Geometry of mixed states and degeneracy structure of geometric phases for multi-level quantum systems. A unitary group approach (Q2783569)

The paper explores some geometric aspects of the Schrödinger evolution of a generic multilevel quantum system, focussing on a three-level one. By cleverly exploiting representation theory, together with a vivid geometric insight, the authors get a grasp at the intricacies of the problem of studying the singularities of the space parametrizing a generic finite level Hamiltonian arising from the degeneracies of the latter, paving the way for further analysis. The basic idea consists in regarding unitary group orbits in the space of density matrices (mixed states) in quantum mechanics (and, a fortiori, the space of pure states, namely, projective space) as coadjoint orbits thereof, so they naturally come equipped with a symplectic structure (Kirillov).NEWLINENEWLINENEWLINEThe three-level case leads to an eight-dimensional parameter space (corresponding to the adjoint representation of \(\text{SU}(3)\)), which comes equipped with a rich singularity structure, encompassing, in a suitable limiting sense, the monopole singularity exhibited by the two-level case. To this aim, appropriate Berry phases are computed via representation theory, after connecting them with the symplectic structures on the orbits, and a relation among them (sum rule) is discussed.NEWLINENEWLINENEWLINEThe article is clearly written and it is complemented by two useful appendices collecting the basics of \(\text{SU}(3)\)-representation theory making it essentially self-contained.