Asymptotic robustness of entanglement in noisy quantum networks and graph connectivity

TL;DR

This work investigates how connectivity aspects of graph-encoded quantum networks govern the asymptotic entanglement properties of isotropic network states under noise. By linking AGME and ABS to graph parameters, it shows that fast degree growth ($\Omega(N)$) suffices for asymptotically robust GME, while slow growth ($o(\log N)$) yields asymptotic biseparability, with explicit constructions proving optimality in certain regimes. It also demonstrates that edge-connectivity alone cannot fully determine AGME/ABS, since minimally connected graphs can still be AGME and that degree growth provides the strongest, though not complete, predictive power. The results highlight a nuanced interplay between graph theory and multipartite entanglement, offering a framework to design network patterns that preserve GME under noise and guiding future exploration of additional graph-analytic criteria.

Abstract

Quantum networks are promising venues for quantum information processing. This motivates the study of the entanglement properties of the particular multipartite quantum states that underpin these structures. In particular, it has been recently shown that when the links are noisy two drastically different behaviors can occur regarding the global entanglement properties of the network. While in certain configurations the network displays genuine multipartite entanglement (GME) for any system size provided the noise level is below a certain threshold, in others GME is washed out if the system size is big enough for any fixed non-zero level of noise. However, this difference has only been established considering the two extreme cases of maximally and minimally connected networks (i.e. complete graphs versus trees, respectively). In this article we investigate this question much more in depth and relate this behavior to the growth of several graph theoretic parameters that measure the connectivity of the graph sequence that codifies the structure of the network as the number of parties increases. The strongest conditions are obtained when considering the degree growth. Our main results are that a sufficiently fast degree growth (i.e. $Ω(N)$, where $N$ is the size of the network) is sufficient for asymptotic robustness of GME, while if it is sufficiently slow (i.e. $o(\log N)$) then the network becomes asymptotically biseparable. We also present several explicit constructions related to the optimality of these results.

Figures (6)

• Figure 1: Isotropic network state of three parties specified by a path graph. The vertices $u,v,w$ specify the parties and the edges $e_1$ and $e_2$ that the corresponding pair of parties share a bipartite isotropic state. Notice that $u,v$ represented in black hold a single qudit, while party $v$ holds two qudits coming from different pairs of isotropic states. We represent this by the two white-dashed circles inside a white vertex representing $v$.
• Figure 2: Critical visibility dividing the isotropic network state given by the graph $G$ into BS and GME regions as a function of $p\in[0,1]$. More specifically, the state is biseparable when $p\leq\overline{p_d(G)}$ (and GME otherwise) and we know, in addition, that $\overline{p_d(G)}\in[1/(d+1),1)$.
• Figure 3: Graph $G_n$ of a minimally edge connected AGME graph sequence ${\mathcal{G}}=(G_n)_n$. Circles represent clusters, which in this case are completely connected.
• Figure 4: Element $G_n$ of the graph sequence ${\mathcal{G}}$ split into three clusters $G_n^{(i)}$, $i=1,2,3.$
• Figure 5: Element $G_n$ of the graph sequence ${\mathcal{G}}$ split into three clusters whose induced subgraphs are complete graphs $K_n$ with a bridge edge joining them.

Theorems & Definitions (54)

• Theorem 1: High degree growth implies AGME; cf. Theorem \ref{['th:degAGME']}
• Theorem 2: Low degree growth implies ABS; cf. Theorem \ref{['th:degABS']}
• Theorem 3: Sublinear degree growth is compatible with ABS; cf. Theorem \ref{['th:highdegABS']}
• Theorem 4: Sublinear degree growth is compatible with AGME; cf. Theorem \ref{['th:lowdegAGME']}
• Definition 2.1
• Proposition 2.2
• Definition 3.1
• Proposition 3.2
• proof
• Proposition 3.3