Geometry of the generalized Bloch sphere for qutrits

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 $T_d$.