When are bosonic Gaussian states classical to learn?

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 Sigma≤(frac12+O(frac1n))I, i.e. close to the vacuum covariance, tomography under single-copy (i.e., non-entangled) measurements fundamentally requires Omega(n3) copies, strictly exceeding the sample complexity Theta(n2) 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 Sigma≥(frac12+nu)I for any parameter nu>0, we prove that single-copy tomography requires N=Thetaleft(n2min(n,1+nu-1)right) copies. This bound is tight and is achieved by simple, non-adaptive, unentangled heterodyne measurements. Crucially, for nu=Omega(1), the sample complexity drops to Theta(n2), 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.