Operational Thresholds for Long-Range Entanglement in Quantum Networks

arXiv:2611.05069v1 Announce Type: new Abstract: Quantum networks distribute entanglement between distant nodes by combining many short entangled links using local quantum operations. These operations only succeed probabilistically, yet existing percolation-based theories of entanglement transmission treat them as perfect. We approach the problem through statistical physics, using a stochastic generalization of concurrence percolation to determine how reliable local operations must be for long-range entanglement to remain possible. On hierarchical $(U,V)$-flower networks, which permit full analytical control across network scales, we derive an exact renormalization recursion for the probability that an end-to-end entangled connection survives. For the $(2,2)$-flower this yields a closed-form operational phase boundary, $\eta_p = 2 - \frac{32}{27}\eta_s$, where $\eta_s$ and $\eta_p$ are the success probabilities of the series (entanglement swapping) and parallel (purification) operations, respectively. At this boundary the survival probability jumps discontinuously from $0$ to $S_\infty = 3/(4\eta_s)$. Numerically propagating the full distribution of end-to-end concurrence yields a second threshold in the initial link quality, which varies continuously with operational reliability. Above both thresholds the large-network state is binary: the connection either fails outright or approaches a perfect Bell pair. Our results thus reveal a two-threshold structure for long-range entanglement, in which operational reliability and initial link quality can compensate one another. Operational success rates therefore emerge alongside initial link quality as a primary design constraint for scalable quantum networks.