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arXiv:quant-ph/0103096 · Physical Review Letters

Asymptotic Relative Entropy of Entanglement

K. Audenaert, Jens Eisert, E. Jané, Martin Bodo Plenio, S. Virmani, Bart De Moor

Published Nov 6, 2001• 97 citations
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Abstract

We present an analytical formula for the asymptotic relative entropy of entanglement with respect to positive partial transpose states for Werner states of arbitrary dimension. We then demonstrate its validity using methods from convex optimization. This is the first case in which the asymptotic value of a subadditive entanglement measure has been calculated.

DOI
10.1103/physrevlett.87.217902

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We present an analytical formula for the asymptotic relative entropy of entanglement with respect to positive partial transpose states for Werner states of arbitrary dimension. We then demonstrate its validity using methods from convex optimization. This is the first case in which the asymptotic value of a subadditive entanglement measure has been calculated.
The study of entanglement inevitably leads to the concept of regularization, where measures are evaluated on infinitely many copies of a state. Single-letter formulas try to make these asymptotic entanglement quantities accessible through a calculation on one copy of a state, but are rarely available. We introduce a decreasing hierarchy of computable upper bounds on the asymptotic relative entropy of entanglement with respect to positive-partial-transpose (PPT) states. The regularization of every fixed hierarchy level equals this asymptotic quantity. Additivity at any level therefore yields a single-letter formula, even when the usual one-copy relative entropy is nonadditive. We establish such additivity at the first nontrivial level for two broad multiparameter families, both containing all Werner states. The upper-bound construction also extends to sandwiched Renyi divergences. The hierarchy motivates kkk-multinegative states, a generalization of binegative states and the associated kkk-multinegativity, which gives explicit upper bounds on both the asymptotic relative entropy and exact PPT entanglement cost. We construct states which are kkk-multinegative at arbitrarily large depths kkk that separate consecutive levels of the previously introduced entanglement-cost hierarchy, disproving its conjectured finite collapse. These results provide state-dependent single-letter formulas while identifying a limitation of universal finite-level characterizations.
For a special class of bipartite states we calculate explicitly the asymptotic relative entropy of entanglement ${E}_{R}^{\ensuremath{\infty}}$ with respect to states having a positive partial transpose. This quantity is an upper bound to distillable entanglement. The states considered are invariant under rotations of the form $\mathrm{O}\ensuremath{\bigotimes}\mathrm{O},$ where O is any orthogonal matrix. We show that in this case ${E}_{R}^{\ensuremath{\infty}}$ is equal to another upper bound on distillable entanglement, constructed by Rains. To perform these calculations, we have introduced a number of results that are interesting in their own right: (i) the Rains bound is convex and continuous; (ii) under some weak assumption, the Rains bound is an upper bound to ${E}_{R}^{\ensuremath{\infty}};$ (iii) for states for which the relative entropy of entanglement ER{E}_{R}ER​ is additive, the Rains bound is equal to ER.{E}_{R}.ER​.
Quantifying entanglement for multipartite quantum state is a crucial task in many aspects of quantum information theory. Among all the entanglement measures, relative entropy of entanglement ERE_{R}ER​ is an important quantity due to its clear geometric meaning, easy compatibility with different system sizes, and various applications in many other related quantity calculations. Lower bounds of ERE_RER​ were previously found based on distance to the set of positive partial transpose states. We propose a method to calculate upper bounds of ERE_RER​ based on the convex hull approximation of the set of separable states. We apply our method to calculate ERE_RER​ for composite systems of various sizes, and compare with the previous known lower bounds, obtaining promising results. Our method adds a new tool for entanglement measure calculation and deepens our understanding for the structure of separable states.
Abstract Quantifying entanglement for multipartite quantum state is a crucial task in many aspects of quantum information theory. Among all the entanglement measures, relative entropy of entanglement E R is an outstanding quantity due to its clear geometric meaning, easy compatibility with different system sizes, and various applications in many other related quantity calculations. Lower bounds of E R were previously found based on distance to the set of positive partial transpose states. We propose a method to calculate upper bounds of E R based on active learning, a subfield in machine learning, to generate an approximation of the set of separable states. We apply our method to calculate E R for composite systems of various sizes, and compare with the previous known lower bounds, obtaining promising results. Our method adds a reliable tool for entanglement measure calculation and deepens our understanding for the structure of separable states.
We demonstrate the irreversibility of asymptotic entanglement manipulation under quantum operations that completely preserve the positivity of partial transpose (PPT), resolving a major open problem in quantum information theory. Our key tool is a new efficiently computable additive lower bound for the asymptotic relative entropy of entanglement with respect to PPT states, which can be used to evaluate the entanglement cost under local operations and classical communication (LOCC). We find that for any rank-two mixed state supporting on the 3⊗3 antisymmetric subspace, the amount of distillable entanglement by PPT operations is strictly smaller than one entanglement bit (ebit) while its entanglement cost under PPT operations is exactly one ebit. As a by-product, we find that for this class of states, both the Rains's bound and its regularization are strictly less than the asymptotic relative entropy of entanglement. So, in general, there is no unique entanglement measure for the manipulation of entanglement by PPT operations. We further show a computable sufficient condition for the irreversibility of entanglement distillation by LOCC (or PPT) operations.