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

TL;DR

This paper characterizes Hamiltonians that generate the fastest possible unitary evolution for pure and quasi-pure quantum states, linking optimal speed to geometric properties of state manifolds.

Contribution

It introduces a geometric framework for identifying Hamiltonians that produce optimal-speed evolution, extending known results to quasi-pure states.

Findings

Hamiltonians are characterized by equigeodesic vectors on the state manifold.

The pure state manifold is modeled as a flag manifold with isometric properties.

Results extend to quasi-pure quantum states, broadening applicability.

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 $\SU(n)$ on the manifold of one-dimensional orthogonal projectors, obtaining an isometry with the flag manifold $\SU(n)/\textnormal{S}(\textnormal{U}(1)\times \textnormal{U}(n-1 ))$. From this construction, we show that Hamiltonians generating optimal-speed time evolution are fully characterized by equigeodesic vectors of $\SU(n)/\textnormal{S}(\textnormal{U}(1)\times \textnormal{U}(n-1))$. We later extend that result to quasi-pure quantum states.