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≤(21+O(n1))I, i.e. close to the vacuum covariance, tomography under single-copy (i.e., non-entangled) measurements fundamentally requires
Ω(n3) copies, strictly exceeding the sample complexity
Θ(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
Σ≥(21+ν)I for any parameter
ν>0, we prove that single-copy tomography requires
N=Θ(n2min(n,1+ν−1)) 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(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.
We report on a novel experiment to generate non-classical atomic states via quantum non-demolition (QND) measurements on cold atomic samples prepared in a high-finesse ring cavity. The heterodyne technique developed for QND detection exhibits an optical shot-noise limited behavior for local oscillator optical power of a few hundred μW, and a detection bandwidth of several GHz. This detection tool is used in a single pass to follow non-destructively the internal state evolution of an atomic sample when subjected to Rabi oscillations or a spin-echo interferometric sequence.
Continuous-variable (CV) quantum states are naturally characterized by their moments, defined as expectation values of products of single- or multimode ladder operators. Many CV Hamiltonians and quantum algorithms are formulated directly in terms of these moments, and therefore an efficient procedure to estimate moments with limited state measurements is necessary. In this paper, we present a protocol for shadow tomography of moment-generating functions (MGFs) of CV states based on randomized homodyne and heterodyne measurements. The resulting shadows enable an efficient and concurrent estimation of many multimode moments. Our complexity analysis shows that the number of measurements required to estimate a certain moment to a given precision grows exponentially in the order of the moments. Finally, we assess the precision of these moment estimators by applying them to two tasks: detecting entanglement in both Gaussian and non-Gaussian states via the Shchukin-Vogel protocol, and characterizing optical loss in a photonic chip. We demonstrate that both tasks can be accomplished with only a few thousand randomized measurements. The small number of required measurements, combined with efficient sample processing, makes our protocol applicable to a wide range of CV simulation tasks.