Geometric phase forN-level systems through unitary integration

Abstract

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 $\mathrm{SU}(N)$ in terms of a $2(N\ensuremath{-}1)$-dimensional base space and reduction to a $(N\ensuremath{-}1)$-level problem, paralleling closely the two-dimensional case. This iteratively solves the time evolution of an $N$-level system and gives $(N\ensuremath{-}1)$ 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).