Entangled symmetric states and copositive matrices

Carlo Marconi1, Albert Aloy2, Jordi Tura3,4, and Anna Sanpera1,5

1Física Teòrica: Informació i Fenòmens Quàntics. Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain
2ICFO - Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
3Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany
4Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
5ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain

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Entanglement in symmetric quantum states and the theory of copositive matrices are intimately related concepts. For the simplest symmetric states, i.e., the diagonal symmetric (DS) states, it has been shown that there exists a correspondence between exceptional (non-exceptional) copositive matrices and non-decomposable (decomposable) Entanglement Witnesses (EWs). Here we show that EWs of symmetric, but not DS, states can also be constructed from extended copositive matrices, providing new examples of bound entangled symmetric states, together with their corresponding EWs, in arbitrary odd dimensions.

Entanglement is one of the most intriguing phenomena in quantum physics whose implications have profound consequences not only from a theoretical point of view but also in light of some computational tasks that would be otherwise unfeasible with classical systems.
For this reason, deciding whether a quantum state is entangled or not, is a problem of paramount importance whose solution, unfortunately, is known to be NP-hard in the general scenario.
In some cases, however, symmetries provide a useful framework to recast the separability problem in a simpler way, thus reducing the original complexity of this task.
In this work we focus on symmetric states, i.e., states that are invariant under permutations of the parties, showing how, in the case of the qudits, the characterization of the entanglement can be accomplished by means of a class of matrices known as copositive. In particular, we establish a connection between entanglement witnesses, i.e., hermitian operators that are able to detect entanglement, and copositive matrices, showing how only a subset of them, dubbed as exceptional, can be used to assess PPT-entanglement in any dimension, with the PPT-entangled edge states detected by the so-called extremal matrices.
Finally we illustrate our findings discussing some examples of families of PPT-entangled states in 3-level and 4-level systems, along with the entanglement witnesses that detect them.
We conjecture that any PPT-entangled state of two qudits can be detected by means of an entanglement witness of the form that we propose.

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[2] Beatrix C. Hiesmayr, "Free versus Bound Entanglement: Machine learning tackling a NP-hard problem", arXiv:2106.03977.

[3] Hari krishnan S V, Ashish Ranjan, and Manik Banik, "State space structure of tripartite quantum systems", Physical Review A 104 2, 022437 (2021).

[4] Joonwoo Bae, Anindita Bera, Dariusz Chruściński, Beatrix C. Hiesmayr, and Daniel McNulty, "How many measurements are needed to detect bound entangled states?", arXiv:2108.01109.

[5] Beatrix C. Hiesmayr, "Free versus bound entanglement, a NP-hard problem tackled by machine learning", Scientific Reports 11, 19739 (2021).

[6] Adam Burchardt, Jakub Czartowski, and Karol Życzkowski, "Entanglement in highly symmetric multipartite quantum states", Physical Review A 104 2, 022426 (2021).

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