We demonstrate that even under positive-partial-transpose-preserving operations in an asymptotic setting Greenberger-Horne-Zeilinger (GHZ) and
W states are not reversibly interconvertible. We investigate the structure of minimal reversible entanglement generating set (MREGS) for tripartite states under positive-partial-transpose- (PPT-) preserving operations. We demonstrate that the set consisting of
W and Einstein-Podolsky-Rosen (EPR) states alone cannot be an MREGS. In this context we prove the surprising result that the relative entropy of entanglement can be strictly subadditive for certain pure tripartite states which is crucial to keep open the possibility that the set of GHZ state and EPR states together constitute an MREGS under PPT-preserving operations.
We present a family of tripartite entangled states that, in an asymptotical sense, can be reversibly converted into Einstein-Podolsky-Rosen (EPR) states, shared by parties B and C, and tripartite Greenberger-Horne-Zeilinger (GHZ) states. Thus we show that a bipartite and a genuine tripartite entanglement can be reversibly combined in a tripartite state. For such states the corresponding fractions of GHZ and EPR states represent a complete quantification of their (asymptotical) entanglement resources. More generally, we show that AB, AC, and BC EPR entanglement and GHZ entanglement can be reversibly combined in a single tripartite state. Finally, we generalize this result to any number of parties.
Hoping to simplify the classification of pure entangled states of multi $(m)\ensuremath{-}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{e}$ quantum systems, we study exactly and asymptotically (in
n) reversible transformations among
nth tensor powers of such states (i.e.,
n copies of the state shared among the same
m parties) under local quantum operations and classical communication (LOCC). For exact transformations, we show that two states whose marginal one-party entropies agree are either locally unitarily equivalent or else LOCC incomparable. In particular we show that two tripartite Greenberger-Horne-Zeilinger states are LOCC incomparable to three bipartite Einstein-Podolsky-Rosen (EPR) states symmetrically shared among the three parties. Asymptotic transformations yield a simpler classification than exact transformations; for example, they allow all pure bipartite states to be characterized by a single parameter---their partial entropy---which may be interpreted as the number of EPR pairs asymptotically interconvertible to the state in question by LOCC transformations. We show that $m\ensuremath{-}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{e}$ pure states having an $m\ensuremath{-}\mathrm{w}\mathrm{a}\mathrm{y}$ Schmidt decomposition are similarly parametrizable, with the partial entropy across any nontrivial partition representing the number of standard quantum superposition or ``cat'' states $|{0}^{\ensuremath{\bigotimes}m}〉+|{1}^{\ensuremath{\bigotimes}m}〉$ asymptotically interconvertible to the state in question. For general $m\ensuremath{-}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{e}$ states, partial entropies across different partitions need not be equal, and since partial entropies are conserved by asymptotically reversible LOCC operations, a multicomponent entanglement measure is needed, with each scalar component representing a different kind of entanglement, not asymptotically interconvertible to the other kinds. In particular we show that the
m=4 cat state is not isentropic to, and therefore not asymptotically interconvertible to, any combination of bipartite and tripartite states shared among the four parties. Thus, although the
m=4 cat state can be prepared from bipartite EPR states, the preparation process is necessarily irreversible, and remains so even asymptotically. For each number of parties
m we define a minimal reversible entanglement generating set (MREGS) as a set of states of minimal cardinality sufficient to generate all $m\ensuremath{-}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{e}$ pure states by asymptotically reversible LOCC transformations. Partial entropy arguments provide lower bounds on the size of the MREGS, but for
m>2 we know no upper bounds. We briefly consider several generalizations of LOCC transformations, including transformations with some probability of failure, transformations with the catalytic assistance of states other than the states we are trying to transform, and asymptotic LOCC transformations supplemented by a negligible
[o(n)] amount of quantum communication.
We find that the asymptotic entanglement of assistance of a general bipartite mixed state is equal to the smaller of its two local entropies. Our protocol gives rise to the asymptotically optimal Einstein-Podolsky-Rosen (EPR) pair distillation procedure for a given tripartite pure state, and we show that it actually yields EPR and Greenberger-Horne-Zeilinger (GHZ) states; in fact, under a restricted class of protocols, which we call ``one-way broadcasting,'' the GHZ rate is shown to be optimal. This result implies a capacity theorem for quantum channels where the environment helps transmission by broadcasting the outcome of an optimally chosen measurement. We discuss generalizations to
m parties and show (for
m=4) that the maximal amount of entanglement that can be localized between two parties is given by the smallest entropy of a group of parties of which the one party is a member, but not the other. This gives an explicit expression for the asymptotic localizable entanglement and shows that any nontrivial ground state of a spin system can be used as a perfect quantum repeater if many copies are available in parallel. Finally, we provide evidence that any unital channel is asymptotically equivalent to a mixture of unitaries and any general channel to a mixture of partial isometries.
We show that not all four-party pure states are Greenberger-Horne-Zeilinger (GHZ) reducible (i.e., can be generated reversibly from a combination of two-, three-, and four-party maximally entangled states by local quantum operations and classical communication asymptotically). We also present some properties of the relative entropy of entanglement for those three-party pure states that are GHZ reducible, and then we relate these properties to the additivity of the relative entropy of entanglement.