1, 2, and 6 qubits, and the Ramanujan-Nagell theorem

TL;DR

This paper explores the symmetry groups of 1, 2, and 6-qubit systems using Ramanujan-Nagell theorem, revealing unique isomorphisms and ruling out certain symmetries through topological analysis.

Contribution

It connects number theory with quantum system symmetries, identifying specific qubit systems with unique symmetry group isomorphisms and applying topological methods to exclude some cases.

Findings

Only 1, 2, and 6-qubit systems share specific symmetry group isomorphisms.

Topological analysis rules out the 6-qubit case.

The Ramanujan-Nagell theorem underpins the classification of these symmetries.

Abstract

A conjecture of Ramanujan that was later proved by Nagell is used to show on the basis of matching dimensions that only three $n$-qubit systems, for $n=1, 2, 6$, can share an isomorphism of their symmetry groups with the rotation group of corresponding dimensions $3, 6, 91$. Topological analysis, however, rules out the last possibility.