Abstract
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 n'th 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). With regard to exact transformations, we show that two states whose 1-party entropies agree are either locally-unitarily (LU) equivalent or else LOCC-incomparable. In particular we show that two tripartite Greenberger-Horne-Zeilinger (GHZ) states are LOCC-incomparable to three bipartite Einstein-Podolsky-Rosen (EPR) states symmetrically shared among the three parties. Asymptotic transformations result in a simpler classification than exact transformations. We show that m-partite pure states having an m-way Schmidt decomposition are simply parameterizable, with the partial entropy across any nontrivial partition representing the number of standard ``Cat'' states (|0^m>+|1^m>) asymptotically interconvertible to the state in question. For general m-partite 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 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.
Highlights
Entanglement, first noted by Einstein-Podolsky-Rosen (EPR) [1] and Schrodinger [2], is an essential feature of quantum mechanics
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 local quantum operations and classical communication (LOCC) transformations
We show that the subclass of multipartite pure states having an m-way Schmidt decomposition is describable by a single parameter, its partial entropy representing the number of standard “Cat” states | 0⊗m + | 1⊗m asymptotically interconvertible to the state in question
Summary
Entanglement, first noted by Einstein-Podolsky-Rosen (EPR) [1] and Schrodinger [2], is an essential feature of quantum mechanics. This framework leads to an additive, multicomponent entanglement measure, based on asymptotically reversible LOCC transformations among tensor powers of such states, and having a number of scalar components equal to the number of states in the MREGS, in other words the number of asymptotically inequivalent kinds of entanglement. Proof: The result follows from the fact that average bipartite entanglement (partial entropy) of bipartite pure states cannot increase under LOCC cf. [23]
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