR) [1] and Schrödinger [2], is an essential feature show that two tripartite Greenberger-Horne-Zeilinger (GHZ) states are LOCC-incomparable to three bipartite Einstein- of quantum mechanics. Entangled two-particle states, Podolsky-Rosen (EPR) states symmetrically shared among by their experimentally verified violations of Bell in- the three parties. Asymptotic transformations result in a equalities, have played an important role in establish- simpler classification than exact transformations; for exam- ing widespread confidence in the correctness of quan- ple, they allow all pure bipartite states to be characterized tum mechanics. Three-particle entangled states, though by a single parameter—their partial entropy—which may be more difficult to produce experimentally, provide even interpreted as the number of EPR pairs asymptotically in- stronger tests of quantum nonlocality. terconvertible to the state in question by LOCC transforma- The canonical two-particle entangled state is the tions. We show that m-partite pure states having an m-way Einstein-Podolsky-Rosen-Bohm (EPR) pair, Schmidt decomposition are similarly parameterizable, with the partial entropy across any nontrivial partition represent- 00 + 11 . (1) m m | i | i ing the number of standard “Cat” states 0⊗ + 1⊗ asymptotically interconvertible to the state in| question.i | Fori (We omit normalization factors when it will cause no general m-partite states, partial entropies across different confusion). The canonical tripartite entangled state is partitions need not be equal, and since partial entropies are the Greenberger-Horne-Zeilinger-Mermin (GHZ) state conserved by asymptotically reversible LOCC operations, a multicomponent entanglement measure is needed, with each 000 + 111 , (2) | i | i scalar component representing a different kind of entanglement, not asymptotically interconvertible to the other kinds. while the corresponding m-partite state In particular we show that the m=4 Cat state is not isentropic m m 0⊗ + 1⊗ . (3) to, and therefore not asymptotically interconvertible to, any | i | i combination of bipartite and tripartite states shared among the four parties. Thus, although the m=4 cat state can be is called an m-particle Cat ( m-Cat) state, in honor of prepared from bipartite EPR states, the preparation process Schrödinger’s cat. is necessarily irreversible, and remains so even asymptoti- More recently it has been realized that entanglement cally. For each number of parties m we define a minimal is a useful resource for various kinds of quantum infor- reversible entanglement generating set (MREGS) as a set mation processing, including quantum state teleporta- of states of minimal cardinality sufficient to generate all m- tion [3], cryptographic key distribution [4], classical com- partite pure states by asymptotically reversible LOCC trans- munication over quantum channels [5–7], quantum error formations. Partial entropy arguments provide lower bounds correction [8], quantum computational speedups [9], and on the size of the MREGS, but for m>2 we know no upper distributed computation [10,11]. In view of its central bounds. We briefly consider several generalizations of LOCC role [12] in quantum information theory, it is important transformations, including transformations with some proba- to have a qualitative and quantitative theory of entan- bility of failure, transformations with the catalytic assistance glement. of states other than the states we are trying to transform, Entanglement only has meaning in the context of a and asymptotic LOCC transformations supplemented by a multipartite quantum system, whose Hilbert space can

1 be viewed as a product of two or more tensor factors cor- Besides the gross distinction between entangled and responding physically to subsystems of the system. We unentangled states, various inequivalent kinds of entan- often think of subsystems as belonging to different ob- glement can be distinguished, in recognition of the fact servers, e.g. Alice has subsystem A, Bob has subsystem that not all entangled states can be interconverted by Bandsoon. local operations and classical communication. For ex- Mathematically, an unentangled or separable state is a ample, bipartite entangled states are further subdivided mixture of product states; operationally it is a state that into distillable and bound entangled states, the former can be made from a pure product state by local opera- being states which are pure or from which some pure en- tions and classical communication (LOCC). Here local tanglement can be produced by LOCC, while the latter operations include unitary transformations, additions of are mixed states which, though inseparable, have zero ancillas (ie enlarging the Hilbert space), measurements, distillable entanglement. and throwing away parts of the system, each performed Within a class of states having the same kind of entan- by one party on his/her subsystem. Mathematically, we glement (eg bipartite pure states) one can seek a scalar represent LOCC by a multilocal superoperator, i.e. a measure of entanglement. Five natural desiderata for completely positive linear map that does not increase such a measure (cf. [17–21]) are: the trace, and can be implemented locally with classical 1 It should be zero for separable states. coordination among the parties. Classical communica- • tion between parties allows local actions by one party to It should be invariant under local unitary transfor- be conditioned on the outcomes of earlier measurements • mations. performed by other parties. This allows, among other things, the creation of mixed states that are classically Its expectation should not increase under LOCC. • correlated but not entangled. It should be additive for tensor products of in- Mathematically speaking, a pure state ΨABC... is sep- • dependent states, shared among the same set of arable if and only if it can be expressed as| a tensori prod- observers (thus if ΨAB and ΦAB are are bipartite uct of states belonging to different parties: states shared between Alice and Bob, and E is an AB AB ΨABC... = αA βB γC ... . (4) entanglement measure, E(Ψ Φ ) should equal | i | i⊗| i⊗| i⊗ E(ΨAB)+E(ΦAB)). ⊗ ABC... A mixed state ρ is separable if and only if it can be It should be stable [22] with respect to transfer of expressed as a mixture of separable pure states: • a subsystem from one party to another, so that in any tripartite state ΨABC, the bipartite entangle- ρABC... = p αA αA βB βB γC γC ... , i| i ih i |⊗| i ih i |⊗| i ih i |⊗ ment of AB with C should differ from that of A i X with BC by at most the entropy of subsystem B. (5) For bipartite pure states it has been shown [17,19,21] that asymptotically there is only one kind of entan- where the probabilities pi 0and ipi=1. States that are not separable≥ are said to be entangled glement and partial entropy is a good entanglement or inseparable. P measure (E) for it. It is equal, both to the state’s entanglement of formation (the number of EPR pairs asymptotically required to prepare the state by LOCC), and the state’s distillable entanglement (the number of EPR pairs asymptotically preparable from the state by 1 General quantum dynamics can be represented mathemat- LOCC). Here partial entropy is the Von Neumann en- ically by completely positive linear maps that do not in- tropy S(ρ)=tr(ρlog ρ) of the reduced density matrix crease trace [13,14]. Such a map say can be written as 2 L obtained by tracing out either of the two parties. (ρ)= LρL†,where L†L 11 . The equality holds i i i i i i In section II to follow, we define exact and asymp- forL trace-preserving superoperators≤ which correspond phys- totic reducibilities and equivalences under LOCC alone, ically toP non-selective dynamics,P e.g. a measurement fol- and with the help of “catalysis”, or asymptotically neg- lowed by forgetting which outcome was produced. In general the superoperators may be trace decreasing and corre- ligible amounts of quantum communication. In section spond to selective operations, e.g. a measurement followed III we use these concepts to develop a framework for by throwing away some outcomes. If is a multilocally im- quantifying tripartite and multipartite pure-state entan- plementable superoperator, it must beL a separable superop- glement, in terms of a canonical set of states which we erator, i.e. a completely positive trace-preserving map of the call a minimal reversible entanglement generating set form shown above, where the Li’s are products of local op- (MREGS). This framework leads to an additive, mul- A B erators – Li = Li Li ... . Note that not all separable ticomponent entanglement measure, based on asymp- superoperators are multilocally⊗ implementable [15,16]. totically reversible LOCC transformations among tensor powers of such states, and having a number of

2 scalar components equal to the number of states in the Now we are ready to show that for any subset X of MREGS, in other words the number of asymptotically parties, the partial entropy SX is nonincreasing under inequivalent kinds of entanglement. LOCC. We state this as a lemma, For general m-partite states, partial entropy arguments give lower bounds on the number of entanglement Lemma 1 : If a multipartite system is initially in a components as a function of m, and allow us to show pure state Ψ, and is subjected to a sequence of LOCC that some states, e.g. the m = 4 Cat state, are not ex- operations resulting in a set of ﬁnal pure states Φi with actly, nor even asymptotically, interconvertible into any probabilities pi, then for any subset X of the parties combination of EPR pairs shared among the parties. S (Ψ) p S (Φ )(8) On the other hand, we show that the subclass of mul- X ≥ i X i i tipartite pure states having an m-way Schmidt decom- X position is describable by a single parameter, its partial entropy representing the number of standard “Cat” Proof: The result follows from the fact that average bi- m m states 0⊗ + 1⊗ asymptotically interconvertible to partite entanglement (partial entropy) of bipartite pure | i | i the state in question. Section III D treats tripartite pure states cannot increase under LOCC cf. [23]. state entanglement, showing in particular that, using exact LOCC transformations, two GHZ states can neither be prepared from nor used to prepare the isentropic combination of three EPR pairs shared symmetrically among A. Reducibilities and equivalences: exact and stochastic the three parties.

We start with LOCC state transformation involving II. REDUCIBILITIES, EQUIVALENCES AND single copies of states. If the state transformation is ex- LOCAL ENTROPIES act we say it is an exact reducibility. If the state transformation suceeds some of the time we say it is stochastic, Reducibility formalizes the notion of a transformation and if the state transformation needs the presence of an- of one state to another being possible under certain con- other state, which is is recovered after the protocol, it ditions, while equivalence formalizes the notion of this is called catalytic reducibility. In this section we define transformation being reversible—possible in both direc- these more precisely. We start with exact reducibility. tions. While studying entanglement it is useful to dis- We say a state Φ is exactly reducible to a state Ψ (written Φ ΨorjustΦ Ψ) by local operations and cuss state transformation under LOCC. This is because ≤LOCC ≤ a good entanglement measure should not increase under classical communication if and only if LOCC. So, if two states are equivalent under LOCC op- Φ= (Ψ) , (9) erations, they will have the same entanglement. This is ∃L L the key idea we will use in section III to quantify entan- where is a multilocally implementable trace preserv- glement. ing superoperator.L Alternatively we may say that the We start by first looking at partial entropies. Partial LOCC protocol corresponding to the superoperator entropies have the nice property that for pure states their L transforms Ψ toP Φ exactly. average does not increase under LOCC. L Intuitively this means that the state transformation Suppose the parties holding a pure state Ψ are num- m from Ψ to Φ can be done by LOCC with probability bered 1,2,... .Let denote a nontrivial subset of the m X one. (Where it will cause no confusion, for pure states parties and let ¯ be the set of remaining parties. Then X we use a plain Greek letter such as Ψ to represent both the reduced density matrix of subset of the parties is X the vector Ψ and the projector Ψ Ψ .) defined as The relation| i of exact LOCC reducibility| ih | for bipartite pure states has been studied in [24] and [25], which give ρX (Ψ) = trX¯ ( Ψ Ψ ). (6) | ih | necessary and sufficient conditions for it in terms of ma- The partial entropy of subset X is the von Neumann jorization of the eigenvalues of the reduced density ma- entropy trix. Nielsen ( [25]) uses notation reminiscent of a chemical reaction: where we say Φ Ψ, he says Ψ Φ. Both SX (Ψ) = tr(ρX (Ψ) log ρX (Ψ))). (7) ≤ → − 2 notations mean that given one copy of Ψ, we can with certainty, by local operations and classical communica- When X = ` consists of a single party, ρ ` is called { } { } tion, make one copy of Φ. the marginal density matrix of party ` and S(ρ ` )its the marginal entropy of party `. Two states are{ said} to Chemical reactions often involve catalysts, molecules be isentropic if for each subset X of the parties SX (Ψ) = which facilitate a reaction without being used up, so it SX (Φ). Two states Ψ and Φ are said to be marginally is natural to look for analogous quantum state transformations. Jonathan and Plenio have recently found an isentropic if S ` (Ψ) = S ` (Φ) for each party `. { } { }

3 example of successful catalysis for bipartite states, where support, i.e., isometric transformations.2 a catalyst allows a transformation to be performed with We now look at some conditions for two states to be certainty which could only be done with some chance of exactly equivalent. By Lemma 1 it is clear that if two failure in the absence of the the catalyst [26]. states are equivalent they must be isentropic, but not all

We say that Φ is catalytically reducible ( LOCCc)toΨ isentropic states are equivalent. Two states Φ and Ψ are if and only if there exists a state Υ such that≤ said to be incomparable if neither is exactly reducible to the other. Φ Υ LOCC Ψ Υ. (10) We are now in a position to demonstrate some impor- ⊗ ≤ ⊗ tant facts about exact LOCC reducibility3. An interesting fact about catalysis is that, because the catalyst is not consumed, one copy of it is suﬃcient to Theorem 1 : If Ψ and Φ are two marginally isentropic transform arbitrarily many copies of Ψ into Φ: pure states, then they are either locally unitarily (LU)- equivalent or else LOCC-incomparable. n n n ΦΥ LOCC ΨΥ Φ Υ LOCC Ψ Υ. (11) ∀ ≤ ⇒ ≤ Corollary 1 : Two states are LOCC equivalent if and Another important form of state transformation in- only if they are LU equivalent. volves probabilistic outcomes, where the procedure for the reducibility may fail some of the time as in “entangle- Ψ,Φ Ψ LOCC Φ Ψ LU Φ. (13) ment gambling” [17]. We capture this idea in stochastic ∀ ≡ ⇔ ≡ reducibility: We say a state Φ is stochastically reducible to a state Corollary 2 : States that are marginally but not fully Ψ under LOCC with yield p if and only if isentropic are necessarily LOCC-incomparable.

(Ψ) Proof: To prove this it suffices to show that for Φ= L , (12) marginally isentropic states Ψ and Φ, if Φ Ψthen ∃L tr( (Ψ)) L they must be local unitarily equivalent. In light≤ of the where is a multilocally implementable superoperator non-increase of partial entropy under LOCC (cf. lemma such thatL tr( (Ψ)) = p. 1) and the fact that these two states are marginally isen- This meansL that a copy of Φ may be obtained from a tropic, a LOCC protocol that converts one state to the copy of Ψ with probability p by LOCC operations. Exact other must conserve the marginal entropies at each step. reducibility corresponds to the case p=1. Suppose the LOCC protocol transforms Ψ to Φ ex- P After looking at reducibilities, we are now in a position actly. In general such a protocol consists of a sequence to consider exact equivalence. of local transformations each done by one party followed Two states Φ and Ψare said to be exactly equivalent by communication of (some of) the information gained to other parties. Without loss of generality assume that ( LOCC or simply )ifΦ ΨandΨ Φ. ≡This means that≡ the two≤ states are≤ exactly intercon- Alice performs the first operation of such a protocol vertible by classically coordinated local operations. In converting Ψ to Φ, which gives the resulting ensemble chemical notation this would be Ψ )* Φ. = pi,ψi . Since Alice’s operation cannot change the E { } BC... Catalytic and stochastic equivalence may be defined density matrix ρ seen by the remaining parties, similarly. BC... In passing we note that many other reducibilities ρ = pitrA( ψi ψi ) . (14) | ih | i (and their corresponding equivalences) can be consid- X ered, e.g. reducibilities via local unitary operations [27] As argued earlier, the average entropy must not change, , stochastic reducibility with catalysis, and reducibil- LU i.e. ≤ities without communication or with one-way communication [28]. Physically, reducibility via local unitary operations and that via local unitary operations along with a change 2 in the local support (corresponding to the increase or Unitary operations are characterized by U †U =11=UU†. decrease in the local Hilbert space dimensions) are the However, if we are want general transformations that preserve same because we could think of the extra dimensions as the norm of vectors, all we need is U †U =11,wheretheU’s being present from the start and extend the local uni- could be rectangular matrices. Such U are called isometric [29]. tary operation to the larger space. Thus, from now on 3 when we say local unitary operations, we mean local uni- These results strengthen Julia Kempe’s result [30] that if two multipartite pure states have isospectral marginal den- tary operations along with a possible change in the local sity matrices, then they are either LU-equivalent or LOCC- incomparable.

4 SBC...(Ψ) = SA(Ψ) = piSBC...(ψi). (15) Exact reducibility is too weak a reducibility to give i a simple classification of entanglement—even for bipar- X tite pure states, there are infinitely many incomparable By the strict concavity of the von Neumann entropy [29] LU equivalence classes, which would lead to infinitely each of the resultant states ψi must have the same re- ≤many distinct kinds of bipartite entanglement. Linden duced density matrix, from the viewpoint of all the other and Popescu [27] have explored the orbits of multipar- parties besides Alice, as the original state Ψ did: tite states under local unitary operations, and shown tr ( ψ ψ )=tr ( Ψ Ψ ). (16) that the number of LU invariants increases exponentially ∀i A | iih i | A | ih | with the number of parties and with the number of qubits Therefore the states ψi must be related by isometries possessed by each party. acting on Alice’s Hilbert space alone: One natural way to strengthen the notion of reducibility is to make it asymptotic. We first consider “asymp- A BC... ψi = U I Ψ . (17) totic LOCC reducibility” [17,28] which expresses the | i i ⊗ | i ability to convert n copies of one pure state into a good where A are unitary transformations acting on Alice’s Ui approximation of n copies of another, in the limit of large Hilbert space, which may have more dimensions than n. A possibly stronger reducibility, which we will call the support of Ψ in Alice’s space (this would corre- | i “asymptotic LOCCq reducibility,” expresses the ability spond to Alice having unilaterally chosen to enlarge her to do the state transformation with the help of a limited Hilbert space, which she is always free to do). Thus (o(n)) amount of quantum communication, in addition Alice’s measurement process, which appears on its face to the unlimited classical communication and local oper- to be a stochastic process not entirely under her control, ations allowed in ordinary LOCC reducibility. Another could in fact be faithfully simulated by having her simply natural way of strengthening asymptotic reducibility is toss a coin to choose a “measurement result” i with prob- to allow catalysis; defining “catalytic asymptotic LOCC ability pi, then perform the deterministic operation Ui reducibility” (LOCCc) in direct analogy with the exact on her portion of the joint state, and then finally report case. We show that asymptotic LOCCc reducibility is at the result i to all the other parties. In the next step of the least as strong as (ie can simulate) LOCCq reducibility. protocol, another party performs similar operations and Ordinary asymptotic LOCC reducibility is enough to sends classical information as to which unitary it per- simplify the classification of all bipartite pure states and formed and so on for each step. Thus the entire protocol some classes of m-partite states, so that, for any given m, consists of local unitary transformations, enlargement of a finite repertoire of standard states (EPR, GHZ, etc), Hilbert space and classical communication, maintaining which we will later call a minimal reversible entangle- at each step the overall state to be pure. The protocol ment generating set or MREGS, can be combined to pre- ends when the state Φ has been obtained. Since this is pare any member of class in an asymptotically reversible an exact reducibility of one pure state to another, for fashion, regardless of the size of the Hilbert spaces of the each possible sequence of local unitaries, the result must parties. Whether this classification can be extended to be Φ. Thus we can define a new protocol 0 that consists P cover general m-partite states for m>2 while maintain- of choosing just one such sequence of local unitaries and ing a finite repertoire size is an open question. it will take Ψ to Φ, showing that the two states are local Let us start by defining ordinary asymptotic LOCC unitarily equivalent. The first corollary follows from the reducibility. fact that if two states are LOCC equivalent, they must State Φ is asymptotically reducible ( LOCC or simply be isentropic and therefore marginally isentropic. The ) to state Ψ by local operations and classical commu- second follows from the fact if that the two states were nication if and only if LU equivalent, they would be fully isentropic, not merely marginally so. δ>0,>0 n,n , (n/n0) 1 <δ and ∀ ∃ 0 L | − | n0 n F ( (Ψ⊗ ), Φ⊗ ) 1 . (18) L ≥ − B. Asymptotic reducibilities and equivalences, and Here is a multi-locally implementable superoperator their relation to partial entropies L that converts n0 copies of Ψ into a high fidelity approximation to n copies of Φ. In chemical notation we can Before we discuss asymptotic reducibilities and equiv- write this as Ψ ; Φ. alences, let us define a quantitative measure of similarity A natural extension of asymptotic LOCC reducibility of two states. One such measure, the fidelity [31] [32] of occurs if we allow catalysis. Thus we define asymptotic a mixed state ρ relative to a pure state ψ,isgivenby LOCCc reducibility as: F(ρ, ψ)= ψ ρ ψ . It is the probability that ρ will pass We say Φ is asymptotically LOCCc reducible ( ) h | | i LOCCc a test for being ψ, conducted by an observer who knows to Ψ if there exists some state Υ such that the state ψ. For mixed states ρ and σ it is given by the 1 2 more symmetric expression F (ρ, σ) = (tr(√σρ√σ) 2 ) . ΦΥ ΨΥ, (19)

5 n where we say the state Υ is a catalyst for this reducibility. prepared from (Ψ Γ)⊗ . This satisﬁes the conditions As with exact catalysis (eq. 11), asymptotic catalysis (eq. 18) for asymptotic⊗ reducibility allows an arbitrarily large ratio of reactant to catalyst:

Φ Γ LOCC Ψ Γ, (22) n n ⊗ ⊗ ΦΥ ΨΥ nΦ Υ Ψ Υ. (20) ⇒∀ or, invoking the definition (eq. 19) of asymptotic cat- Another way of extending asymptotic LOCC re- alytic reducibility, ducibility is to allow a sublinear amount of quantum communication during the transformation process. Φ LOCCc Ψ, (23) State Φ is said to be asymptotically LOCCq reducible which was to be demonstrated. While the converse (i.e. ( LOCCq) to state Ψ iff that asymptotic catalytic reducibility can be simulated δ>0,>0 n,k, (k/n) <δ and by LOCCq transformations) seems plausible, we have ∀ ∃ L k n n not been able to prove it except in special cases. F ( (Γ⊗ Ψ⊗ ), Φ⊗ ) 1 , (21) L ⊗ ≥ − Asymptotic reducibilities and equivalences can have m m non-integer yields. This can be expressed using tensor where Γ denotes the m-Cat state 0⊗ + 1⊗ . The m-Cat states used here are a| convenienti | wayi of al- exponents that take on any nonnegative real value, so x y that Φ⊗ Ψ⊗ denotes lowing a sublinear amount o(n) of quantum communication, since they can be used as described in section III D (n/n0) x/y <δ and to generate EPR pairs between any two parties which in ∀δ>0 ∃n,n0, | − | n0 n turn can be used to teleport quantum data between the F ( (Ψ⊗ ), Φ⊗ ) 1 . (24) parties. The o(n) quantum communication allows the L ≥ − definition to be simpler in one respect: a single tensor In this case we say x/y is the asymptotic efficiency or power n can be used for the input state Ψ and output yield with which Φ can be obtained from Ψ. In chemical x state Φ, rather than the separate powers n and n0 used notation this could be expressed by Ψ ; y Φ, keeping in the definition of ordinary asymptotic LOCC reducibil- in mind that the coefficient x represents an asymptotic ity without quantum communication, because any o(n) yield or number of copies of the state Φ, not a scalar shortfall in number of copies of the output state can be factor multiplying the state vector. made up by using the Cat states to synthesize the extra Clearly, if a stochastic state transformation with yield output states de novo. This definition is more natural p ispossiblefromΨtoΦthenΨ;pΦ because of the than that for ordinary asymptotic LOCC reducibility in law of large numbers and the central limit theorem. that the input and output states are allowed to differ in We are now in a position to define the most impor- any way that can be repaired by an o(n) expenditure of tant tool in quantifying entanglement, namely asymp- x y quantum communication, rather than only in the spe- totic equivalence. We say that Ψ⊗ and Φ⊗ , with x y cific way of being n versus n0 copies of the desired state x, y 0, are asymptotically equivalent (Ψ⊗ Φ⊗ )if ≥ y ≈ x where n n0 is o(n). and only if Φ⊗ is asymptotically reducible to Ψ⊗ and − Clearly LOCC implies LOCCq and LOCCc, because or- vice versa. Two states are said to be asymptotically in- dinary asymptotic reducibility is a special case of the two comparable if neither is asymptotically reducible to the other kinds of asymptotic reducibility. We can also show other. that asymptotic LOCCq reducibility implies asymptotic Although we will mainly be concerned with asymp-

LOCCc reducibility, because any LOCCq protocol can be totic equivalence ( ), two possibly stronger reducibil- simulated by a protocol with the m-Cat state Γ ities mentioned earlier—asymptotic≈ LOCC reducibility LOCCc as catalyst, only a sublinear (and therefore asymptoti- with a catalyst ( LOCCc) and asymptotic LOCC re- cally negligible) amount of which is consumed. In more ducibility with a small amount of quantum commu- detail, if Φ LOCCq Ψ, then from eq. 21, for each and nication ( LOCCq)—give rise to their own correspond- n δ, there exist n and k such that Ψ⊗ can be converted ing versions of equivalence and incomparability. Since n to a 1 faithful approximation to Φ⊗ with the help LOCCc transformations can simulate both LOCCq and − of k

6 into Φ, even asymptotically and even with the help of a For reducibilities and entropy inequalities (omitting catalyst, then for each subset X of the parties, SX (Φ) mention of the states Ψ and Φ where it will create no cannot exceed SX (Ψ); otherwise an increase of partial confusion) we have entropy could be made to occur in violation of Lemma (Φ Ψ) 1.) LU LOCC LOCC LOCCq LOCCc ≡S (Φ) ⇒≤S (Ψ). ⇒ ⇒ ⇒ ⇒ ∀X X ≤ X Not Marginally Isentropic (25)

Marginally Isentropic 6 For equivalences and entropy equalities we have

5 (Φ LU Ψ) LOCC LOCC LOCCq LOCCc Isentropic ≡ ⇔≡ ⇒≈ ⇒≈ ⇒≈ ⇒ X SX (Φ) = SX (Ψ) (i.e. Φ and Ψ are Isentropic) 4 ∀Φ and Ψ are Marginally Isentropic ⇒ ⇒ (Φ LU Ψ) or Φ and Ψ are LOCC incomparable. 3 ~ ≡ ~LOCC Isospectral (26) Figure 1 illustrates several of these relations. LOCC 2

1 C. Bipartite entanglement: a reinterpretation

LU As an example of the usefulness of these concepts let us reexpress the bipartite pure-state entanglement result [17] in terms of asymptotic equivalence. In this new lan- FIG. 1. Relation of exact and asymptotic equivalences to guage, any bipartite pure state ΨAB is asymptotically equality of local entropies. Two states are exactly equivalent equivalent to (ΨAB) EPR pairs: this is the number of under local operations and classical communication (LOCC) SA if and only if they are equivalent under local unitary opera- EPR pairs that, asymptotically, can be obtained from AB tions alone (LU). An example (circled 1) is the LU intercon- and are required to prepare Ψ by classically coordi- vertibility of the two Bell states 00 + 11 and 00 11 . nated local operations. Exact equivalence of course implies| i asymptotic| i | equivalencei−| i In proving this result, the concepts of entanglement (dotted region) including (circled 2) the asymptotic equiva- concentration and dilution [17] are central. The process lence between an EPR pair and an isentropic but not isospec- of asymptotically reducing a given bipartite pure state tral two-trit state of the form α 00 + β 11 + γ 22 .Asym- to EPR singlet form is entanglement dilution and that potitically equivalent states are| necessarilyi | i isentropic,| i but of reducing EPR singlets to an arbitrary bipartite pure not conversely. For example (circled 3) the isentropic—and state is entanglement concentration. Then the above re- indeed isospectral—tripartite states 2GHZ and 3EPR (see sult means that entanglement concentration and dilution section III D) have very recently been shown [40] to be in- are reversible in the sense of asymptotic equivalence, i.e., comparable with respect asymptotic LOCC reducibility. This they approach unit efficiency and fidelity in the limit of example also illustrates the fact (cf [30]) that isospectral large number of copies n. The crucial requirement for states of three or more parties need not be LU-equivalent. these methods to work is the existence of the Schmidt A tensor product of circled 2 with circled 3 type states yields biorthogonal (normal or polar) form for bipartite pure isentropic states (circled 4) that are neither asymptotically states [37], that is, the fact that any bipartite pure state LOCC equivalent nor isospectral. States that are marginally ΨAB can be written in a biorthogonal form: but not fully isentropic (circled 5) must be incomparable with | i respect to exact LOCC reducibility. Finally, at the periph- ΨAB = λ iA iB , (27) i| i⊗| i ery (circled 6) are states that are not even marginally isen- i tropic. These include incomparable pairs such as AB-EPR vs X where iA and iB form orthonormal bases in Alice’s BC-EPR, and properly reducible pairs such as GHZ vs EPR, | i | i but no cases of exact or even asymptotic equivalence. and Bob’s Hilbert space respectively, where by choice of phases of local bases the coefficients λi can be made real and non-negative. We collect the relations we have proved in this section from the definitions of the various reducibilities, using Lemma 1 and Theorem 1, and express them as III. TRIPARTITE AND MULTIPARTITE PURE-STATE ENTANGLEMENT Theorem 2 : The following implications hold among the reducibilities, equivalences, and partial entropies of a pair of multipartite pure states: In this section we use the tools we developed earlier to propose a framework for quantifying multipartite pure-

7 state entanglement. Discussions in the last section were bipartite states, generalize in a straightforward manner, valid for pure as well as mixed states. However from now so that for an m-partite state ΨABC..., the local entropy on we will restrict out attention to pure states. as seen by any party, or indeed any nontrivial subset of In section III A we consider the natural generaliza- the parties, gives the asymptotic number of m-partite tion of the bipartite states, namely the m-party states cat states into which it can be asymptotically intercon- with an m-way Schmidt decomposition which we call m- verted. That is, if ΨABC...M is a Schmidt decomposable orthogonal states. We show that for each m such states multipartite state, then can be characterized by a scalar entanglement measure, S (ΨABC...M) which may be iterpreted as the number of m-Cat states ΨABC...M CatABC...M ⊗ A . (29) asymptotically equivalent to the state in question.single | i≈| i parameter. In section III B we introduce the concepts of Entanglement concentration on an m-orthogonal state ABC...M entanglement span, entanglement coefficients and mini- Ψ , like its bipartite counterpart, can be done by mal entanglement generating sets (MREGS), as elements parallel local actions of the observers, without any com- of a general framework for quantifying multipartite pure- munication. Starting with a number n of copies of the state entanglement. In section III C we derive lower state to be concentrated, each party makes an incom- bounds on the cardinality of MREGS. In section III D plete von Neumann measurement, collapsing the system where we study the question of interconversion between onto a uniform superposition over an eigenspace of one m-Cat and EPR states. In section III E we show unique- eigenvalue in the product Schmidt basis. After enough ness of the entanglement coefficients for natural MREGS such states have been accumulated to span a Hilbert possibilities for tripartite states. space of dimension slightly more than some power k of 2m, another measurement suffices, with high probability, to collapse the state onto a maximally entangled m- mk A. Schmidt-decomposable or m-orthogonal states partite state in a Hilbert space of dimension 2 ,which can then be transformed by local operations into a tensor product of km-partite Cat states. We consider Alice, Bob, Claire, ..., Matt as m ob- Entanglement dilution (cf. Fig. 2) proceeds in the servers who have one subsystem each of a m-part system same way as for bipartite states, except that Alice lo- in a joint m-partite pure state. Some m-partite pure cally prepares a supply of bipartite pure states ΦA,A0 states, but not all, can be written in a m-orthogonal having the same Schmidt spectrum as the multipartite form analogous to the Schmidt biorthogonal form. We Schmidt-decomposable state ΨABC...M which she wishes call such states m-orthogonal or Schmidt decomposable. to share with the other parties. Here the superscript Thus an m-partite pure state ΨABC... is Schmidt decom- A,A signifies that both parts of this state are in Alice’s posable or m-orthogonal if and| only ifi it can be written 0 laboratory, whereas her goal is to end up with states in a form shared among all the parties. As in bipartite entan- ABC...M A B C M glement dilution, Alice then Schumacher-compresses the Ψ = λi i i i ... i , (28) | i | i⊗| i⊗| i ⊗| i part of a tensor product of copies of ΦAA0 ,result- i A0 n X ing in approximately k compressed qubits, where k/n A B C M where i , i , i , ..., i are orthonormal bases asymptotically approaches SA(Φ) = SA(Ψ), the local for the| correspondingi | i | i party.| Noticei that by change of entropy of the Schmidt-decomposable state she wishes phases of local bases, each of the Schmidt coefficients λi to share. She then teleports these k compressed qubits can be made real and non-negative. In any m-orthogonal to the other parties—Bob, Claire, etc. The teleportation state, the reduced entropy seen by any observer, indeed is performed not with k EPR pairs, as in ordinary tele- by any nontrivial subset of observers, is the same, be- portation, but with km-partite Cat states, which she ing given by the Shannon entropy of the squares of the has shared beforehand with the other parties. For each Schmidt coefficients. Already this makes it obvious that of the compressed qubits, Alice performs a Bell mea- not all tripartite and higher states are Schmidt decom- surement on that qubit and one leg of an m-partite Cat posable since, for any m>2 it is clear that there are state, and broadcasts the two-bit classical result to all pure m-partite states having unequal partial entropies the other parties, who then each apply the correspond- for the different observers. Peres [33] gives necessary and ing Pauli rotation to their leg of the shared Cat state. sufficient conditions for a multipartite pure state to be Finally all the other parties besides Alice apply Schu- Schmidt decomposable. Thapliyal [34] recently gave an- macher decompression to their legs of the rotated Cat other characterization, showing that an m-partite pure states, leaving the m parties in a high-fidelity approxi- ABC...M n state is Schmidt-decomposable if and only if each of the mation to the m-partite state (Ψ )⊗ which they m 1 partite mixed states obtained by tracing out one wished to share. party− is separable. This entanglement dilution protocol requires 2k/n bits For such Schmidt decomposable states, the notions of of classical information per copy (of the target state) to entanglement concentration and dilution, developed for be communicated from Alice to the other two parties.

8 Lo and Popescu in [39] show a bipartite entanglement We start by looking at the concept of the entanglement dilution protocol which requires O(1/√n) bits of com- span of a set of states. munication per copy, thus asymptotically, the classical Given the set of states = ψ1,ψ2, ..., ψk ,theiren- communication cost per copy goes to zero for their pro- tanglement span ( ( )) is definedG { as the set of} states that tocol. The question then is whether a similar protocol they can reversiblyS G generate under asymptotic LOCC. can be found for the dilution of m-Cat states into m- That is, orthogonal states. It is easy to see that replacing telepor- k tation through EPR states with teleportation through xi ( )= Ψ Ψ ψi ⊗ , with xi 0 . (30) the m-partite Cat states in their protocol gives us a S G { | ≈ | i ≥ } i=1 protocol for entanglement dilution of the m-Cat states O into m-orthogonal states. This protocol again uses only Notice that the xi give a quantitative amount of entan- O(1/√n) classical communication per copy, an asymp- glement in terms of the spanning states. They are called totically vanishing amount. the entanglement coefficients. In general these coefficients may be non-unique, for example if two states in the set are locally unitarily related. Loosely speaking B. Framework for quantifying entanglement of these coefficients may be non-unique if the “kinds of en- multipartite pure states tanglement” they correspond to are not “independent”. Let us look at some examples. The entanglement span under LOCC of any bipartite state is the set of all bi- Now we apply concepts of reducibilities and equiv- partite states. Another example is provided by the set alences II in attempting to quantifying entangle- of m-orthogonal states. Any such state in general and ment. For general m-partite states, there will be sev- in particular the -Cat state spans the set of all the eral inequivalent kinds of entanglement under asymp- m -orthogonal states. totically reversible LOCC (or LOCCq or LOCCc) m Let us now introduce the concept of reversible entan- transformations—at least as many the number of inde- glement generating sets (REGS), which is dual to the pendently variable partial entropies for such states—and concept of entanglement span. A set = perhaps more. However, a good entanglement measure ψ1,ψ2, ..., ψn of states is said to be a reversible entanglementG { gener-} ought to be defined so as to assign equal entanglement (in ating set (REGS) for a class of states if and only if the case of a multicomponent measure, equal in all com- ( ). C ponents) to asymptotically equivalent states. This forms C⊆SClearly,G every REGS for the class of +1 partite states the basis of our framework for quantifying entanglement. m is a REGS for each of its m-partite subsystems. In particular any REGS for the full class of -partite states A m A must be capable of generating an EPR pair between any two of the parties. One might suspect that the set of Φ1n all m(m 1)/2 EPR pairs would be a sufficient REGS − A' C M for generating all m-partite states, but as we will see in A B 1n k σ Ψ section III C, that is not the case for m 4. GHZ D ≥ B 0 To quantify entanglement, one would like to know the fewest kinds of entanglement needed to make all states in C C σ a given class. To this end we define a minimal reversible D 0 entanglement generating set (MREGS) as a REGS of minimal cardinality. Again the set = EPR is an G2 { } FIG. 2. Entanglement dilution for Schmidt-decomposable example of a MREGS for bipartite entanglement which tripartite states. Alice prepares a local supply of n bipartite induces the entanglement measure given by the partial states ΦAA0 isospectral to the Schmidt-decomposable tripar- entropy in bits. tite state ΨA,B,C she wishes to share, and Schumacher com- Thus we have reduced the problem of quantifying en- presses their A0 halves ( )tok nS(ρA) qubits. Then, tanglement to the problem of finding the MREGS and C ≈ using k previously shared GHZ states, she teleports the com- the corresponding entanglement coefficients. The en- pressed qubits to Bob and Charlie simultaneously (Here tanglement coefficients give us the entanglement mea- M denotes a Bell measurement, the thick lines a 2k-bit classical sure in terms of how many of the states in the MREGS message Alice broadcasts to both Bob and Claire, and σ the are required to reversibly make the state by asymptotic conditional Pauli rotation which completes the teleportation LOCC. process). Finally, Bob and Claire Schumacher-decompress If we drop the requirement of reversibility, we get the ( )theirkqubits to recover n qubits each, in a state closely D notion of a entanglement generating set (EGS), a set of approximating n copies of the diluted Schmidt-decomposable ABC states which can generate every state in under exact tripartite state Ψ they wished to share. or asymptotic LOCC. An EGS needs onlyC one member, since the m-partite Cat state by itself is sufficient

9 to generate all m-partite entangled states, though not party to one-party partial entropy in state Υ. It is easy reversibly. This can be seen because the m-Cat state to see that r21 = 1 for Cat states, independent of m, but can give an EPR pair between any two parties by exact r21 =2(m 2)/(m 1) for Toast states, the numerator of LOCC. So Alice can make the desired multipartite state the latter expression− − being the number of edges, in an m- in her lab and then teleport it using these EPR pairs, partite complete graph, joining a two-vertex subset X to thus generating an arbitrary multipartite state exactly its complement, while the denominator is the number of by LOCC, starting from the appropriate number of m- edges incident on any single vertex. Thus Cat and Toast Cat states. To see that the transformation is irreversible, states have equal r21 for m = 3, but for Toast states note that an m-partite Cat state can be used to prepare with larger m, the ratio exceeds 1, as shown in the table at most one EPR state, say between Alice and Bob, but below. Therefore the 4-Cat, unlike the 4-Toast state, m 1 EPR states, say connecting Alice to every other cannot be asymptotically equivalent to any combination party,− are needed to prepare the Cat state again. Thap- of the six EPR pairs, and the MREGS for m =4 must liyal [35] has shown that a pure m-partite state Ψ is an have at least seven members. EGS (can be transformed into a cat state by LOCC) if and only if its partial entropies SX are positive across Parties State r21 all nontrivial partitions X. 3 Cat (GHZ) 1 The following section exhibits some simple lower Toast (3 EPRs) 1 bounds on the cardinality of the MREGS for tripartite 4 Cat 1 and higher entangled pure states. Unfortunately we do Toast (6 EPRs) 4/3 not know any corresponding upper bounds. We can- 5 Cat 1 not exclude the possibility that for tripartite and higher Toast (10 EPRs) 3/2 states an inﬁnite number of asympotitically inequivalent 5-Qubit Codeword 2 kinds of entanglement might exist. 6 Cat 1 Toast (15 EPRs) 8/5

C. Lower Bounds on the size of MREGS based on local entropies For m = 5, the table also includes an entry for It is easy to see that the Alice-Bob EPR state EPRAB the maximally-entangled state of ﬁve qubits, (e.g. a (regarded as a special case of an m-partite state in which codeword in the well-known 5-qubit error-correcting all the parties besides Alice and Bob are unentangled code [38,23]) which has maximal entropy across any par- bystanders in a standard 0 state) is an MREGS for tition X. Since this state has an r21 even greater than the class containing all and| onlyi those states which have the Toast state, the MREGS for m=5 must have at least AB entanglement but no other entanglement, more pre- 12 states. Similarly, the MREGS for m =6 musthave at least 31 members, without considering other entropy cisely states for which SX is zero if X includes both A and B or neither A and B, and has a constant nonzero ratios besides r21 or other states besides the EPR, 4-Cat, value for all other X. Therefore, in order to generate and 6-Cat states. all possible bipartite EPR pairs, the MREGS for general m-partite pure states must have at least m(m 1)/2 members, which can be taken without loss of generality− D. Exact Reducibilities between GHZ and EPR to be the m(m 1)/2 bipartite EPR states themselves. However, for− all m>3 the partial entropy argument At this point it is natural to ask whether three EPR requires the MREGS to include other states as well. pairs (shared symmetrically among Alice, Bob, and Without pursuing it exhaustively [36], we will sketch Claire) can be reversibly interconverted to two GHZ how local entropy arguments can be used to derive other states. Partial entropy arguments do not resolve the lower bounds on the size of the MREGS for general m- question because, for both the 3EPR state and the 2GHZ partite states. state, the partial entropy of any nontrivial subset of the Let us restrict our attention to m-partite pure states Υ parties is 2 bits. Nevertheless, the impossibility of per- in which the partial entropy S(trX( Υ Υ )) of a subset forming this conversion follows from the fact that two X depends only on the number of| membersih | of X,not states are LOCC equivalent if and only if they are equiv- on which parties are members of X. Two examples of alent under local unitary operations. such as state are the m-way Cat state, and a tensor To see that 2GHZ and 3EPR states are LOCC in- product of m(m 1)/2 EPR pairs, one shared between comparable, ﬁrst observe that, since the two states are each pair of parties.− We shall call the latter the m-way isentropic, they must, by Theorem 1, either be LOCC Toast state, after the custom of clinking glasses during a incomparable or LU equivalent. To see that they are toast. Let r21(Υ) = SAB(Υ)/SA(Υ) be the ratio of two- not LU equivalent, observe that the mixed state ob-

10 tained by tracing out Alice from the 2GHZ state, namely reducibilities follow: To get an EPR pair say between ρBC(2GHZ), a maximally mixed, separable state of the Bob and Claire, Alice performs a measurement in the two parties Bob and Claire, while the corresponding Hadamard basis namely, 0+1 , 0 1 and informs {| i | − i} mixed state obtained from the 3EPR states, ρBC(3EPR) Bob and Claire about the outcome. Using this informa- is a distillable entangled state, consisting of the tensor tion, Bob and Claire can perform conditioned rotations product of an intact BC EPR pair with another random that give them and EPR pair. Clearly, this LOCC pro- qubit held by each party. But if 3EPR and 2GHZ were tocol can be generalized to many parties, to transform a LU equivalent, Bob and Claire, by performing their own m-Cat state into an EPR pair between any two parties, local unitary transformations without reference to Alice, by having the remaining m 2 parties measure in the could make ρBC(3EPR) from ρBC(2GHZ). Since they Hadamard basis, and communicate− the result to the two cannot do this (otherwise they would be generating en- parties, who then perform appropriate conditioned local tanglement by LOCC), 3EPR and 2GHZ states cannot unitary operations. be LU equivalent; therefore, by corollary 1 they must be To get a GHZ state from two EPR pairs say EPRAB LOCC incomparable. and EPRAB , Alice makes a GHZ state in her| lab andi then| uses thei EPR pairs to teleport Bob’s and Charlie’s parts to them. Clearly, this protocol can be generalized to make a m-Cat state from a set of m EPR pairs, shared C Cby one party with all the rest. EPR In passing we note that any set of EPR pairs that de- EPR scribe a connected graph, the nodes representing parties and the edges representing the shared EPR pairs, is an ABCEGS. This is easy to prove using teleportation as done above.

ABGHZ E. Uniqueness of entanglement coefficients One key question about this framework for quantifying entanglement is whether entanglement coefficients are unique. Surely this is to be desired if we are to interpret the values of the coefficients as representing the amounts ABof different kinds of entanglement present in the given EPR state. We do not know how to show uniqueness in general, but we can show this for some cases of interest. For concreteness let us consider the case of three parties, say Alice, Bob and Claire. We noted earlier that X all EPR pairs shared between two parties must be in AB BC CA X the MREGS, so EPR , EPR and EPR must be in the MREGS. Let us consider the entanglement span of 2GHZ these three EPR pairs. Assume that there exists a state 3EPR Ψ in this span such that the entanglement coefficients are FIG. 3. Top: Two EPR pairs, together involving three not unique, say (x, y, z)and(a, b, c), where x, y and z parties, can be exactly transformed to one GHZ state. A (resp. a,b,andc) denote the amounts of EPRAB, EPRBC GHZ state can be transformed into any one of the three EPR and EPRCA in the two decompositions. Then using the pairs. These transformations are exact and irreversible, in- fact that asymptotically LOCC equivalent states must volving loss of entanglement across some bipartite boundary. be isentropic, we have, Bottom: The transformations between the symmetric 3EPR state and the 2GHZ state, marked by an X, cannot be done x + y = a + b, y + z = b + c, z + x = c + a. (31) exactly, even though the partial entropies agree, by the arguments of this section. Very recently it has been shown [40] This implies that (x, y, z)=(a, b, c) and thus proves that these transformation cannot even be done asymptoti- uniqueness. Clearly such an argument works for the en- cally. tanglement span of EPR pairs of more parties, because there are at most m(m 1)/2 EPR pairs shared by dif- Figure 3 shows the exact reducibilities that hold ferent parties and the isentropic− condition gives the same among EPR and GHZ states. The protocols for these number of independent constraints.

11 Now we look at the entanglement span of the above of entanglement under asymptotically reversible LOCC three EPR pairs and the GHZ state. If we assume the transformations, we know of no nontrivial upper bounds. GHZ belongs to the span of the EPRs then uniqueness As noted earlier, even for tripartite states we do not has already been proved. Thus let us assume that the know that the number is finite. One possible ap- GHZ is asymptotically not equivalent to the EPRs. Let proach to this problem, which we do not explore in de- the non-unique entanglement coefficients be (x, y, z, w) tail here, would be to further generalize the notion of and (x δx,y δy,z δz,w+δw), with the first three state by allowing tensor factors to appear with negative coefficients− representing− − the amount of the EPRs and the as well as nonintegral exponents. A generalized state AB 2 0.3 last representing the amount of GHZ. Without loss of such as (EPR )⊗ (GHZ)⊗− (in chemical notation, AB ⊗ generality we can assume δw =2δ>0. Again using the 2EPR 0.3GHZ) would thus represent a quantum fact that asymptotically LOCC equivalent states must “contract”− comprising a license, asymptotically, to con- be isentropic we have sume two Alice-Bob EPR pairs along with an obliga- tion to produce 0.3 GHZ states. Allowing negative en- δ δ δ = δ δ δ = δ δ δ =0 . (32) w − x − y w − y − z w − z − x tanglement coefficients would also solve the problem of nonuniqueness of entanglemnet coefficients, allowing any Solving these equations we find that, state to be described as a unique, but not necessarily positive, linear combination of states in a smaller MREGS. δx = δy = δz = δw/2=δ. (33) The most powerful result we could hope for from ap- This implies that proaches of this kind would be to show that under some appropriately strengthened (but still natural) notion of AB BC CA 2 EPR EPR EPR LOCCc GHZ⊗ . (34) asymptotic reducibility, all isentropic states are asymp- ⊗ ⊗ ≈ totically equivalent. A less ambitious result would be For more complicated sets of states, the requirement to show that for simple asymptotic reducibility, or some that entanglement coefficientsS be positive may lead to strengthened version of it, all isentropic states are either nonuniqueness. Because of positivity, all extremal points equivalent or incomparable, in analogy with the fact that of must be in the MREGS, and for some ,thenumber all isentropic states must be either equivalent or incom- ofS extremal points may considerably exceedS the dimen- parable under exact LOCC reducibility (corollary 1). sionality of (For example, for n 3, each interior point of a regularSn-gon can be expressed≥ in multiple ways as a convex combination of vertices). ACKNOWLEDGEMENTS Note that there may be many MREGS, for example any bipartite state is as MREGS for bipartite entanglement. So how do we decide upon a canonical We thank David DiVincenzo, Julia Kempe, Noah Lin- MREGS? Possible criteria include requiring the states den, Barbara Terhal, Armin Uhlmann, and Bill Wootters in the MREGS to be of as low Hilbert space dimension for helpful discussions. CHB, JAS, and AVT acknowl- as possible, and as high in partial entropy within that edge support from the USA Army Research office, grant Hilbert space as possible. Thus for the bipartite case the DAAG55-98-C-0041 and AVT also under DAAG55-98- EPR state is the canonical MREGS, up to local unitary 1-0366. DR acknowledges support from the Giladi pro- operations. gram.

IV. DISCUSSION AND OPEN PROBLEMS

Very recently [40] Linden, Popescu, Schumacher and Westmoreland, using a relative entropy argument, have [1] A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47, 777 strengthened the result of section III D by showing that (1935). asymptotically reversible transformations are insufficient [2] E. Schrödinger, Naturwissenschaften 23, 807-812, 823- to interconvert 2GHZ and 3EPR (indeed the states re- 828, 844-849 (1935). Translation: Proc of APS, 124, 323 main asymptotically incomparable even with the help of (1980). a catalyst). Therefore the MREGS for m=3 must con- [3] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, tain at least four states (without loss of generality the A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, GHZ and the three bipartite EPR states). Of course we 1895 (1993). would like to know whether these resources are sufficient [4] C.H. Bennett and G. Brassard, “Quantum Cryptogra- to prepare all tripartite pure states in an asymptotically phy: Public Key Distribution and Coin Tossing”, Pro- reversible fashion. ceedings of IEEE International Conference on Comput- A more fundamental problem is that although we ers Systems and Signal Processing, Bangalore India, have lower bounds on the number of inequivalent kinds December 1984, pp 175-179.; D. Deutsch, A. Ekert,

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