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Multipartite Entanglement Detection Via Projective Tensor Norms

Abstract

We introduce and study a class of entanglement criteria based on the idea of applying local contractions to an input multipartite state, and then computing the projective tensor norm of the output. More precisely, we apply to a mixed quantum state a tensor product of contractions from the Schatten class \(S_1\) to the Euclidean space \(\ell _2\), which we call entanglement testers. We analyze the performance of this type of criteria on bipartite and multipartite systems, for general pure and mixed quantum states, as well as on some important classes of symmetric quantum states. We also show that previously studied entanglement criteria, such as the realignment and the SIC POVM criteria, can be viewed inside this framework. This allows us to answer in the positive two conjectures of Shang, Asadian, Zhu, and Gühne by deriving systematic relations between the performance of these two criteria.

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Notes

  1. Note that this norm relation holds in general for any tester \({\mathcal {E}}\): \(\Vert {\mathcal {E}}^{\otimes 2}(\rho )\Vert _{\ell _2^{d^2} \otimes _\pi \ell _2^{d^2}} = \Vert {\hat{E}} X {\hat{E}}^*\Vert _1\).

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Acknowledgements

MAJ acknowledges the support of Université Toulouse III Paul Sabatier in the form of an invited professorship, during which this work was initiated. We would like to thank Guillaume Aubrun for the extremely valuable help in understanding map factorization questions.

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Correspondence to Cécilia Lancien.

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Communicated by Matthias Christandl.

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Jivulescu, M.A., Lancien, C. & Nechita, I. Multipartite Entanglement Detection Via Projective Tensor Norms. Ann. Henri Poincaré (2022). https://doi.org/10.1007/s00023-022-01187-9

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  • DOI: https://doi.org/10.1007/s00023-022-01187-9