Geometry of the generalized Bloch sphere for qutrits

Source abstract

Abstract The geometry of the generalized Bloch sphere Ω 3 , the state space of a qutrit, is studied. Closed form expressions for Ω 3 , its boundary ∂Ω 3 , and the set of extremals 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 Ω 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 Ω 3 , into direct sum of the two-dimensional and the two (inequivalent) three-dimensional irreducible representations of T d .