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Dimension-Free Entanglement Detection in Multipartite Werner States

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Abstract

Werner states are multipartite quantum states that are invariant under the diagonal conjugate action of the unitary group. This paper gives a complete characterization of their entanglement that is independent of the underlying local Hilbert space: for every entangled Werner state there exists a dimension-free entanglement witness. The construction of such a witness is formulated as an optimization problem. To solve it, two semidefinite programming hierarchies are introduced. The first one is derived using real algebraic geometry applied to positive polynomials in the entries of a Gram matrix, and is complete in the sense that for every entangled Werner state it converges to a witness. The second one is based on a sum-of-squares certificate for the positivity of trace polynomials in noncommuting variables, and is a relaxation that involves smaller semidefinite constraints.

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Authors and Affiliations

  1. Institute of Theoretical Physics, Jagiellonian University, 30-348, Kraków, Poland

    Felix Huber

  2. Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia

    Igor Klep

  3. LAAS-CNRS, Institute of Mathematics, Toulouse, France

    Victor Magron

  4. Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark

    Jurij Volčič

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  1. Felix Huber
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  2. Igor Klep
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  4. Jurij Volčič
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Correspondence to Jurij Volčič.

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Communicated by G. Chiribella.

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Felix Huber was supported by the FNP through TEAM-NET (POIR.04.04.00-00-17C1/18-00). Igor Klep was supported by the Slovenian Research Agency grants J1-2453, J1-8132, N1-0217 and P1-0222. Victor Magron was supported by the French Research Agency grants ANR-18-ERC2-0004-01 and ANR-19-PI3A-0004, the EU’s Horizon 2020 research and innovation programme 813211, and the PHC Proteus grant 46195TA. Jurij Volčič was supported by the National Science Foundation grant DMS-1954709, and Villum Fonden via the Villum Young Investigator grant (No. 37532).

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Huber, F., Klep, I., Magron, V. et al. Dimension-Free Entanglement Detection in Multipartite Werner States. Commun. Math. Phys. 396, 1051–1070 (2022). https://doi.org/10.1007/s00220-022-04485-9

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