Bosonic Gaussian states show when quantum learning matches classical sample sizes
When are bosonic Gaussian states classical to learn?
Information TheoryMachine Learning
Summary
This paper investigates when certain quantum states called bosonic Gaussian states can be learned as easily as classical Gaussian data. The authors find that cold states close to the quantum vacuum require many more measurements to learn, showing a distinctly quantum challenge. In contrast, warmer states with more thermal noise can be learned with about as many samples as a classical distribution, bridging a gap between quantum and classical learning. This work links physics and learning theory to show when quantum problems behave like classical ones.
What this means in practice
- •For quantum sensing engineers: Optimize the number of measurements required for characterizing bosonic states based on thermal noise levels in sensing devices.
- •For quantum communication developers: Predict when classical measurement methods suffice for learning light states in optical communication channels, reducing complexity.
A theory result. No direct application yet.
Authors
Senrui Chen, Antonio Anna Mele, Francesco Anna Mele, John Preskill
Abstract
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.