Quantum Correlations in Three-Beam Symmetric Gaussian States Accessed via Photon-Number-Resolving Detection and Quantum Universal Invariants

TL;DR

This work addresses the problem of characterizing multipartite quantum correlations in three-beam symmetric Gaussian states using quantum universal invariants (the 1-, 2-, and 3-beam purities $\mu_1$, $\mu_2$, $\mu_3$ and the Seralian $\Delta_2$) derived from intensity moments up to sixth order. The authors present a complete framework for parametrizing the covariance matrix, deriving physicality constraints, and relating $\mu_3$ to $\mu_1$, $\mu_2$, and $\Delta_2$, enabling identification of STBGS without full state tomography. Through PPT-based entanglement analysis, Gaussian steering measures, and experimental validation with photon-number-resolving detection, they demonstrate genuine tripartite entanglement and regions of coexisting bipartite and tripartite correlations, closely resembling noisy GHZ/W states. The results highlight the practical utility of quantum universal invariants for diagnosing complex CV entanglement in multipartite systems and point to robust applications in quantum networks, metrology, and state characterization under realistic noise. The study also shows how experimental design (e.g., number of modes $M$) and higher-order moments influence the detectability and robustness of these quantum correlations.

Abstract

Quantum correlations of 3-beam symmetric Gaussian states are analyzed using their quantum universal invariants. These invariants, 1-, 2-, and 3-beam purities, are expressed in terms of the beams' intensity moments up to sixth order. The 3-beam symmetric Gaussian states with varying amounts of the noise are experimentally generated using entangled photon pairs from down-conversion, their invariants are determined, and their quantum correlations are quantified. The coexistence of bi- and tripartite entanglement and genuine tripartite entanglement are observed in these states that resemble the noisy GHZ/W states.

Figures (11)

• Figure 1: Two- and three-beam entanglement analyzed in the plane spanned by 1- and 2-beam purities $\mu_1$ and $\mu_2$. Panel (a) highlights areas differing in 3-beam entanglement properties: Region I (coexistence of separable and bound entangled states) is shown in blue, Region II (coexistence of separable, bound entangled, and fully entangled states) is depicted in orange, the fully entangled region is drawn in green. Panel (b) focuses on 2-beam entanglement: area with 2-beam separable (entangled) states is shown in light blue (green). In the light-orange area separable and entangled states coexist. These regions are classified using the criteria of Ref. Adesso2004b. Isolated black boxes (black triangles) refer to the experimental data discussed for $M = 6.7$ ($M = 40$) independent modes Upper and lower bounds on three-beam (two-beam) logarithmic negativity $E_{N,3}$ [$E_{N,2}$] are shown in panels (c) [(d)] and (e) [(f)], respectively, as they depend on 'rotated' marginal purities $(\mu_1 + \mu_2)/2$ and $(\mu_1 - \mu_2)/2$. In panels (c) and (e), the black solid curves indicate contour levels of $E_{N,3}$ equal to 0.5, 1, and 2. In panels (d) and (f), the black solid curves correspond to $E_{N,2} = 0.5$.
• Figure 2: Three-mode Gaussian steering: Steerable region, coexistence region, and unsteerable region for steering parameters (a) ${\cal G}^{2\rightarrow 1}$ and (b) ${\cal G}^{1\rightarrow 2}$ are drawn in turn in green, orange, and blue color in the plane spanned by 1- and 2-beam purities $\mu_1$ and $\mu_2$. Isolated black boxes (black triangles) refer to the experimental data for $M = 6.7$ ($M = 40$). Upper and lower bounds ${\cal G}^{2\rightarrow 1}_{\rm max}$ [${\cal G}^{1\rightarrow 2}_{\rm max}$] and ${\cal G}^{2\rightarrow 1}_{\rm min}$ [${\cal G}^{1\rightarrow 2}_{\rm min}$] on the steering parameter as they depend on 'rotated' marginal purities $(\mu_1+\mu_2)/2$ and $(\mu_1-\mu_2)/2$ are drawn in (c) [(d)] and (e) [(f)], respectively. In (c) -- (f), black solid curves correspond to ${\cal G}^{2\rightarrow 1}_{\rm max}$, ${\cal G}^{1\rightarrow 2}_{\rm max}$, ${\cal G}^{2\rightarrow 1}_{\rm min}$, and ${\cal G}^{1\rightarrow 2}_{\rm min}$ equal to 0.5, 1, and 2.
• Figure 3: Structure of 3-beam Gaussian states containing photon-pair (red $\infty$) and single-photon (green $\rightarrow$) fields. Both fields can be symmetrized by alternating the signal (s) and the idler (i) beams (double $\infty$ and $\rightarrow$)
• Figure 4: (a) One-beam mean photon number $\langle n\rangle$, (b) its Fano factor $F$, (c) 2-beam noise-reduction parameter $R$, and 2-beam Kullback-Leibler divergence per one typical spatio-temporal mode $H_2/M$ as they depend on 1-beam mean noise photon number $\langle n_{\rm n} \rangle$. Used symbols and curves are described in the caption to Fig. \ref{['fig5']}. In (b) [(c)], the horizontal dashed line $F =1$ [$R = 1$] defines the classical-quantum border.
• Figure 5: (a) One-, (b) two-, and (c) 3-beam purities $\mu_1$, $\mu_2$, and $\mu_3$ and (d) seralian $\Delta_2$ as they depend on 1-beam mean noise photon number $\langle n_{\rm n} \rangle$. Isolated symbols with error bars give experimental data, curves originate in the general Gaussian model. Black curves and symbols $\circ$ denote exact values obtained from the intensity moments up to sixth order. The use of moments only up to second order gives blue curves and symbols $\ast$. Fourth-order moments provide the lower (green curves and symbols +) and upper (red curves and symbols $\triangle$) estimates for the determined quantities.