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.
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.
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.
Efficiently testing whether a quantum state possesses a given structure is both a fundamental and a practical task in quantum information. Among the most important structured families are Gaussian states, which underpin quantum optics and many-body physics while defining paradigmatic regimes of efficient classical simulation. Despite the central role of Gaussian states, optimal tests of Gaussianity have remained unknown. Here we give optimal and robust algorithms for testing fermionic and bosonic Gaussian states, with sample complexity independent of system size, particle number, and energy. Remarkably, the protocols implement experimentally feasible measurements associated with known exact symmetry characterizations of Gaussianity, using Bell sampling on two copies for fermions and passive interferometry with photon counting on three copies for bosons. We further obtain tolerant testers requiring joint measurements on a larger, but still constant, number of copies. At the heart of our analysis is a unified approach---which we call the gradient-flow method---that applies across all structured families considered here and underlies our optimal testers: representation theory turns approximate satisfaction of the defining symmetry into quantitative control of the gradient of the rejection probability, allowing a continuous descent from the input state to the target family whose length bounds their distance.
Bosonic and fermionic Gaussian quantum states are families of quantum states that play a central role in quantum physics and quantum information processing. We study the problem of testing whether a given state is either Gaussian, or has bounded overlap with any Gaussian state. Previous analyses of this testing problem led to tests that require a number of copies of the state that scales with the number of modes (the system size). In this work we show that a constant number of copies of the state suffices. We derive these results based on the symmetries of identical copies of Gaussian states, and by bounding the distance to the set of Gaussian states through a gradient flow and a spectral bound studied previously for the Kac random walk on a sphere.
We study the problem of property testing quantum states: given a pure state ∣ψ⟩ that either (A) belongs to some class C of pure states, or (B) is ε-far in trace distance from all states in C, determine which is the case with high probability. It is known that there exists classes C for which the sample complexity of property testing necessarily scales with the size of the system. In this work, we identify classes for which testing only requires a size-independent number of samples. We prove a tight sample complexity of Θ(1/ε2) for the property testing of five important classes of Gaussian quantum states: 1) Fermionic Gaussian states, 2) Slater determinants, 3) bosonic Gaussian states, 4) zero-mean bosonic Gaussian states, and 5) bosonic coherent states.
While property testers for some of these classes have been studied previously, those analyses only gave upper bounds scaling polynomially in the number of modes. In each case, our tester is the projection onto the ``top'' irrep of two or three copies. We get to constant sample complexity by lower bounding how quickly the rejection probability increases with distance from the class; the main technical component involves bounding the spectral gap of symmetric extension of the test projector. We then establish optimality by proving matching lower bounds based on binary state discrimination for each class. Our testers and optimality claims also extend to the tolerant setting, wherein ψ may be O(ε)-close to C in case (A), rather than lying exactly inside.
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.