Qubit Geodesics on the Bloch Sphere from Optimal-Speed Hamiltonian Evolutions

TL;DR

This paper analyzes the shortest paths (geodesics) of quantum state evolution on the Bloch sphere driven by optimal-speed Hamiltonians, exploring their geometric properties and implications for quantum dynamics.

Contribution

It provides an explicit geodesic analysis for single-qubit evolutions under Hamiltonians that achieve maximal speed between states, including geometric efficiency and phase considerations.

Findings

Identifies geodesic paths as minimal statistical length trajectories.

Verifies geodesicity through geometric efficiency and phase analysis.

Discusses challenges in extending to open quantum systems.

Abstract

In the geometry of quantum evolutions, a geodesic path is viewed as a path of minimal statistical length connecting two pure quantum states along which the maximal number of statistically distinguishable states is minimum. In this paper, we present an explicit geodesic analysis of the dynamical trajectories that emerge from the quantum evolution of a single-qubit quantum state. The evolution is governed by an Hermitian Hamiltonian operator that achieves the fastest possible unitary evolution between given initial and final pure states. Furthermore, in addition to viewing geodesics in ray space as paths of minimal length, we also verify the geodesicity of paths in terms of unit geometric efficiency and vanishing geometric phase. Finally, based on our analysis, we briefly address the main hurdles in moving to the geometry of quantum evolutions for open quantum systems in mixed quantum states.