Dimension-Free Entanglement Detection in Multipartite Werner States

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  • Felix Huber
  • Igor Klep
  • Victor Magron
  • Jurij Volčič

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.

OriginalsprogEngelsk
TidsskriftCommunications in Mathematical Physics
Vol/bind396
Sider (fra-til)1051–107
ISSN0010-3616
DOI
StatusUdgivet - 2022

Bibliografisk note

Funding Information:
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).

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

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