Geometric phase for N-level systems through unitary integration

Geometric phases are important in quantum physics and are now central to fault-tolerant quantum computation. For spin 1/2, the Bloch sphere S-2, together with a U(1) phase, provides a complete SU(2) description. We generalize to N-level systems and SU(N) in terms of a 2(N-1)-dimensional base space and reduction to a (N-1)-level problem, paralleling closely the two-dimensional case. This iteratively solves the time evolution of an N-level system and gives (N-1) geometric phases explicitly. A complete analytical construction of an S-4 Bloch-like sphere for two qubits is given for the Spin(5) or SO(5) subgroup of SU(4).