发表机构
Freie Universität Berlin（柏林自由大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文从学习理论角度研究玻色高斯态的量子-经典边界，发现其可学习性由热涨落决定：冷态需要Ω(n^3)样本（非经典），暖态仅需Θ(n^2)样本（经典），并刻画了平滑的量子-经典交叉。
AI 中文摘要
物理学中的一个基本问题是：经典行为何时从量子系统中涌现？玻色高斯态为探索这一量子-经典边界提供了自然的设定，因为它们同时捕捉了光的经典场行为和内在的量子本质。在此，我们从学习理论的角度解决这个问题，提出疑问：玻色高斯态何时在可学习性上是经典的？也就是说，在什么条件下（如果有的话），一个n模玻色高斯态可以用与学习经典2n元高斯分布所需的同样少的样本和同样简单的操作来学习？我们建立了一个由态的热涨落控制的平滑的可学习性交叉：- 冷高斯态在可学习性上是非经典的：当协方差矩阵满足Σ≤(1/2+O(1/n))I，即接近真空协方差时，在单副本（即非纠缠）测量下的层析成像从根本上需要Ω(n^3)个副本，严格超过学习经典高斯分布的样本复杂度Θ(n^2)。我们证明，即使允许少副本纠缠测量，这种困难仍然存在。- 暖高斯态在可学习性上是经典的：当热涨落超过真空噪声，由Σ≥(1/2+ν)I（对任意参数ν>0）参数化时，我们证明单副本层析成像需要N=Θ(n^2 min(n,1+ν^{-1}))个副本。这个界限是紧的，并且由简单的、非自适应的、非纠缠的外差测量实现。关键的是，对于ν=Ω(1)，样本复杂度降至Θ(n^2)，与经典情形匹配。我们的结果紧密刻画了玻色高斯态可学习性中的量子到经典的交叉，揭示了基础物理学与统计学习理论之间的新颖联系，并对现实世界的传感实验有影响。
英文摘要
A fundamental question in physics is: When does classical behavior emerge from quantum systems? Bosonic Gaussian states provide a natural setting to explore this quantum-classical boundary, as they capture both the classical field behavior and the intrinsic quantum nature of light. Here, we address this problem from a learning-theoretic perspective by asking: When are bosonic Gaussian states classical to learn? That is, under what conditions (if any) can an n-mode bosonic Gaussian state be learned with as few samples, and with operations as simple, as are needed to learn a classical 2n-variate Gaussian distribution? We establish a smooth crossover in learnability governed by the state's thermal fluctuations: - Cold Gaussian states are non-classical to learn: When the covariance matrix satisfies $Σ\le(\frac12+O(\frac1n))I$, i.e. close to the vacuum covariance, tomography under single-copy (i.e., non-entangled) measurements fundamentally requires $Ω(n^3)$ copies, strictly exceeding the sample complexity $Θ(n^2)$ of learning classical Gaussian distributions. We show that this hardness persists even when few-copy entangled measurements are allowed. - Warm Gaussian states are classical to learn: When thermal fluctuations exceed the vacuum noise, parameterized by $Σ\ge(\frac12+ν)I$ for any parameter $ν>0$, we prove that single-copy tomography requires $N=Θ\left(n^2\min(n,1+ν^{-1})\right)$ copies. This bound is tight and is achieved by simple, non-adaptive, unentangled heterodyne measurements. Crucially, for $ν=Ω(1)$, the sample complexity drops to $Θ(n^2)$, matching the classical case. Our results tightly characterize a quantum-to-classical crossover in the learnability of bosonic Gaussian states, reveal a novel connection between fundamental physics and statistical learning theory, and have implications for real-world sensing experiments.
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