# Locally Maximally Entangled States of Multipart Quantum Systems

Jim Bryan1, Samuel Leutheusser2,3, Zinovy Reichstein1, and Mark Van Raamsdonk2

1Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, B.C., V6T 1Z1, Canada
2Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, B.C., V6T 1Z2, Canada
3Center for Theoretical Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

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### Abstract

For a multipart quantum system, a locally maximally entangled (LME) state is one where each elementary subsystem is maximally entangled with its complement. This paper is a sequel to~[J. Bryan, Z. Reichstein and M. Van Raamsdonk, Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory, Ann. Henri Poincaré 19 (2018), no. 8, 2491-2511. MR3830220], which gives necessary and sufficient conditions for a system to admit LME states in terms of its subsystem dimensions $(d_1, d_2, \dots, d_n)$, and computes the dimension of the space ${\cal S}_{LME}/K$ of LME states up to local unitary transformations for all non-empty cases. Here we provide a pedagogical overview and physical interpretation of the underlying mathematics that leads to these results and give a large class of explicit constructions for LME states. In particular, we construct all LME states for tripartite systems with subsystem dimensions $(2,A,B)$ and give a general representation-theoretic construction for a special class of stabilizer LME states. The latter construction provides a common framework for many known LME states. Our results have direct implications for the problem of characterizing SLOCC equivalence classes of quantum states, since points in ${\cal S}_{LME}/K$ correspond to natural families of SLOCC classes. Finally, we give the dimension of the stabilizer subgroup $S \subset \operatorname{SL}(d_1, \mathbb{C}) \times \cdots \times \operatorname{SL}(d_n, \mathbb{C})$ for a generic state in an arbitrary multipart system and identify all cases where this stabilizer is trivial.

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[2] Robin Reuvers, "Generalized Pauli constraints in large systems: The Pauli principle dominates", Journal of Mathematical Physics 62 3, 032204 (2021).

[3] Xiao-Dong Yu, Timo Simnacher, Nikolai Wyderka, H. Chau Nguyen, and Otfried Gühne, "A complete hierarchy for the pure state marginal problem in quantum mechanics", Nature Communications 12 1, 1012 (2021).

[4] Ning Bao, Newton Cheng, Sergio Hernández-Cuenca, and Vincent P. Su, "The quantum entropy cone of hypergraphs", SciPost Physics 9 5, 067 (2020).

[5] Oskar Słowik, Martin Hebenstreit, Barbara Kraus, and Adam Sawicki, "A link between symmetries of critical states and the structure of SLOCC classes in multipartite systems", Quantum 4, 300 (2020).

[6] David Sauerwein, Nolan R. Wallach, Gilad Gour, and Barbara Kraus, "Transformations among Pure Multipartite Entangled States via Local Operations are Almost Never Possible", Physical Review X 8 3, 031020 (2018).

[7] Jim Bryan, Zinovy Reichstein, and Mark Van Raamsdonk, "Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory", Annales Henri Poincaré 19 8, 2491 (2018).

[8] Xin Li, "Maximal lexicographic spectra and ranks for states with fixed uniform margins", arXiv:1801.00986.

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