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 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.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 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.

arXiv:1406.5330Open Systems & Information DynamicsJun 1, 201213 citations

Remarks on quantum symmetry

Karl-Henning Rehren

Czechoslovak Journal of PhysicsDec 1, 19920 citations

Quantum statistical mechanics and Galois symmetries

Alain Connes A, Matilde Marcolli

Colloquium Publications - American Mathematical Society/Colloquium PublicationsDec 20, 20070 citations

Bethe Ansatz, Galois symmetries, and finite quantum systems

Tadeusz Lulek

We demonstrate some applications of arithmetic structures in Bethe Ansatz — the famous substitution which yields an exact solution of a quantum $N$ -body problem for the linear magnetic ring of $N$ spins $1/2$ within the XXX model. We point out the purelyWe demonstrate some applications of arithmetic structures in Bethe Ansatz — the famous substitution which yields an exact solution of a quantum $N$-body problem for the linear magnetic ring of $N$ spins $1/2$ within the XXX model. We point out the purely

Banach Center PublicationsJan 1, 20161 citation

PHASE SYMMETRIES OF COHERENT STATES IN GALOIS QUANTUM MECHANICS

Julien Page, Gabriel Catren

Nov 20, 20181 citation

We demonstrate some applications of arithmetic structures in Bethe Ansatz — the famous substitution which yields an exact solution of a quantum

N

-body problem for the linear magnetic ring of

N

spins

1/2

within the XXX model. We point out the purely