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Entangleability of cones

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Abstract

We solve a long-standing conjecture by Barker, proving that the minimal and maximal tensor products of two finite-dimensional proper cones coincide if and only if one of the two cones is generated by a linearly independent set. Here, given two proper cones \({\mathcal {C}}_1\), \({\mathcal {C}}_2\), their minimal tensor product is the cone generated by products of the form \(x_1\otimes x_2\), where \(x_1\in {\mathcal {C}}_1\) and \(x_2\in {\mathcal {C}}_2\), while their maximal tensor product is the set of tensors that are positive under all product functionals \(\varphi _1\otimes \varphi _2\), where \(\varphi _1|_{{\mathcal {C}}_1}\geqslant 0\) and \(\varphi _2|_{{\mathcal {C}}_2}\geqslant 0\). Our proof techniques involve a mix of convex geometry, elementary algebraic topology, and computations inspired by quantum information theory. Our motivation comes from the foundations of physics: as an application, we show that any two non-classical systems modelled by general probabilistic theories can be entangled.

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Notes

  1. Alternatively, the reader will find at https://github.com/gaubrun/entangleability a SageMath script which checks the correctness of (7).

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Acknowledgements

We are very grateful to Kyung Hoon Han for several remarks which helped us clarifying the manuscript. We thank also Alexander Müller-Hermes for useful comments on some of the results. GA was supported in part by ANR (France) under the grant StoQ (2014-CE25-0003). LL acknowledges financial support from the European Research Council under the Starting Grant GQCOP (Grant no. 637352), from the Foundational Questions Institute under the grant FQXi-RFP-IPW-1907, and from the Alexander von Humboldt Foundation. CP is partially supported by Spanish MINECO through Grant No. MTM2017-88385-P, by the Comunidad de Madrid through grant QUITEMAD-CM P2018/TCS4342 and by SEV-2015-0554-16-3. MP acknowledges support from grant VEGA 2/0142/20, from the grant of the Slovak Research and Development Agency under contract APVV-16-0073, from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation - 447948357) and the ERC (Consolidator Grant 683107/TempoQ).

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Authors and Affiliations

  1. Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622, Villeurbanne, France

    Guillaume Aubrun

  2. School of Mathematical Sciences and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, University Park, NG7 2RD, Nottingham, UK

    Ludovico Lami

  3. Institute of Theoretical Physics and IQST, Universität Ulm, Albert-Einstein-Allee 11D, 89069, Ulm, Germany

    Ludovico Lami

  4. Dpto. Análisis Matemático y Matemática Aplicada, Fac. Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias s/n, 28040, Madrid, Spain

    Carlos Palazuelos

  5. Instituto de Ciencias Matemáticas, C/ Nicolás Cabrera, 13–15, 28049, Madrid, Spain

    Carlos Palazuelos

  6. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, Bratislava, Slovakia

    Martin Plávala

  7. Naturwissenschaftlich-Technische Fakultät, Universität Siegen, 57068, Siegen, Germany

    Martin Plávala

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  1. Guillaume Aubrun
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Correspondence to Guillaume Aubrun.

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Aubrun, G., Lami, L., Palazuelos, C. et al. Entangleability of cones. Geom. Funct. Anal. 31, 181–205 (2021). https://doi.org/10.1007/s00039-021-00565-5

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  • DOI: https://doi.org/10.1007/s00039-021-00565-5

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