Fractional Contribution of Dynamical and Geometric Phases in Quantum Evolution

TL;DR

The paper establishes a universal law linking the instantaneous partition of a quantum state's total phase into dynamical and geometric components to the Bargmann angle $S_0(t)$ between the initial and evolved states. It proves a unitary-evolution decomposition in a two-dimensional subspace and derives explicit infinitesimal relations $d\Phi_D$ and $d\Phi_G$, yielding a simple, gauge-invariant geometric-phase fraction $f_g(t) = \sin^2\left(\frac{S_0}{2}\right)$. By connecting the evolution to the Fubini-Study metric, it defines quantum circuitousness and derives a geometric-phase–dependent quantum speed limit $T \ge \frac{\hbar |\Phi_G(T)|}{\Delta H \bar{F}}$ with $\bar{F} = \frac{1}{T}\int_0^T \tan\left(\frac{S_0}{2}\right) dt$. The results enable real-time assessment of an evolution's geometricity, informing robust geometric quantum gates and optimal control strategies, and they offer new insights into quantum speed limits beyond traditional bounds.

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.