Characterizing Optimal-speed unitary time evolution of pure and quasi-pure quantum states

Abstract

We present a characterization of the Hamiltonians that generate optimal-speed unitary time evolution and the associated dynamical trajectory, where the initial states are either pure states or quasi-pure quantum states. We construct the manifold of pure states as an orbit under the conjugation action of the Lie group $\text{SU}(n)$ on the manifold of one-dimensional orthogonal projectors, obtaining an isometry with the flag manifold $\text{SU}(n)/\text{S}(\text{U}(1)\times \text{U}(n-1))$. From this construction, we show that Hamiltonians generating optimal-speed time evolution are fully characterized by equigeodesic vectors of $\text{SU}(n)/\text{S}(\text{U}(1)\times \text{U}(n-1))$. We later extend that result to quasi-pure quantum states.