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 Nth tensor power of any irreducible representation of 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.
Highlights
Multipartite entangled states [1, 2] play important roles in different areas of physics including quantum computation, quantum communication [4, 5] and quantum metrology [6] as well as condensed matter physics [7]
To make calculations more traceable, we introduce the moving distinguished point denoted by O
We introduce the parameter = j − n, i.e. the distance from O
Summary
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. Local symmetries of the designed quantum state are equal to the unitary group of local mode operations acting diagonally on all traps. Our main technical result states that the N th tensor power of any irreducible representation of 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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