Witnessing quantum non-Gaussianity from intensity moments

TL;DR

We introduce a moment-based quantum non-Gaussianity (QNG) witness that uses only the photon-number mean $m$ and variance $s^2$, enabling QNG verification from intensity-like measurements without photon-number resolving detectors. The main result proves that for fixed $m$, any mixture of Gaussian states has a lower bound on the non-centered second moment that is achieved by a single optimal displaced squeezed vacuum, yielding a parametric QNG boundary $m_{ ext{NG}}(r), s_{ ext{NG}}^2(r)$ and establishing convexity to justify a single-mode witness. The framework is reformulated in terms of integrated intensity moments and $g^{(2)}$, and is illustrated with practical measurement schemes such as homodyning and amplification-based methods, including robust loss and additive-noise corrections. The witness applies to single and multimode scenarios, offers resilience to imperfections, and provides substantial advantage for bright states, opening pathways to proof-of-principle tests and applications in quantum sensing and computation. Overall, the moment-based QNG criterion broadens the accessible regime for verifying non-Gaussian resources with standard intensity-detection techniques and amplification protocols, while maintaining rigorous bounds and practical correction strategies.

Abstract

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.

Figures (13)

• Figure 1: Comparison of QNG witnesses. (a) In terms of photon number mean and variance. The blue line shows the border of the converted Fock probability-based witness, and the orange line the new moment-based witness: the latter has an extended applicability towards bright states. The areas below either line correspond to pairs of values indicating quantum non-Gaussianity. The dashed black line shows the border of the non-classicality witness, $s^2<m$. (b) The same lines in terms of $s^2-m$ and $m$ for a narrower range of mean values, showing that the two witnesses are equivalent for low values of $m$.
• Figure 2: Steps of optimizing the photon number variance of a Gaussian state without changing the mean photon number. We start off with an arbitrary Gaussian state represented by ellipse (0), and first rotate it about its center $(d_0, 0)$ (1), and then increase its distance from the origin while keeping its squeezing parameter unchanged until we arrive at the displaced squeezed vacuum marked (2). State (3) represents the minimal photon number variance displaced squeezed vacuum given the mean photon number, and has the moments described in Eqs. (\ref{['eq:m']}-\ref{['eq:s2']}).
• Figure 3: Behavior of the QNG witness (\ref{['eq:m']}-\ref{['eq:s2']}) for a mixture of two optimal displaced squeezed vacua. The purple line shows the non-centered second moment of an optimal displaced squeezed vacuum as a function of its mean photon number, that is, $s^2_{\mathrm{NG}}(r)+m^2_{\mathrm{NG}}(r)$ as a function of $m_{\mathrm{NG}}(r)$ ($r\geqslant 0$). The mixture components are shown with black disks, with mean photon numbers one and three. Any state corresponding to their mixture lies in this representation on the thin black line connecting the two black disks. The black circle corresponds to a 60%-40% mixture of the initial states, which has a mean photon number of 1.8. However, since the purple line represents a convex function, we obtain a lower second moment for a single optimal displaced squeezed vacuum with the same mean, 1.8 (black triangle).
• Figure 4: Behavior of the QNG witness (\ref{['eq:m']}-\ref{['eq:s2']}) for two independent modes: number variance per mode and mean photon number per mode. The black disk corresponds to two identical optimally squeezed modes with $m_1 = m_2 = 15$. The black filled triangle corresponds to the average from the two states denoted by the empty triangles ($m_1 = 5$, $m_2 = 25$), the average photon number per mode is again 15; however, the variance is lower than in the identical case. The black filled diamond shows the average from $m_1 = 0$ and $m_2 = 30$ (empty diamonds): due to the convexity of the optimal variance curve, this is the setup that provides the lowest variance for $\overline m = 15$.
• Figure 5: Mean and variance of photon numbers for optimal displaced squeezed vacua in the case of $M$ identical modes. This illustrates that assuming $M = 1$ provides the worst-case scenario in the identical multimode case, that is, if such a state can be represented by a point below the $M = 1$ orange line, it also qualifies as QNG for any $M > 1$.