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An Asymptotic Property of Factorizable Completely Positive Maps and the Connes Embedding Problem

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Abstract

We establish a reformulation of the Connes embedding problem in terms of an asymptotic property of factorizable completely positive maps. We also prove that the Holevo–Werner channels \({W_n^-}\) are factorizable, for all odd integers \({n\neq 3}\). Furthermore, we investigate factorizability of convex combinations of \({W_3^+}\) and \({W_3^-}\), a family of channels studied by Mendl and Wolf, and discuss asymptotic properties for these channels.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen Ø, Denmark

    Uffe Haagerup & Magdalena Musat

Authors
  1. Uffe Haagerup
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  2. Magdalena Musat
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Correspondence to Magdalena Musat.

Additional information

Communicated by Y. Kawahigashi

U. Haagerup is supported by the ERC Advanced Grant no. OAFPG 247321, and partially supported by the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation at the University of Copenhagen, and the Danish Council for Independent Research, Natural Sciences.

M. Musat is supported by the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation at the University of Copenhagen, and the Danish Council for Independent Research, Natural Sciences.

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Haagerup, U., Musat, M. An Asymptotic Property of Factorizable Completely Positive Maps and the Connes Embedding Problem. Commun. Math. Phys. 338, 721–752 (2015). https://doi.org/10.1007/s00220-015-2325-9

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  • DOI: https://doi.org/10.1007/s00220-015-2325-9

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