We demonstrate that even under positive partial transpose preserving operations in an asymptotic setting GHZ and W states are not reversibly interconvertible. We investigate the structure of minimal reversible entanglement generating set (MREGS) for tri-partite states under positive partial transpose (ppt) preserving operations. We demonstrate that the set consisting of W and EPR states alone cannot be an MREGS. In this context we prove the surprising result that the relative entropy of entanglement can be strictly sub-additive for certain pure tri-partite 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 consider the transformation of multipartite states in the single-copy setting under positive-partial-transpose-preserving operations (PPT operations) and obtain both qualitative and quantitative results. First, for some pure-state transformations that are impossible under local operations and classical communication (LOCC), we demonstrate that they become possible with a surprisingly large success probability under PPT operations. Furthermore, we clarify the convertibility of arbitrary multipartite pure states under PPT operations and show that a drastic simplification in the classification of pure-state entanglement occurs when the set of operations is switched from LOCC to PPT operations. Indeed, the infinitely many types of LOCC-incomparable entanglement are reduced to only one type under the action of PPT operations. This is a clear manifestation of the increased power afforded by the use of PPT-bound entanglement. In addition, we further enlarge the set of operations to clarify the effect of another type of bound entanglement, multipartite unlockable bound entanglement, and show that a further simplification occurs. As compared to pure states a more complicated situation emerges in mixed-state settings. While single-copy distillation becomes possible under PPT operations for some mixed states it remains impossible for other mixed states.