Wreath products and multipartite quantum systems

Abstract

A natural symmetry group of a multicomponent quantum system is a special combination of a symmetry group acting within a single component (“local group”) and a group that permutes the components (“spatial symmetry group”). This combination is called the wreath product. Unitary representations of wreath products describe quantum evolutions of multipartite systems. It is known that any unitary representation of a finite group is contained in some permutation representation. We describe an algorithm for decomposing permutation representations of wreath products into irreducible components. This decomposition makes it possible to study the quantum behavior (entanglement, non-local correlations, etc.) of multipartite systems in invariant subspaces of the permutation Hilbert space.