Fractional Contribution of Dynamical and Geometric Phases in Quantum Evolution

TL;DR

This paper establishes a universal law linking the division of quantum evolution phases into geometric and dynamical parts directly to the Bargmann angle, enabling precise real-time quantification of geometricity in quantum processes.

Contribution

It introduces a simple, universal law connecting phase partitioning in quantum evolution to the Bargmann angle, providing a rigorous and practical measure of geometricity.

Findings

Phase division governed solely by Bargmann angle

Provides real-time measure of geometricity

Facilitates design of robust geometric quantum gates

Abstract

The fundamental division of the total quantum evolution phase into geometric and dynamical components is a central problem in quantum physics. Here, we prove a remarkably simple and universal law demonstrating that this partitioning is governed, at every instant, solely by a single geometric quantity: the Bargmann angle (Bures angle). This result provides a universally applicable and rigorous way to define the exact fraction of the total phase that is geometric versus dynamical in origin, thereby establishing a new quantitative link between the dynamics of quantum evolution and the geometry of the state space. This finding has immediate practical consequences, furnishing a real-time measure of the geometricity of an evolution for designing high-fidelity geometric quantum gates with optimized robustness, and opening new avenues for quantum speed limit and coherent control.