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.
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.
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.
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
Discrete structures in product Hilbert spaces are investigated. For monopartite systems of size
d one relies on the Weyl--Heisenberg group
WH(d), while in the case of composite Hilbert spaces with the local dimensions
di we identify designs covariant with respect to the product group,
⨂iWH(di). In analogy with magic -- a quantity attaining its maximum for states fiducial with respect to
WH(d) -- we introduce a similar quantifier of {\sl product magic}, defined with respect to the product group. The maximum of this quantity over all equimodular vectors yields fiducial states that generate
d=∏idi \textit{a priori} isoentangled mutually unbiased bases (MUBs), which, when supplemented by the identity, form their complete set. Such fiducial states are explicitly constructed in all prime-power dimensions
d=pn with
p≥3. The result for
p≥5 extends the construction of Klappenecker and R{ö}tteler, whereas for
p=3 it is mathematically distinct and is based on Galois rings. The global maximum of the quantifier of product magic for
d=23 yields fiducial states corresponding to the symmetric informationally complete (SIC) generalized measurement of Hoggar. Our approach feeds into a unifying perspective in which highly symmetric quantum designs emerge from fiducial states with extremal properties via structured group-orbit constructions.
The connection between maximal sets of mutually unbiased bases (MUBs) in a prime-power dimensional Hilbert space and finite phase-space geometries is well known. In this article, we classify MUBs according to their degree of covariance with respect to the natural symmetries of a finite phase-space, which are the group of its affine symplectic transformations. We prove that there exist maximal sets of MUBs that are covariant with respect to the full group only in odd prime-power dimensional spaces, and in this case, their equivalence class is actually unique. Despite this limitation, we show that in dimension [Formula: see text] covariance can still be achieved by restricting to proper subgroups of the symplectic group, that constitute the finite analogues of the oscillator group. For these subgroups, we explicitly construct the unitary operators yielding the covariance.