On the separation of quantum time evolution into holonomic and dynamical parts

TL;DR

The work investigates whether non-adiabatic, non-Abelian quantum evolution on a subspace can be decomposed into holonomic and dynamical parts. Using an operator-based framework with the restricted Schrödinger equation, Anandan’s equation, and a frame-based construction, it shows that the evolution on a subspace is governed by a time-ordered exponential of a sum $oldsymbol{ ext{A}}(t)+oldsymbol{ ext{K}}(t)$, and that, in general, $oldsymbol{ ext{A}}$ and $oldsymbol{ ext{K}}$ do not commute, so a product separation is not possible. The paper analyzes Yu and Tong’s proposed factorization, demonstrating that the purported dynamical component depends on the holonomy itself, rendering the separation non-physical except in three highly restrictive special cases. It then presents three concrete Lambda-system scenarios to illustrate when holonomy, dynamics, or their special combinations arise, and discusses gauge covariance to show the robustness of the conclusions. Overall, the results close a gap in understanding of non-adiabatic quantum holonomies and clarify when holonomic quantum computation schemes can rely on a genuine holonomic-dynamical split.

Abstract

The issue of separating Schrödinger-type quantum time evolution into a product of holonomic and dynamical parts in the non-adiabatic non-Abelian case is addressed. Contrary to the recent claim in [Phys. Rev. Lett. 131, 200202 (2023)], we establish that such separation is generally invalid.