Geometric phases and Bloch-sphere constructions for SU(N) groups with a complete description of the SU(4) group

摘要

A two-sphere ("Bloch" or "Poincare") is familiar for describing the dynamics of a spin-1 /2 particle or lightpolarization. Analogous objects are derived for unitary groups larger than SU(2) through an iterative procedurethat constructs evolution operators for higher-dimensional SU(N) in terms of lower-dimensional ones. Wefocus, in particular, on the SU(4) of two qubits which describes all possible logic gates in quantum computa-tion and entangled states in quantum-information sciences. For a general Hamiltonian of SU(4) with 15parameters, and for Hamiltonians of its various subgroups so that fewer parameters suffice, we derive Bloch-like rotation of unit vectors analogous to the one familiar for a single spin in a magnetic field. The unitaryevolution of a quantal spin pair is thereby expressed as rotations of real, many-dimensional vectors. Corre-spondingly, the manifolds involved are Bloch two-spheres along with higher dimensional manifolds such as afour-sphere for the SO(5) subgroup and an eight-dimensional Grassmannian manifold for the general SU(4).The latter may also be viewed as two, mutually orthogonal, real six-dimensional unit vectors moving on afive-sphere with an additional phase constraint. This geometrical picture for two spins provides the extensionand generalization of the Bloch sphere that has proved invaluable for the understanding of the dynamics of asingle spin.