Designing locally maximally entangled quantum states with arbitrary local symmetries

TL;DR

This paper presents a method to design multiparticle quantum states with arbitrarily large local symmetries, enhancing robustness and applicability in quantum information protocols.

Contribution

It introduces a systematic way to create critical states with extensive local symmetries using representation theory and combinatorial analysis, advancing quantum state design.

Findings

States can have arbitrarily large local unitary symmetry groups.

Designed states are realizable in systems of distinguishable traps with bosons or fermions.

Theoretical proof that tensor powers of irreducible representations contain trivial representations.

Abstract

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.