A Quantum Non-Gaussianity Criterion Based on Photon Correlations $g^{(2)}$ and $g^{(3)}$

TL;DR

The paper develops a robust, attenuation-resistant criterion for quantum non-Gaussianity based on photon correlations $g^{(2)}$ and $g^{(3)}$, with the key bound $g^{(3)} olinebreak olinebreak olinebreak olinebreak olinebreak olinebreak olinebreak olinebreak igl(2 - 3 ext{sqrt}{g^{(2)}}igr)^2$ for Gaussian states and the practical test $ ext{sqrt}{g^{(3)}} + 3 ext{sqrt}{g^{(2)}} < 2$. The supplemental material provides detailed derivations for Gaussian pure and mixed states, extends the bound to multi-mode fields via Bloch–Messiah, and introduces tangent-line linear bounds; it also documents three-fold coincidence measurements and a highly significant p-value against Gaussianity. Experimentally, a quantum dot single-photon source in a photonic crystal waveguide yields $g^{(2)} = 0.00334(4)$ and $g^{(3)} = 0$, giving $ ext{sqrt}{g^{(3)}} + 3 ext{sqrt}{g^{(2)}} = 0.174(13)$, exceeding the Gaussian bound by more than $100$ standard deviations and a p-value of $4 imes10^{-4793}$. The approach is advantageous in lossy settings, offering a practical benchmark for quantum advantage in continuous-variable systems and paving the way for higher-order correlation analyses in the future.

Abstract

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 $\sqrt{g^{(3)}} + 3 \sqrt{g^{(2)}} \geq 2$. 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 $\sqrt{g^{(3)}} + 3 \sqrt{g^{(2)}} = 0.174 (13)$, which represents a statistical significance of more than $100$ standard deviations.

Figures (7)

• Figure 1: Second- and third-order correlation function for Gaussian states. The blue points represent $g^{(2)}$ and $g^{(3)}$ for a large set of combinations of $\alpha$, $r$ and $\theta$. The dashed lines represent fixed values of $r$ ($\theta=0$), indicating that the boundary is reached for $r \rightarrow 0$. For $|\alpha| \rightarrow \infty$ the coherent states are dominant corresponding to $g^{(2)} = g^{(3)} = 1$ (red circle). The dotted lines show the analytic upper ($g^{(3)} > 4$) and lower ($g^{(3)} < 4$) boundary for Gaussian pure states, see Eq. \ref{['eq:g3_g2_th']}. For the scatter plot, we use 1000 equally distributed values in the interval $\alpha \in (0, 1]$, 501 values for $r \in [0, 1]$ and 21 values for $\theta \in [0, \pi ]$.
• Figure 2: Non-Gaussianity bound based on $g^{(2)}$ and $g^{(3)}$. The white region cannot be reached by any incoherent superposition of Gaussian states, see Eq. \ref{['eq:g2g3_s49']}. Gaussian pure states are always above the bound in Eq. \ref{['eq:g3_g2_bound']} (black solid line, blue region). An incoherent superposition of Gaussian states can be below the bound but only for $g^{(2)} > 4/9$ (red region). The blue cross close to the boundary corresponds to the Gaussian pure state $| \xi = 1/100, \alpha = 1/5 \rangle$ and the red cross to the mixed state $0.75 | \xi = 1/100, \alpha = 1/5 \rangle \langle \xi = 1/100, \alpha = 1/5 | + 0.25 | 0 \rangle \langle 0 |$.
• Figure 3: Experimental validation with a single-photon source. (a) Scanning electron microscopy image of the device. The quantum dot, pulsed laser excitation and emitted photons are schematically indicated. The emitted photons are routed to three single-photon detectors via a pair of beam splitters to measure coincidence counts and extract $g^{(n)}$. (b) Measured two-fold coincidence counts to extract the second-order intensity correlations, $g^{(2)}$. The coincidences at zero delay are strongly suppressed in comparison to the peaks with detections from separate excitation pulses. (c), (d) Measured three-fold coincidences for (c) the same excitation pulse and (d) three separate pulses after transformation and projection onto Jacobi coordinates $j_1 = (2t_1 - t_2 - t_3)/\sqrt{6}$ and $j_2 = (t_2-t_3)/\sqrt{2}$ with $t_i$ the time at detector $i$. The main panels show data with intentionally increased laser leakage, while the insets show the data for optimized conditions.
• Figure S1: Linear bound - tangent lines. The dashed and dotted lines represent different (simpler) versions of the general bound, see Eq. \ref{['eq:tangent_lines']} and Eq. \ref{['eq:g3_cumulant']}.
• Figure S2: Measured three-fold coincidences. The detected coincidences for (a) the same pulse, (b) a pair of subsequent pulses, and (c) three subsequent pulses, each after transformation into Jacobi coordinates $j_1 = (2t_1 - t_2 - t_3)/\sqrt{6}$ and $j_2 = (t_2-t_3)/\sqrt{2}$ with $t_i$ the time at detector $i$.