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Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask)

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Abstract

In this paper we study the subset of generalized quantum measurements on finite dimensional systems known as local operations and classical communication (LOCC). While LOCC emerges as the natural class of operations in many important quantum information tasks, its mathematical structure is complex and difficult to characterize. Here we provide a precise description of LOCC and related operational classes in terms of quantum instruments. Our formalism captures both finite round protocols as well as those that utilize an unbounded number of communication rounds. While the set of LOCC is not topologically closed, we show that finite round LOCC constitutes a compact subset of quantum operations. Additionally we show the existence of an open ball around the completely depolarizing map that consists entirely of LOCC implementable maps. Finally, we demonstrate a two-qubit map whose action can be approached arbitrarily close using LOCC, but nevertheless cannot be implemented perfectly.

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Authors and Affiliations

  1. Department of Physics, Southern Illinois University, Carbondale, IL, 62901, USA

    Eric Chitambar

  2. The Perimeter Institute for Theoretical Physics, Waterloo, ON, N2L 2Y5, Canada

    Eric Chitambar

  3. Department of Combinatorics and Optimization, Institute for Quantum Computing, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Debbie Leung, Laura Mančinska & Maris Ozols

  4. IBM TJ Watson Research Center, 1101 Kitchawan Road, Yorktown Heights, NY, 10598, USA

    Maris Ozols

  5. Institució Catalana de Recerca i Estudis Avançats (ICREA), Pg. Lluis Companys 23, 08010, Barcelona, Spain

    Andreas Winter

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

    Andreas Winter

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

    Andreas Winter

  8. Centre for Quantum Technologies, National University of Singapore, 2 Science Drive 3, Singapore, 117542, Singapore

    Andreas Winter

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  1. Eric Chitambar
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Correspondence to Eric Chitambar.

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Communicated by M. B. Ruskai

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Chitambar, E., Leung, D., Mančinska, L. et al. Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask). Commun. Math. Phys. 328, 303–326 (2014). https://doi.org/10.1007/s00220-014-1953-9

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