On the geometry and invariants of qubits, quartits and octits

TL;DR

This paper explores the geometric structures and invariants of multi-level quantum systems like qubits, quartits, and octits, using group theory and connections to complex reflection groups and the Weyl group of E8.

Contribution

It introduces a group-theoretic framework for understanding invariants of qubits, quartits, and octits, linking quantum systems to classical geometric objects and symmetry groups.

Findings

Invariants of complex reflection groups related to qubits and quartits are identified.

Real gates over octits are derived, involving the Weyl group of E8.

Connections between multilevel quantum systems and solid state NMR are discussed.

Abstract

Four level quantum systems, known as quartits, and their relation to two- qubit systems are investigated group theoretically. Following the spirit of Klein's lectures on the icosahedron and their relation to Hopf sphere bra- tions, invariants of complex re ection groups occuring in the theory of qubits and quartits are displayed. Then, real gates over octits leading to the Weyl group of E8 and its invariants are derived. Even multilevel systems are of interest in the context of solid state nuclear magnetic resonance.