Geometric phases and Bloch sphere constructions for SU(N), with a complete description of SU(4)

Abstract

A two-sphere ("Bloch" or "Poincare") is familiar for describing the dynamics of a spin-1/2 particle or light polarization. Analogous objects are derived for unitary groups larger than SU(2) through an iterative procedure that constructs evolution operators for higher-dimensional SU in terms of lower-dimensional ones. We focus, in particular, on the SU(4) of two qubits which describes all possible logic gates in quantum computation. For a general Hamiltonian of SU(4) with 15 parameters, and for Hamiltonians of its various sub-groups 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 unitary evolution of a quantal spin pair is thereby expressed as rotations of real vectors. Correspondingly, the manifolds involved are Bloch two-spheres along with higher dimensional manifolds such as a four-sphere for the SO(5) sub-group and an eight-dimensional Grassmannian manifold for the general SU(4). This latter may also be viewed as two, mutually orthogonal, real six-dimensional unit vectors moving on a five-sphere with an additional phase constraint.

9 pages

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• Dynamical invariants for quantum control of four-level systems
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• Calculation of quantum discord in arbitrary dimensions, especially for X- and other specialized states
• An exact mapping between the states of arbitrary N-level quantum systems and the positions of classical coupled oscillators
• Bloch sphere like construction of SU(3) Hamiltonians using unitary integration
• Vector Properties of Entanglement in a Three-Qubit System
• Entanglement and entropy in multipartite systems: a useful approach
• Hilbert-Schmidt measure of pairwise quantum discord for three-qubit $X$ states
• Nonlinear Bogolyubov-Valatin transformations: 2 modes
• SU(4) based classification of four-level systems and their semiclassical solution