Gaussian states are an essential building block for various applications in quantum optics and quantum information science, yet the precise relation between their second- and third-order correlation functions remains not fully explored. We discuss connections between these correlation functions by constructing an explicit decomposition formula for arbitrary sixth-order moments of ladder operators for general Gaussian states and demonstrate how the derived relations enable state classification from correlation data alone. Whereas violating these relations certifies non-Gaussianity, satisfying them provides evidence for a Gaussian-state description and allows a direct distinction among nondisplaced, nonsqueezed, and displaced-squeezed sectors of the Gaussian state space. Furthermore, we show that it is not possible to uniquely extract state parameters solely from correlation-function measurements without prior assumptions about the Gaussian state. Resolving this ambiguity requires additional loss-sensitive information, e.g., measuring the mean intensity or the vacuum overlap of each mode. In particular, we show under which circumstances these measurements can be used to reconstruct a generic Gaussian state.
Gaussian states are an essential building block for various applications in quantum optics and quantum information science, yet the precise relation between their second- and third-order correlation functions remains not fully explored. We discuss connections between these correlation functions by constructing an explicit decomposition formula for arbitrary sixth-order moments of ladder operators for general Gaussian states and demonstrate how the derived relations enable state classification from correlation data alone. Whereas violating these relations certifies non-Gaussianity, satisfying them provides evidence for a Gaussian-state description and allows a direct distinction among nondisplaced, nonsqueezed, and displaced-squeezed sectors of the Gaussian state space. Furthermore, we show that it is not possible to uniquely extract state parameters solely from correlation-function measurements without prior assumptions about the Gaussian state. Resolving this ambiguity requires additional loss-sensitive information, e.g., measuring the mean intensity or the vacuum overlap of each mode. In particular, we show under which circumstances these measurements can be used to reconstruct a generic Gaussian state.
Direct measurement of quantum non-Gaussianity requires some variant of a discrete photon-resolving detection, which is feasible only for low mean photon numbers. For a large mean photon number, intensity detection by linear photodiodes provides a continuous signal; therefore, the Fock probabilities of the unknown input state are not directly available. On the other hand, intensity moments can be measured directly, and photon-number moments can be estimated. Therefore, we derive and analyze a quantum non-Gaussianity witness based solely on the photon-number mean and variance (or alternatively, the second-order correlation g ( 2 ) ) of an unknown state. Due to the simplicity of the used photon-number moments, the measurement results are easy to correct for losses and additive noise. We provide examples of simple amplification-based measurement schemes where our witness can be applied directly, thereby opening pathways to proof-of-principle tests and applications.
A Quantum Non-Gaussianity Criterion Based on Photon Correlations g(2) and g(3)
Quantum non-Gaussian states, which cannot be written as mixtures of Gaussian states, are necessary to achieve a quantum advantage in continuous variable systems. They represent an important benchmark for the realization of an advanced quantum light source, as they cannot be made by simple means such as displacement and squeezing. We introduce an attenuation-resistant sufficient criterion for quantum non-Gaussian states based on the second- and third-order correlation functions, g(2) and g(3). The general non-linear bound for classical mixtures of Gaussian states is g(3)+3g(2)≥2. Any mixture of Gaussian states must fulfill this inequality, thus, the violation of it represents a direct confirmation of quantum non-Gaussianity. We experimentally show the non-Gaussianity of the state produced by a quantum dot single-photon source, where we obtain g(3)+3g(2)=0.174(13), which represents a statistical significance of more than 100 standard deviations.
Identification, and subsequent quantification of quantum correlations, is critical for understanding, controlling, and engineering quantum devices and processes. We derive and implement a general method to quantify various forms of quantum correlations using solely the experimental intensity moments up to the fourth order. This is possible as these moments allow for an exact determination of the global and marginal impurities of two-beam Gaussian fields. This leads to the determination of steering, tight lower and upper bounds for the negativity, and the Kullback-Leibler divergence used as a quantifier of state nonseparability. The principal squeezing variances are determined as well using the intensity moments. The approach is demonstrated on the experimental twin beams with increasing intensity and the squeezed super-Gaussian beams composed of photon pairs. Our method is readily applicable to multibeam Gaussian fields to characterize their quantum correlations.
Non-Gaussian states of light are essential for numerous quantum information protocols; thus, certifying non-Gaussianity is crucial. Full quantum state tomography, commonly used for this purpose, is a complicated procedure and yields inconclusive results for strongly mixed states. Certifying non-Gaussianity through directly measurable parameters is a simpler alternative, typically achieved by measuring photon-number probabilities - either directly, using photon-number resolving detectors, or through Hanbury Brown--Twiss type measurements with single-photon detectors. Here, we demonstrate theoretically and experimentally that optical parametric amplification combined with conventional intensity detectors can effectively replace this approach without the need for photon-number resolution. In our method, we measure the mean photon number and the second-order correlation function for the amplified state. Using it, we successfully certify the non-Gaussianity of a heralded quasi-single-photon state. Since optical parametric amplification is a broadband and multimode process, our method provides a foundation for developing high-dimensional quantum technologies utilizing broadband multimode non-Gaussian states.
We analyze the impact of photon loss on the photon-number statistics of Gaussian states. Specifically, we propose and carefully evaluate several methods to mitigate deviations in the photon-number distributions of lossy (displaced) squeezed vacuum states from those of their lossless counterparts. These methods rely on appropriately redefining the parameters of Gaussian states when the loss budget is known in order to recover, as closely as possible, the desired photon-number distribution associated with each target state. While it is intrinsically hard to directly optimize the photon-number distribution of high-dimensional, correlated multimode Gaussian states, the proposed methods are instead based on optimizing specific key properties such as fidelity, phase-space functions, low-order moments of the underlying photon-number statistics, or overlap with the vacuum state. In particular, our results show that optimizing the fidelity between a pure Gaussian target state and a modified Gaussian state that has passed through a loss channel does typically not result in closeness of the corresponding photon-number distributions. Furthermore, we show that correcting for the vacuum overlap minimizes the deviation in the photon-number distribution for large parameter ranges which we explicitly prove for single-mode squeezed vacuum and provide numerical evidence for general (displaced) squeezed vacuum states. As photon loss is a key limitation for Gaussian boson sampling, our results provide insights into the feasibility and limitations of such photonic quantum simulations in lossy environments and offer guidelines for mitigating these imperfections.
Non-Gaussian states are essential for achieving a quantum advantage in continuous-variable (CV) information processing. Among these, coherent superpositions of squeezed states are a foundational resource. While exact higher-order statistics are available in the undisplaced case, a complete and analytically tractable treatment with a common displacement has been missing. We introduce and solve the displaced Janus state---a coherent superposition of two squeezed coherent states that share the same displacement---and develop an analytical framework, based on a family of generalized squeezing polynomials, that yields closed-form expressions for arbitrary-order factorial moments and coherence functions g(k)(0), the full Wigner function, and the quantum Fisher information. The analysis shows how interference at a fixed mean, driven by a mismatch of the component covariances rather than by mean separation, can be precisely engineered to transform the extreme photon bunching of the constituents into strong sub-Poissonian and even perfect multiphoton suppression. We further provide a rigorous quantum Fisher information analysis, proving that parameters encoded by linear generators (for example, the number operator) are bounded by the standard quantum limit, whereas parameters encoded by quadratic generators (for example, squeezing transformations) achieve Heisenberg-limited scaling. Together, these results furnish a complete analytical toolkit for a versatile class of non-Gaussian states, establishing the displaced Janus state as a key primitive for hybrid quantum protocols, quantum metrology, and fault-tolerant continuous-variable computation.