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Positive Maps and Entanglement in Real Hilbert Spaces

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Abstract

The theory of positive maps plays a central role in operator algebras and functional analysis and has countless applications in quantum information science. The theory was originally developed for operators acting on complex Hilbert spaces and less is known about its variant on real Hilbert spaces. In this article, we study positive maps acting on a full matrix algebra over the reals, pointing out a number of fundamental differences with the complex case and discussing their implications in quantum information. We provide a necessary and sufficient condition for a real map to admit a positive complexification and connect the existence of positive maps with non-positive complexification with the existence of mixed states that are entangled in real Hilbert space quantum mechanics, but separable in the complex version, providing explicit examples both for the maps and for the states. Finally, we discuss entanglement breaking and PPT maps, and we show that a straightforward real version of the PPT-squared conjecture is false even in dimension 2. Nevertheless, we show that the original PPT-squared conjecture implies a different conjecture for real maps, in which the PPT property is replaced by a stronger property of invariance under partial transposition (IPT). When the IPT property is assumed, we prove an asymptotic version of the conjecture.

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Acknowledgements

GC was supported by the Hong Kong Research Grant Council through Grant 17300920 and through the Senior Research Fellowship Scheme via SRFS2021-7S02, and by the Croucher foundation. KRD and VIP were partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). MR is supported by the European Research Council (ERC Grant Agreement No. 851716). The authors would like to thank the anonymous referees for many helpful suggestions in this article.

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Authors and Affiliations

  1. Department of Computer Science, The University of Hong Kong, Pok Fu Lam Road, Pok Fu Lam, Hong Kong

    Giulio Chiribella

  2. Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1 , Canada

    Kenneth R. Davidson

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

    Vern I. Paulsen

  4. Univ Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP, 69342, Lyon Cedex 07, France

    Mizanur Rahaman

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  1. Giulio Chiribella
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  4. Mizanur Rahaman
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Correspondence to Mizanur Rahaman.

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Communicated by David Pérez-García.

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Chiribella, G., Davidson, K.R., Paulsen, V.I. et al. Positive Maps and Entanglement in Real Hilbert Spaces. Ann. Henri Poincaré 24, 4139–4168 (2023). https://doi.org/10.1007/s00023-023-01325-x

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