Gaussian superpositions for bosonic encodings

Abstract

Non-Gaussian bosonic states are ubiquitous in interacting light--matter systems, many-body platforms, and relativistic quantum field settings, but their quantitative characterization is hindered by the infinite-dimensional Hilbert space and by the poor scalability of Fock-space truncation methods. We introduce an exact finite-manifold encoding for states supported on a finite span of Gaussian branches, enabling the use of standard finite-dimensional quantum-information tools directly on an effective density matrix whose entries are determined by Gaussian overlaps. As demonstrations, we obtain closed-form and numerically stable evaluations of entropies and relative-entropy non-Gaussianity, and derive an analytic expression for the bipartite entanglement negativity of arbitrary multimode two-branch Gaussian superpositions, including a minimal which-branch dephasing model. Our framework provides a practical bridge between experimentally accessible continuous-variable resources (e.g., cat-like and measurement-conditioned states) and discrete-variable information measures, with immediate applications to benchmarking non-Gaussian resources in several quantum technology platforms.

Figures (2)

• Figure 1: Relative-entropy non-Gaussianity $\delta_{\rm nG}(\rho)=S(\tau(\rho))-S(\rho)$ for phase-randomized two-branch single-mode encodings (no Fock truncation). We fix a coherence-loss parameter $p=0.1$ in \ref{['eq:phase_randomization_model']} and evaluate $\delta_{\rm nG}$ as a function of displacement $\alpha$ for several amplitude ratios $\kappa\in\{0,10^{-3},10^{-2},0.2,0.5,1\}$ (curves). The entropy $S(\rho)$ is obtained from the exact $2\times2$ reduced matrix on ${\mathsf{Span}}\{| g_1{\rangle},| g_2{\rangle}\}$ determined by Gaussian overlaps (Appendix B), while $S(\tau(\rho))$ is obtained from the covariance matrix of $\rho$ computed analytically from Gaussian cross moments (Appendix C).
• Figure 2: Negativity of a bosonic encoded Bell-like state $| \Psi_\varphi{\rangle}\propto | g_1{\rangle}_A| g_1{\rangle}_B+e^{i\varphi}| g_2{\rangle}_A| g_2{\rangle}_B$, evaluated in closed form from the local overlaps $a=\langle g_1|g_2\rangle_A$ and $b=\langle g_1|g_2\rangle_B$ as $\mathcal{N}=\sqrt{(1-a^2)(1-b^2)}/(2(1+ab\cos\varphi))$ ($a,b\ge0$). The branches are displaced squeezed states $| g_{1,2}{\rangle}=D(\pm\alpha)S(r)| 0{\rangle}$ (single mode per party), so that $a=b$ follows directly from the pure-Gaussian fidelity in terms of first and second moments. The plot shows $\mathcal{N}$ as a function of displacement amplitude $\alpha$ and squeezing $r$ for fixed $\varphi$; negative $r$ corresponds to opposite squeezing angles across the two branches.