dorsal/arxiv
View SchemaExact and Asymptotic Measures of Multipartite Pure State Entanglement
| Authors | Charles H. Bennett, Sandu Popescu, Daniel Rohrlich, John A. Smolin, Ashish V. Thapliyal |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9908073 |
| URL | https://arxiv.org/abs/quant-ph/9908073 |
| DOI | 10.1103/PhysRevA.63.012307 |
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.
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"abstract": "In an effort to simplify the classification of pure entangled states of multi\n(m) -partite quantum systems, we study exactly and asymptotically (in n)\nreversible transformations among n\u0027th tensor powers of such states (ie n copies\nof the state shared among the same m parties) under local quantum operations\nand classical communication (LOCC). With regard to exact transformations, we\nshow that two states whose 1-party entropies agree are either locally-unitarily\n(LU) equivalent or else LOCC-incomparable. In particular we show that two\ntripartite Greenberger-Horne-Zeilinger (GHZ) states are LOCC-incomparable to\nthree bipartite Einstein-Podolsky-Rosen (EPR) states symmetrically shared among\nthe three parties. Asymptotic transformations result in a simpler\nclassification than exact transformations. We show that m-partite pure states\nhaving an m-way Schmidt decomposition are simply parameterizable, with the\npartial entropy across any nontrivial partition representing the number of\nstandard ``Cat\u0027\u0027 states (|0^m\u003e+|1^m\u003e) asymptotically interconvertible to the\nstate in question. For general m-partite states, partial entropies across\ndifferent partitions need not be equal, and since partial entropies are\nconserved by asymptotically reversible LOCC operations, a multicomponent\nentanglement measure is needed, with each scalar component representing a\ndifferent kind of entanglement, not asymptotically interconvertible to the\nother kinds. In particular the m=4 Cat state is not isentropic to, and\ntherefore not asymptotically interconvertible to, any combination of bipartite\nand tripartite states shared among the four parties. Thus, although the m=4 cat\nstate can be prepared from bipartite EPR states, the preparation process is\nnecessarily irreversible, and remains so even asymptotically.",
"arxiv_id": "quant-ph/9908073",
"authors": [
"Charles H. Bennett",
"Sandu Popescu",
"Daniel Rohrlich",
"John A. Smolin",
"Ashish V. Thapliyal"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.63.012307",
"title": "Exact and Asymptotic Measures of Multipartite Pure State Entanglement",
"url": "https://arxiv.org/abs/quant-ph/9908073"
},
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