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.
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.
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 ε trace distance with high probability. We prove a lower bound of Ω(n3/ε2) for Gaussian measurements, matching the best known upper bound up to doubly-log energy dependence, and Ω(n2/ε2) for arbitrary measurements. We further show an upper bound of O(n2/ε2) 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 Θ(E/ε2) 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 ε total variation distance. Our results significantly advance quantum learning theory in the bosonic regimes and have practical impact in quantum sensing and benchmarking applications.
Unlike in finite dimensions, quantum information in continuous-variable systems has the peculiar feature that without imposing physical constraints, the sample complexity of state tomography can be unbounded. Remarkably, this is even the case for state-of-the-art protocols for learning Gaussian states, which have finite-dimensional descriptions: the best known rates scale with loglogE, where E is the energy of the system. We prove this is not an artifact of existing analyses, but a fundamental limitation of the measurements used. We show: (1) Any protocol that uses Gaussian measurements, even entangled or adaptively chosen ones, must incur a loglogE dependence. This answers an open question posed by a number of previous works. (2) There is a smooth tradeoff between the number of rounds of adaptivity and the energy dependence, and we give a matching protocol achieving this interpolated rate. (3) With highly entangled, non-Gaussian measurements, one can learn n-mode pure Gaussian states with O(n2/ε2) samples, independent of E. This answers an open question posed by Chen et al. (4) A simple protocol based on the single-copy canonical phase POVM of Holevo and Helstrom learns single-mode pure Gaussian states with O(1/ε2) samples, again independent of E.
Our results clarify the role of energy in bosonic state tomography and shed new light on the intriguing interplay between adaptivity, entanglement, and magic in quantum learning.
This paper reviews recent advances in quantum learning theory for continuous-variable (CV) systems. Quantum learning theory investigates how to extract classical information from quantum systems as efficiently as possible. CV systems are ubiquitous in nature and in quantum technologies, as they describe bosonic and quantum-optical systems. While quantum learning theory for finite-dimensional systems has been extensively studied, the corresponding theory for CV systems has only recently begun to develop; here we provide a concise review. We focus on the following questions: what is the minimum number of copies (the sample complexity) required to learn a non-Gaussian state, possibly under energy constraints? What is the sample complexity for learning Gaussian states? How does the performance of CV state learning depend on non-Gaussianity? How can one test whether a state is Gaussian or far from the set of Gaussian states? And how can Gaussian processes be learned efficiently? Central to these topics, we also review several bounds on the trace distance between CV states in terms of their covariance matrices, which may be of independent interest. Overall, this work summarises selected developments in tomography of CV systems and highlights a selection of open problems.
Gaussian states are widely regarded as one of the most relevant classes of continuous-variable (CV) quantum states, as they naturally arise in physical systems and play a key role in quantum technologies. This motivates a fundamental question: given copies of an unknown CV state, how can we efficiently test whether it is Gaussian? We address this problem from the perspective of representation theory and quantum learning theory, characterizing the sample complexity of Gaussianity testing as a function of the number of modes. For pure states, we prove that just a constant number of copies is sufficient to decide whether the state is exactly Gaussian. We then extend this to the tolerant setting, showing that a polynomial number of copies suffices to distinguish states that are close to Gaussian from those that are far. In contrast, we establish that testing Gaussianity of general mixed states necessarily requires exponentially many copies, thereby identifying a fundamental limitation in testing CV systems. Our approach relies on rotation-invariant symmetries of Gaussian states together with the recently introduced toolbox of CV trace-distance bounds.
The sample complexity is the minimum number of copies required to learn an accurate classical description of a quantum state. Bosonic and fermionic Gaussian quantum states are families of quantum states that play a key role in quantum science and technology, from quantum optics and many-body physics to quantum chemistry, quantum computing, and quantum information theory. Despite their importance, their sample complexity had not been fully determined. We settle this open problem and show that both bosonic and fermionic Gaussian states can be learned using a number of copies that scales quadratically in the number of modes, regardless of whether the state is pure or mixed, and independently of any energy bound on the state. We derive these results by using the representation theory of Gaussian unitaries and by putting forth a generalization of the random purification channel to this setting and beyond.
We prove that learning an unknown mixed fermionic Gaussian state on m modes to trace distance ε requires Ω(m3/ε2) copies when measurements act on one copy at a time, even with arbitrary POVMs, fresh ancillas and classical adaptivity, but without quantum memory between copies. The bound holds for 0<ε≤1/3600 and separates this model from the known collective rate Θ(m2/ε2). It follows from a uniform single-copy Fisher-information budget and a dimension-independent comparison between trace distance and Gaussian parameters. On covariance matrices of operator norm at most 1−c, we prove a Frobenius-to-trace-norm continuity bound with constant [2c(2−c)]−1/2; matchgate shadows then attain O(m3/(cε2)) copies. This also gives explicit error certificates for thermal free-fermion states. For a class of mixed passive bosonic Gaussian states with total mean photon number at most one, a reduction to bounded-block qudit tomography gives a single-copy lower bound Ω(m3/(ε2log(m/ε))), versus collective complexity Θ(m2/ε2). These separations answer the mixed-state measurement-resource question posed by Chen et al. A two-mode example has optimal two-copy Fisher trace 5, versus the separable ceiling 4. An IBM replication gave a pre-registered witness lower bound 4.12 at one-sided 95% shot-noise confidence; its interpretation assumes the prescribed preparations and a common measurement channel and does not bound preparation systematics.