De Finetti Structures in Networks of Quantum Systems

Consider $N$ identical quantum systems at the vertices of a graph, interacting via pairwise edge couplings. We designate $K$ vertices as a system of interest $\mathrm{S}$ and the remaining $N-K$ as a reservoir $\mathrm{R}$. Under generic connectivity assumptions between $\mathrm{S}$ and $\mathrm{R}$, we show that for $N\to \mathrm{\infty }$, the reduced state of $\mathrm{S}$ converges to a convex combination of symmetric product states of $K$ single-vertex density matrices – called a de Finetti representation. Unlike in standard derivations for symmetric (exchangeable) systems, we establish de Finetti representations for quantum networks, which are inherently non-exchangeable. We construct the representations explicitly and quantify the convergence rate. We show that Erdős–Rényi random graphs exhibit the de Finetti structure: as the reservoir grows, the reduced state of any fixed finite subgraph converges almost surely to an explicit deterministic product state. As applications, we establish a general area law for the entanglement entropy of $\mathrm{S}$, analyze mean-field and fluctuation scalings of the $\mathrm{S}$-$\mathrm{R}$ interaction, and illustrate our results on qubit networks. We also derive quantitative convergence estimates for information-theoretic quantities, including von Neumann entropy, mutual information, and entanglement within $\mathrm{S}$ and between $\mathrm{S}$ and $\mathrm{R}$.