Galois actions on the eigenproblem of the Heisenberg heptagon

TL;DR

This paper explores the eigenproblem of the Heisenberg heptagon using Galois theory, revealing a rich arithmetic structure that introduces Galois qubits as potential quantum memory units, linking algebraic extensions to quantum states.

Contribution

It introduces Galois qubits derived from the eigenproblem of the Heisenberg heptagon, connecting algebraic field extensions with quantum information concepts.

Findings

Identification of Galois qubits as elementary memory units.

Analysis of the lattice of Galois groups via Kummer theory.

Connection between field extensions and quantum state permutations.

Abstract

We analyse the exact solution of the eigenproblem for the Heisenberg Hamiltonian of magnetic heptagon, i.e. the ring of N=7 nodes, each with spin 1/2, within the XXX model with nearest neighbour interactions, from the point of view of finite extensions of the field $\mathbb{Q}$ of rationals. We point out, as the main result, that the associated arithmetic structure of these extensions makes natural an introduction of some Galois qubits. They are two-dimensional subspaces of the Hilbert space of the model, which admit a quantum informatic interpretation as elementary memory units for a (hypothetical) computer, based on their distinctive properties with respect to the action of related Galois group for indecomposable factors of the secular determinant. These Galois qubits are nested on the lattice of subfields which involves several minimal fields for determination of eigenstates (the complex Heisenberg field), spectrum (the real Heisenberg field), and Fourier transforms of magnetic configurations (the cyclotomic field, based on the simple 7th root of unity). The structure of the corresponding lattice of Galois groups is presented in terms of Kummer theory, and its physical interpretation is indicated in terms of appropriate permutations of eigenstates, energies, and density matrices.