Abstract
The multipartite quantum systems are of particular interest for the study of such phenomena as entanglement and non-local correlations. The symmetry group of the whole multipartite system is the wreath product of the group acting in the “local” Hilbert space and the group of permutations of the constituents. The dimension of the Hilbert space of a multipartite system depends exponentially on the number of constituents, which leads to computational difficulties. We describe an algorithm for decomposing representations of wreath products into irreducible components. The C implementation of the algorithm copes with representations of dimensions in quadrillions. The program, in particular, builds irreducible invariant projectors in the Hilbert space of a multipartite system. The expressions for these projectors are tensor product polynomials. This structure is convenient for efficient computation of quantum correlations in multipartite systems with a large number of constituents.
Highlights
The Hilbert space of a multipartite quantum system is the tensor product of the Hilbert spaces of the components: H = N ⊗ HxThe states of the multipartite system that can be represented as a x=1 weighted sum of tensor products of the states of the components are called separable
To reproduce the usual property of the space, homogeneity, we will assume that the group of spatial symmetries G = G(X) permutes transitively the components of a multipartite system
K is the number of i=1 irreducible components of the wreath product representation, k(i) denotes some numbering of the orbits of G on LX, 1MN is the identity matrix in the Hilbert space (1)
Summary
The Hilbert space of a multipartite quantum system is the tensor product of the Hilbert spaces of the components: H. To reproduce the usual property of the space, homogeneity, we will assume that the group of spatial symmetries G = G(X) permutes transitively the components of a multipartite system. In this case, the Hilbert space of the system can be written as. We can embed any constructive quantum model into a suitable invariant subspace of some permutation representation [3, 4]. If we can decompose permutation representations of a group into irreducible components, we can construct any representations of the group. We describe an algorithm for decomposing Hilbert space (1) into invariant subspaces with respect to the natural symmetry group of a multipartite quantum system. The algorithm outputs a complete set of projectors to irreducible invariant subspaces
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