Exploiting higher-order correlation functions for photon-statistics-based characterization and reconstruction of arbitrary Gaussian states

TL;DR

This work derives general analytical connections between the second- and third-order photon correlations, $g^{(2)}$ and $g^{(3)}$, for arbitrary multimode Gaussian states by constructing an explicit sixth-order moment decomposition. These relations enable loss-robust tests of Gaussianity and allow classification into non-displaced, non-squeezed, and displaced-squeezed sectors, while showing that full parameter reconstruction requires at least one loss-sensitive observable per mode. The authors extend the framework to single- and multimode cases, providing practical strategies for state reconstruction, including interference-based methods to remove displacement and Williamson decomposition to extract squeezing and thermal parameters. The approach offers a scalable alternative to full Gaussian tomography, leveraging passive linear optics and standard intensity/click measurements, with clear guidance on when and how phase information must be supplied for complete characterization.

Abstract

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 non-displaced, non-squeezed, and displaced-squeezed sectors of the Gaussian state space. Further, 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.

Figures (2)

• Figure 1: Functional dependence $g^{(3)}(g^{(2)})$ for different subclasses of the Gaussian sector. Coherent states (point $C$, green) and thermal states (point $T$, red) are shown together with the relations of non-squeezed ($1 \leq g^{(2)} \leq 2$) and non-displaced ($2 \leq g^{(2)}$) states. Note that the domains of these classes are bounded, cf. Eqs. \ref{['eq:g2SingleMode']} for $\mathrm{cov}_{}=0$ (note that $\bar{n} = |\alpha|^{2}+N$ in this case) and $\alpha=0$, respectively. A squeezed-displaced example following Eq. \ref{['eq:g3-g2singleDST']} with $N = 0.5$, $\alpha = 5$, and $r = 1.6$ is plotted in gray. The two black points correspond to the Fock-state mixtures $F_1 = \tfrac{5}{8}\op{0} + \tfrac{3}{8}\op{2}$ and $F_2 \approx 0.9\op{2} + 0.1\op{18}$, both lying exactly on the linear relation for non-displaced Gaussian states, Eq. \ref{['eq:g3-g2singleSV']}. The dashed continuation of the blue curve illustrates that this functional dependence may also be fulfilled by certain non-Gaussian states outside the allowed domain for Gaussian states (point $F_{1}$).
• Figure 2: Complete graph symbolizing the set of equations to determine the displacement phases for $M=4$. The global phase shift has been set to $\varphi_1=0$. Edges in red constitute a spanning tree of the graph.