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 $Ω(n^3/\varepsilon^2)$ for Gaussian measurements, matching the best known upper bound up to doubly-log energy dependence, and $Ω(n^2/\varepsilon^2)$ 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 $\widetildeΘ(E/\varepsilon^2)$ 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.

翻译：连续变量系统支撑着量子计算、量子通信和量子传感等关键量子技术。高斯玻色量子态在引力波探测、暗物质搜索等各类应用中自然出现。一个基本问题是：如何用尽可能少的样本刻画未知的高斯玻色量子态？尽管已有数十年探索，其终极效率极限仍不明确。本文研究了学习一个能量小于 $E$ 的 $n$ 模高斯态所需的最小样本数量（以高概率达到 $\varepsilon$ 迹距离精度）。我们证明：对高斯测量而言，下界为 $\Omega(n^3/\varepsilon^2)$，与已知最优上界仅差能量的双对数因子；对任意测量而言，下界为 $\Omega(n^2/\varepsilon^2)$。此外，我们进一步证明：若高斯态保证为纯态或被动态，则上界可改进至 $\widetilde{O}(n^2/\varepsilon^2)$。有趣的是，高斯测量足以实现对纯高斯态的近乎最优学习，而学习被动高斯态时，非高斯测量被证明是达到最优学习所必需的。最后，针对通过非纠缠高斯测量学习单模高斯态，我们对任意非自适应方案给出近乎紧致的界 $\widetildeΘ(E/\varepsilon^2)$，表明自适应学习对实现能量无关的缩放至关重要。作为副产品，我们建立了高斯态迹距离与其Wigner分布总变差距离之间的紧致关系，并获得了学习任意高斯态Wigner分布（达到 $\varepsilon$ 总变差距离）的近乎紧致样本复杂度界。我们的结果显著推进了玻色体系下的量子学习理论，并对量子传感和量子基准测试具有实际应用价值。