Geometry of the generalized Bloch sphere for qutrits

TL;DR

This paper explores the geometry of the state space of a qutrit, providing explicit formulas and classifying its sections, revealing new geometric and symmetry features not previously recognized.

Contribution

It offers closed-form descriptions of the qutrit Bloch sphere, classifies its sections into equivalence classes, and uncovers new geometric and symmetry properties.

Findings

Explicit formulas for the qutrit Bloch sphere and its boundary.

Classification of 28 two-sections and 56 three-sections into equivalence classes.

Identification of geometrically equivalent but unitarily inequivalent sections.

Abstract

The geometry of the generalized Bloch sphere $\Omega_3$, the state space of a qutrit, is studied. Closed form expressions for $\Omega_3$, its boundary $\partial \Omega_3$, and the set of extremals $\Omega_3^{\rm ext}$ are obtained by use of an elementary observation. These expressions and analytic methods are used to classify the 28 two-sections and the 56 three-sections of $\Omega_3$ into unitary equivalence classes, completing the works of earlier authors. It is shown, in particular, that there are families of two-sections and of three-sections which are equivalent geometrically but not unitarily, a feature that does not appear to have been appreciated earlier. A family of three-sections of obese-tetrahedral shape whose symmetry corresponds to the 24-element tetrahedral point group $T_d$ is examined in detail. This symmetry is traced to the natural reduction of the adjoint representation of $SU(3)$, the symmetry underlying $\Omega_3$, into direct sum of the two-dimensional and the two (inequivalent) three-dimensional irreducible representations of $T_d$.