Towards sample-optimal learning of bosonic Gaussian quantum states

Quantum Physics [Submitted on 18 Mar 2026] Towards sample-optimal learning of bosonic Gaussian quantum states Senrui Chen, Francesco Anna Mele, Marco Fanizza, Alfred Li, Zachary Mann, Hsin-Yuan Huang, Yanbei Chen, John Preskill Continuous-variable systems enable key quantum technologies in computation, communication, and sensing. Bosonic Gaussian states emerge naturally in various such applications, including gravitational-wave and dark-matter detection. A fundamental question is how to characterize an unknown bosonic Gaussian state from as few samples as possible. Despite decades-long exploration, the ultimate efficiency limit remains unclear. In this work, we study the necessary and sufficient number of copies to learn an n-mode Gaussian state, with energy less than E, to \varepsilon trace distance with high probability. We prove a lower bound of \Omega(n^3/\varepsilon^2) for Gaussian measurements, matching the best known upper bound up to doubly-log energy dependence, and {\Omega}(n^2/\varepsilon^2) for arbitrary measurements. We further show an upper bound of \widetilde{O}(n^2/\varepsilon^2) given that the Gaussian state is promised to be either pure or passive. Interestingly, while Gaussian measurements suffice for nearly optimal learning of pure Gaussian states, non-Gaussian measurements are provably required for optimal learning of passive Gaussian states. Finally, focusing on learning single-mode Gaussian states via non-entangling Gaussian measurements, we provide a nearly tight bound of \widetilde\Theta(E/\varepsilon^2) for any non-adaptive schemes, showing adaptivity is indispensable for nearly energy-independent scaling. As a byproduct, we establish sharp bounds on the trace distance between Gaussian states in terms of the total variation distance between their Wigner distributions, and obtain a nearly tight sample complexity bound for learning the Wigner distribution of any Gaussian state to \varepsilon total variation distance. Our results greatly advance quantum learning theory in the bosonic regimes and have practical impact in quantum sensing and benchmarking applications. Comments: 59 pages, 3 figures, 1 table. Comments welcome Subjects: Quantum Physics (quant-ph); Information Theory (cs.IT); Machine Learning (cs.LG); Mathematical Physics (math-ph) Cite as: arXiv:2603.18136 [quant-ph]   (or arXiv:2603.18136v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2603.18136 Focus to learn more Submission history From: Senrui Chen [view email] [v1] Wed, 18 Mar 2026 18:00:00 UTC (629 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-03 Change to browse by: cs cs.IT cs.LG math math-ph math.IT math.MP References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)