Negativity Percolation in Continuous-Variable Quantum Networks

TL;DR

A Gaussian-to-Gaussian entanglement distribution scheme is introduced that deterministically transports two-mode squeezed vacuum states across large CV networks and reveals a new form of entanglement percolation--negativity percolation theory (NegPT)--characterized by a bounded entanglement measure called the ratio negativity.

Abstract

Quantum networks (QNs) have been predominantly driven by discrete-variable (DV) architectures. Yet, optical platforms naturally generate Gaussian states--the common states of continuous-variable (CV) systems, making CV-based QNs an attractive route toward scalable, chip-integrated quantum computation and communication. To bridge the gap between well-studied DV entanglement percolation theories and their CV counterpart, we introduce a Gaussian-to-Gaussian entanglement distribution scheme that deterministically transports two-mode squeezed vacuum states across large CV networks. Analysis of the scheme's collective behavior using statistical-physics methods reveals a new form of entanglement percolation--negativity percolation theory (NegPT)--characterized by a bounded entanglement measure called the ratio negativity. We discover that NegPT exhibits a mixed-order phase transition, marked simultaneously by both an abrupt change in global entanglement and a long-range correlation between nodes. This distinctive behavior places CV-based QNs in a new universality class, fundamentally distinct from DV systems. Additionally, the abruptness of this transition introduces a critical vulnerability of CV-based QNs: conventional feedback mechanism becomes inherently unstable near the threshold, highlighting practical implications for stabilizing large-scale CV-based QNs. Our results unify statistical models for CV-based entanglement distribution and uncover previously unexplored critical phenomena unique to CV systems, providing valuable insights and guidelines essential for developing robust, feedback-stabilized QNs.

Figures (8)

• Figure 1: Gaussian-to-Gaussian deterministic entanglement transmission (G-G DET) scheme. Applicable to Gaussian quantum networks (QN), the scheme consists of two LOCC protocols: (a) Entanglement swapping, facilitated by homodyne detection and displacement P2002; (b) Entanglement concentration, facilitated by non-standard optical components MRN. Both protocols are deterministic, taking two (or more) TMSVS $|\psi^{r_{i}}\rangle$ as input and a new TMSVS $|\psi^{r}\rangle$ as output. (c) The two LOCC protocols map to series and parallel rules, respectively, to construct G-G DET. (d) Consider a QN example built upon three node. The G-G DET scheme consists of two steps: First, the parallel rule converts the states $|\psi^{r_1}\rangle$ and $|\psi^{r_2}\rangle$ ($r_1\geq r_2$) into $|\psi^{r_{1,2}}\rangle$ with $\sinh r_{1,2} = \sinh r_1 \cosh r_2$ between $S$ and $R$; second, the series rule transforms $|\psi^{r_{1,2}}\rangle$ and $|\psi^{r_{3}}\rangle$ to the final state $|\psi^{r}\rangle$ with the ratio negativity $\text{X}_{\text{SC}}=\tanh r_{1,2} \tanh r_3$ between $S$ and $T$.
• Figure 2: Bethe lattice. (a) A Bethe lattice of degree $k$ (i.e., each node is incident to $k$ links) and network depth $l$ (the path length from the yellow node to the red nodes). (b) The sponge-crossing ratio negativity $\mathrm{X}_\text{SC}$ between $S$ and $T$ for various $k$ (right panel), satisfying the power law $\mathrm{X}_\text{SC}-\mathrm{X}_\text{SC}^{+}\sim |\chi-\chi_{\text{th}}|^{0.47(5)}$ as $\chi\to\chi_{\text{th}}^+$ (left panel). The numerical value $0.47\pm0.05$ is derived by a linear least-squares fit to the sixteen data points. (c) When $\chi\to\chi_{\text{th}}^-$, $\mathrm{X}_\text{SC}$ exhibits a plateau behavior until the network depth $l$ exceeds the correlation length $l^*$ (defined as the depth $l$ at which $\mathrm{X}_\text{SC}=0.5$), after which $\mathrm{X}_\text{SC}$ abruptly drops to zero. (d) Near the critical threshold, we observe $l^* \sim \left|\chi - \chi_{\text{th}}\right|^{-0.508(9)}$, indicating $z\nu \approx 1/2$.
• Figure 3: Entanglement percolation in two-dimensional square lattices. (a) $\text{X}_{\text{SC}}$ for square lattices with different side length $L$. (b) Scaling of the correlation length $\xi$ near the critical threshold $\chi_{\text{th}}\approx0.715$ follows $\xi \sim |\chi - \chi_{\text{th}}|^{-\nu}$, with a fitted critical exponent $\nu = 0.02\pm0.02$.
• Figure 4: Feedback stabilization of QN against entanglement decay. (a) Under the same feedback control [Eq. \ref{['eq-FOPTD']}], the DV-based QN ($k=3$ Bethe lattice) exhibits rapid stabilization; (b) whereas the CV-based QN exhibits long-term "on/off" instability, a direct result of the abrupt drop in Fig. \ref{['Figure2']}(b).
• Figure 5: Continuous-variable entanglement swapping. Given $N$ TMSVSs $|\psi^{r_{1}}\rangle,|\psi^{r_{2}}\rangle,\dots,|\psi^{r_{N}}\rangle$, the final state $|\psi^{r}\rangle$ between $S$ and $T$ derived by entanglement swapping is a TMSVS, where the squeezing parameter $r$ satisfies Eq. \ref{['1D']}.