Geometry of the generalized Bloch sphere for qutrits

Abstract

The geometry of the generalized Bloch sphere $\\Omega_3$, the state space of a\nqutrit, is studied. Closed form expressions for $\\Omega_3$, its boundary\n$\\partial \\Omega_3$, and the set of extremals $\\Omega_3^{\\rm ext}$ are obtained\nby use of an elementary observation. These expressions and analytic methods are\nused to classify the 28 two-sections and the 56 three-sections of $\\Omega_3$\ninto unitary equivalence classes, completing the works of earlier authors. It\nis shown, in particular, that there are families of two-sections and of\nthree-sections which are equivalent geometrically but not unitarily, a feature\nthat does not appear to have been appreciated earlier. A family of\nthree-sections of obese-tetrahedral shape whose symmetry corresponds to the\n24-element tetrahedral point group $T_d$ is examined in detail. This symmetry\nis traced to the natural reduction of the adjoint representation of $SU(3)$,\nthe symmetry underlying $\\Omega_3$, into direct sum of the two-dimensional and\nthe two (inequivalent) three-dimensional irreducible representations of $T_d$.\n