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 entan- gled 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 quan- tum 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 Schr¨odinger [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 entangle- ment, 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 Schr¨odinger’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 gen- eral 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