The Fer expansion of a driven spin under su(2) operations cannot terminate beyond first order
Abstract
The Fer expansion writes the propagator of a driven quantum system as an ordered product of exponentials, each generated by the time integral of an effective Hamiltonian that governs the evolution left over by the previous factors. It can be stopped after finitely many factors whenever this leftover evolution is known in closed form, for instance when the effective Hamiltonian commutes with itself at all times, or precesses about a fixed axis with constant angular velocity. For su(2), in any representation, from a qubit to a spin of any size, we resum the Fer recursion into a closed map on curves in ℝ3. We prove that, for a real-analytic drive, in both cases the expansion stops beyond the first factor only trivially, when the first exponential is already exact.
Related
- Beyond Verified Answers: Solver-Informed Self-Distillation for Bootstrapping Operations Research Language Models
- Beyond Marginal Monitoring: Distributed Joint-Distribution Testing for Data Concept Drift in Large Scale E-Commerce Operations
- Beyond Expected Profit: How Risk Aversion Alters Learning in the Data-Driven Newsvendor
- Lens spaces that cannot be obtained by two integral surgeries
- Families of knots that cannot be made Legendrian parametrically
BibTeX
@misc{bagnasacco2026expansion,
title = {{The Fer expansion of a driven spin under $\mathfraksu(2)$ operations cannot terminate beyond first order}},
author = {Lorenzo Bagnasacco and Daniel Burgarth and Vittorio Giovannetti and Kazuya Yuasa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27979},
eprint = {2610.27979},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27979},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Fer expansion of a driven spin under su(2) operations cannot terminate beyond first order
AU - Lorenzo Bagnasacco
AU - Daniel Burgarth
AU - Vittorio Giovannetti
AU - Kazuya Yuasa
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27979
UR - https://arxiv.org/abs/2610.27979
AB - The Fer expansion writes the propagator of a driven quantum system as an ordered product of exponentials, each generated by the time integral of an effective Hamiltonian that governs the evolution left over by the previous factors. It can be stopped after finitely many factors whenever this leftover evolution is known in closed form, for instance when the effective Hamiltonian commutes with itself at all times, or precesses about a fixed axis with constant angular velocity. For su(2), in any representation, from a qubit to a spin of any size, we resum the Fer recursion into a closed map on curves in ℝ^3. We prove that, for a real-analytic drive, in both cases the expansion stops beyond the first factor only trivially, when the first exponential is already exact.
ER -