Geometric phase for SU(N) through fiber bundles and unitary integration

Abstract

Geometric phases are important in quantum physics and now central to fault tolerant quantum computation. For spin-1/2 and SU(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 a S^4 Bloch-like sphere for two qubits is given for the Spin(5) or SO(5) subgroup of SU(4).