Geometry of the generalized Bloch sphere for qutrits

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. Omega(3), and the set of extremals Omega(ext)(3) 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.