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 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.
Abstract. A composite quantum system comprising a finite number k of subsystems which are described with position and momentum variables in Zn i , 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 . 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.
We prove that a commutative parasemifield S is additively idempotent provided that it is finitely generated as a semiring. Consequently, every proper commutative semifield T that is finitely generated as a semiring is either additively constant or additively idempotent. As part of the proof, we use the classification of finitely generated lattice-ordered groups to prove that a certain monoid associated to the parasemifield S has a distinguished geometrical property called prismality.2010 Mathematics Subject Classification. primary: 12K10, 16Y60, 20M14, secondary: 11H06, 52A20.This statement can be now understood as an extension of the folklore theorem referred in the beginning, i.e., that every (commutative) field that is finitely generated as a ring is finite. Of course, the greatest difference is the existence of additively idempotent semifields.Since every semifield is clearly ideal-simple, part (b) of the theorem follows immediately from (c). Also, one can be slightly more precise in the additively constant case: this occurs if and only if there is a finitely generated (multiplicative) abelian group G(·) and the semiring is the semifield S := G ∪ {o}, where o is a new element. Operations that extend the multiplication on G are defined by a+b = o and a·o = o for all a, b ∈ S.
Commutative semirings with divisible additive semigroup are studied. We show that an additively divisible commutative semiring is idempotent, provided that it is finitely generated and torsion. In case that a one-generated additively divisible semiring posseses no unit, it must contain an ideal of idempotent elements. We also present a series of open questions about finitely generated commutative semirings and their equivalent versions.2010 Mathematics Subject Classification. 16Y60, 20M14.
Let A be a commutative nilpotent finitely-dimensional algebra overis the subalgebra of A generated by elements a p , a ∈ A. We show that the conjecture holds if A (p) is at most 2-generated. We give a complete characterization of 2-generated nilpotent commutative algebras in the terms of standard basis with respect to the reverse lexicographical ordering.
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