Geometry of quantum dynamics and optimal control for mixed states

TL;DR

This paper explores the geometric structure of quantum state evolution, establishing a link between Riemannian distances and energy dispersions, and applies these insights to optimize quantum control and evolution times.

Contribution

It introduces a geometric framework connecting Hamiltonian dynamics with Riemannian structures on quantum state spaces, enabling control of evolution paths and times.

Findings

Riemannian distance equals the infimum of energy dispersions for connecting states

Derived a geometric version of the Mandelstam-Tamm bound

Provided conditions for Hamiltonians to transport states along geodesics

Abstract

Geometric effects make evolution time vary for different evolution curves that connect the same two quantum states. Thus, it is important to be able to control along which path a quantum state evolve to achieve maximal speed in quantum calculations. In this paper we establish fundamental relations between Hamiltonian dynamics and Riemannian structures on the phase spaces of unitarily evolving finite-level quantum systems. In particular, we show that the Riemannian distance between two density operators equals the infimum of the energy dispersions of all possible evolution curves connecting the two density operators. This means, essentially, that the evolution time is a controllable quantity. The paper also contains two applied sections. First, we give a geometric derivation of the Mandelstam-Tamm estimate for the evolution time between two distinguishable mixed states. Secondly, we show how to equip the Hamiltonians acting on systems whose states are represented by invertible density operators with control parameters, and we formulate conditions for these that, when met, makes the Hamiltonians transport density operators along geodesics.