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Relative Entropy and Squashed Entanglement

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Abstract

We are interested in the properties and relations of entanglement measures. Especially, we focus on the squashed entanglement and relative entropy of entanglement, as well as their analogues and variants.

Our first result is a monogamy-like inequality involving the relative entropy of entanglement and its one-way LOCC variant. The proof is accomplished by exploring the properties of relative entropy in the context of hypothesis testing via one-way LOCC operations, and by making use of an argument resembling that by Piani on the faithfulness of regularized relative entropy of entanglement.

Following this, we obtain a commensurate and faithful lower bound for squashed entanglement, in the form of one-way LOCC relative entropy of entanglement. This gives a strengthening to the strong subadditivity of von Neumann entropy. Our result improves the trace-distance-type bound derived in Brandão et al. (Commun Math Phys, 306:805–830, 2011), where faithfulness of squashed entanglement was first proved. Applying Pinsker’s inequality, we are able to recover the trace-distance-type bound, even with slightly better constant factor. However, the main improvement is that our new lower bound can be much larger than the old one and it is almost a genuine entanglement measure.

We evaluate exactly the relative entropy of entanglement under various restricted measurement classes, for maximally entangled states. Then, by proving asymptotic continuity, we extend the exact evaluation to their regularized versions for all pure states. Finally, we consider comparisons and separations between some important entanglement measures and obtain several new results on these, too.

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Authors and Affiliations

  1. Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore, 117543, Singapore

    Ke Li & Andreas Winter

  2. ICREA-Institució Catalana de Recerca i Estudis Avançats, Pg. Lluis Companys 23, 08010, Barcelona, Spain

    Andreas Winter

  3. Física Teòrica: Informació i Fenomens Quàntics, Universitat Autònoma de Barcelona, 08193, Bellaterra (Barcelona), Spain

    Andreas Winter

  4. Department of Mathematics, University of Bristol, Bristol, BS8 1TW, UK

    Andreas Winter

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  1. Ke Li
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  2. Andreas Winter
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Correspondence to Andreas Winter.

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Communicated by M. B. Ruskai

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Li, K., Winter, A. Relative Entropy and Squashed Entanglement. Commun. Math. Phys. 326, 63–80 (2014). https://doi.org/10.1007/s00220-013-1871-2

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  • DOI: https://doi.org/10.1007/s00220-013-1871-2

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