Designing locally maximally entangled quantum states with arbitrary local symmetries

Oskar Słowik1, Adam Sawicki1, and Tomasz Maciążek2

1Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland
2School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol BS8 1UG, UK

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One of the key ingredients of many LOCC protocols in quantum information is a multiparticle (locally) maximally entangled quantum state, aka a critical state, that possesses local symmetries. We show how to design critical states with arbitrarily large local unitary symmetry. We explain that such states can be realised in a quantum system of distinguishable traps with bosons or fermions occupying a finite number of modes. Then, local symmetries of the designed quantum state are equal to the unitary group of local mode operations acting diagonally on all traps. Therefore, such a group of symmetries is naturally protected against errors that occur in a physical realisation of mode operators. We also link our results with the existence of so-called strictly semistable states with particular asymptotic diagonal symmetries. Our main technical result states that the $N$th tensor power of any irreducible representation of $\mathrm{SU}(N)$ contains a copy of the trivial representation. This is established via a direct combinatorial analysis of Littlewood-Richardson rules utilising certain combinatorial objects which we call telescopes.

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[2] 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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