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arXiv:1201.3903 · Journal of Physics Conference Series

Symmetries of finite Heisenberg groups forκ-partite systems

M Korbelář, J Tolar

Published Feb 8, 2012• 3 citations
arXivHTMLPDF

Abstract

Symmetries of finite Heisenberg groups represent an important tool for the study of deeper structure of finite-dimensional quantum mechanics. This short contribution presents extension of previous investigations to composite quantum systems comprised of k subsystems which are described with position and momentum variables in n i , i = 1,..., κ . Their Hilbert spaces are given by κ -fold tensor products of Hilbert spaces of dimensions n 1 ,..., n κ . Symmetry group of the corresponding finite Heisenberg group is given by the quotient group of a certain normalizer. We provide the description of the symmetry groups for arbitrary multipartite cases. The new class of symmetry groups represents very specific generalization of finite symplectic groups over modular rings.

DOI
10.1088/1742-6596/343/1/012122

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•Nov 21, 2012•1 citation
Symmetries of finite Heisenberg groups represent an important tool for the study of deeper structure of finite-dimensional quantum mechanics. This short contribution presents extension of previous investigations to composite quantum systems comprised of k subsystems which are described with position and momentum variables in n i , i = 1,..., κ . Their Hilbert spaces are given by κ -fold tensor products of Hilbert spaces of dimensions n 1 ,..., n κ . Symmetry group of the corresponding finite Heisenberg group is given by the quotient group of a certain normalizer. We provide the description of the symmetry groups for arbitrary multipartite cases. The new class of symmetry groups represents very specific generalization of finite symplectic groups over modular rings.
A composite quantum system comprising a finite number k of subsystems which are described with position and momentum variables in , i = 1, ..., k , is considered. Its Hilbert space is given by a k -fold tensor product of Hilbert spaces of dimensions n 1 , ..., n k . The symmetry group of the respective finite Heisenberg group is given by the quotient group of certain normalizer. This paper extends our previous investigation of bipartite quantum systems to arbitrary multipartite systems of the above type. It provides detailed description of the normalizers and the corresponding symmetry groups. The new class of symmetry groups represents a very specific generalization of symplectic groups over modular rings. As an application, a new proof of existence of the maximal set of mutually unbiased bases in Hilbert spaces of prime power dimensions is provided.
Abstract. Symmetries of the finite Heisenberg group represent an im-portant tool for the study of deeper structure of finite-dimensional quan-tum mechanics. As is well known, these symmetries are properly ex-pressed in terms of certain normalizer. This paper extends previous investigations to composite quantum systems consisting of two subsys-tems — qudits — with arbitrary dimensions n and m. In this paper we present detailed descriptions — in the group of inner automorphisms of GL(nm,C) — of the normalizer of the Abelian subgroup generated by tensor products of generalized Pauli matrices of orders n and m. The symmetry group is then given by the quotient group of the normalizer. Contents
Symmetries of the finite Heisenberg group represent an important tool for the study of deeper structure of finite-dimensional quantum mechanics. As is well known, these symmetries are properly expressed in terms of certain normalizer. This paper extends previous investigations to composite quantum systems consisting of two subsystems—qudits—with arbitrary dimensions n and m . In this paper, we present detailed descriptions—in the group of inner automorphisms of —of the normalizer of the Abelian subgroup generated by tensor products of generalized Pauli matrices of orders n and m . The symmetry group is then given by the quotient group of the normalizer.
In this paper the Clifford groups of general multipartite quantum systems with configuration space Zn1⊕⋯⊕Znk\mathbb{Z}_{n_1}\oplus\cdots\oplus\mathbb{Z}_{n_k}Zn1​​⊕⋯⊕Znk​​, k≥1k\geq 1k≥1 are studied. It is known that the Clifford group is a natural semidirect product provided the dimension N=n1⋯nkN=n_1\cdots n_kN=n1​⋯nk​ of the corresponding Hilbert space is an odd number. For even NNN special results on the Clifford groups are scattered in the mathematical literature, but they do not concern the semidirect product structure. Using the relation of generators of the associated symplectic group Sp[n1,…,nk]\mathrm{Sp}_{[n_1,\dots,n_k]}Sp[n1​,…,nk​]​ we prove that for even N=n1⋯nkN=n_1\cdots n_kN=n1​⋯nk​ both Clifford group and the projective Clifford group are natural semidirect products if and only if NNN is not divisible by four.
It is well-known (see [1]) that for a Heisenberg magnet symmetry operators and symmetry classes can be defined in a very similar way as for tensors (see e.g. [2, 3, 4]). Newer papers which consider the action of permutations on the Hilbert space Ti of the Heisenberg magnet are [5, 6, 7, 8]. We define symmetry classes and commutation symmetries in the Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites and investigate them by means of tools from the representation theory of symmetric groups C[ S N ] such as decompositions of ideals of the group ring C[ S N ], idempotents of C[ S N ], discrete Fourier transforms of S N , Littlewood-Richardson products. In particular, we determine smallest symmetry classes and stability subgroups of both single eigenvectors v and subspaces U of eigenvectors of the Hamiltonian H of the magnet. Expectedly, the symmetry classes defined by stability subgroups of v or U are bigger than the corresponding smallest symmetry classes of v or U , respectively. The determination of the smallest symmetry class for U bases on an algorithm which calculates explicitely a generating idempotent for a non-direct sum of right ideals of C[ S N ]. Let U (r1,r2) μ be a subspace of eigenvectors of a fixed eigenvalue μ of H with weight ( r 1 , r 2 ). If one determines the smallest symmetry class for every v U (r1,r2) μ then one can observe jumps of the symmetry behaviour. For "generic" v U (r1,r2) μ all smallest symmetry classes have the same maximal dimension d and 'structure'. But U (r1,r2) μ can contain linear subspaces on which the dimension of the smallest symmetry class of v jumps to a value smaller than d. Then the stability subgroup of v can increase. We can calculate such jumps explicitely. In our investigations we use computer calculations by means of the Mathematica packages PERMS and HRing.
Abstract. It is well-known (see [1]) that for a Heisenberg magnet symmetry operators and symmetry classes can be defined in a very similar way as for tensors (see e.g. [2, 3, 4]). Newer papers which consider the action of permutations on the Hilbert space H of the Heisenberg magnet are [5, 6, 7, 8]. We define symmetry classes and commutation symmetries in the Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites and investigate them by means of tools from the representation theory of symmetric groups SN such as decompositions of ideals of the group ring C[SN], idempotents of C[SN], discrete Fourier transforms of SN, Littlewood-Richardson products. In particular, we determine smallest symmetry classes and stability subgroups of both single eigenvectors v and subspaces U of eigenvectors of the Hamiltonian H of the magnet. Expectedly, the symmetry classes defined by stability subgroups of v or U are bigger than the corresponding smallest symmetry classes of v or U, respectively. The determination of the smallest symmetry class for U bases on an algorithm which calculates explicitely a generating idempotent for a non-direct sum of right ideals of C[SN]. Let U (r1,r2)