The exploration of tomography of bosonic Gaussian states is presumably as old as quantum optics, but only recently, their precise and rigorous study have been moving into the focus of attention, motivated by technological developments. In this work, we present an efficient and experimentally feasible Gaussian state tomography algorithm with provable recovery trace-distance guarantees, whose sample complexity depends only on the number of modes, and - remarkably - is independent of the state's photon number or energy, up to doubly logarithmic factors. Our algorithm yields a doubly-exponential improvement over existing methods, and it employs operations that are readily accessible in experimental settings: the preparation of an auxiliary squeezed vacuum, passive Gaussian unitaries, and homodyne detection. At its core lies an adaptive strategy that systematically reduces the total squeezing of the system, enabling efficient tomography. Quite surprisingly, this proves that estimating a Gaussian state in trace distance is generally more efficient than directly estimating its covariance matrix. Our algorithm is particularly well-suited for applications in quantum metrology and sensing, where highly squeezed - and hence high-energy - states are commonly employed. As a further contribution, we establish improved sample complexity bounds for standard heterodyne tomography, equipping this widely used protocol with rigorous trace-norm guarantees.
The exploration of tomography of bosonic Gaussian states is presumably as old as quantum optics, but only recently, their precise and rigorous study have been moving into the focus of attention, motivated by technological developments. In this work, we present an efficient and experimentally feasible Gaussian state tomography algorithm with provable recovery trace-distance guarantees, whose sample complexity depends only on the number of modes, and - remarkably - is independent of the state's photon number or energy, up to doubly logarithmic factors. Our algorithm yields a doubly-exponential improvement over existing methods, and it employs operations that are readily accessible in experimental settings: the preparation of an auxiliary squeezed vacuum, passive Gaussian unitaries, and homodyne detection. At its core lies an adaptive strategy that systematically reduces the total squeezing of the system, enabling efficient tomography. Quite surprisingly, this proves that estimating a Gaussian state in trace distance is generally more efficient than directly estimating its covariance matrix. Our algorithm is particularly well-suited for applications in quantum metrology and sensing, where highly squeezed - and hence high-energy - states are commonly employed. As a further contribution, we establish improved sample complexity bounds for standard heterodyne tomography, equipping this widely used protocol with rigorous trace-norm guarantees.
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.
Quantum measurements are probabilistic and, in general, provide only partial information about the underlying quantum state. Obtaining a full classical description of an unknown quantum state requires the analysis of several different measurements, a task known as quantum-state tomography. Here we analyse the ultimate achievable performance in the tomography of continuous-variable systems, such as bosonic and quantum optical systems. We prove that tomography of these systems is extremely inefficient in terms of time resources, much more so than tomography of finite-dimensional systems such as qubits. Not only does the minimum number of state copies needed for tomography scale exponentially with the number of modes, but, even for low-energy states, it also scales unfavourably with the trace-distance error between the original state and its estimated classical description. On a more positive note, we prove that the tomography of Gaussian states is efficient by establishing a bound on the trace-distance error made when approximating a Gaussian state from knowledge of the first and second moments within a specified error bound. Last, we demonstrate that the tomography of non-Gaussian states prepared through Gaussian unitaries and a few local non-Gaussian evolutions is efficient and experimentally feasible.
Multimode Gaussian states are a versatile resource for quantum information technologies and have been realized across a wide range of physical platforms. Recent progress in the large-scale generation of such states provides a key ingredient for scalable quantum technologies. Despite the importance of accurately characterizing these states, conventional tomography methods are often impractical because they require large sample sizes and can yield unphysical states. Here we present a reliable and efficient tomography method for multimode Gaussian states based on maximum-likelihood estimation. By directly operating on covariance matrices, the method avoids the exponential overhead associated with density-matrix reconstruction. We consider two commonly used detection schemes--single and joint homodyne detection--and systematically analyze the reconstruction performance. Our method outperforms conventional approaches by ensuring physical covariance matrices and achieving better agreement with the true states. To demonstrate the experimental applicability of the method, we experimentally generate various multipartite entangled states--six-mode graph states with different connectivity, a six-mode GHZ state, and a fully connected ten-mode graph state--and reconstruct their covariance matrices. Using the reconstructed covariance matrices, we quantify fidelities, detect entanglement, and reveal the multimode structure of squeezing and noise. Our technique offers a practical diagnostic tool for developing scalable quantum technologies.
Quantum measurements are probabilistic and, in general, provide only partial information about the underlying quantum state. Obtaining a full classical description of an unknown quantum state requires the analysis of several different measurements, a task known as quantum-state tomography. Here we analyse the ultimate achievable performance in the tomography of continuous-variable systems, such as bosonic and quantum optical systems. We prove that tomography of these systems is extremely inefficient in terms of time resources, much more so than tomography of finite-dimensional systems such as qubits. Not only does the minimum number of state copies needed for tomography scale exponentially with the number of modes, but, even for low-energy states, it also scales unfavourably with the trace-distance error between the original state and its estimated classical description. On a more positive note, we prove that the tomography of Gaussian states is efficient by establishing a bound on the trace-distance error made when approximating a Gaussian state from knowledge of the first and second moments within a specified error bound. Last, we demonstrate that the tomography of non-Gaussian states prepared through Gaussian unitaries and a few local non-Gaussian evolutions is efficient and experimentally feasible.
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.
Gaussian states of bosonic quantum systems enjoy numerous technological applications and are ubiquitous in nature. Their significance lies in their simplicity, which in turn rests on the fact that they are uniquely determined by two experimentally accessible quantities, their first and second moments. But what if these moments are only known approximately, as is inevitable in any realistic experiment? What is the resulting error on the Gaussian state itself, as measured by the most operationally meaningful metric for distinguishing quantum states, namely, the trace distance? In this work, we fully resolve this question by demonstrating that if the first and second moments are known up to an error ε , the trace distance error on the state also scales as ε , and this functional dependence is optimal. To prove this, we establish tight bounds on the trace distance between two Gaussian states in terms of the norm distance of their first and second moments. As an application, we improve existing bounds on the sample complexity of tomography of Gaussian states.