Asymptotic Limits of Entanglement Distribution

Reliable distribution of quantum entanglement over long distances is a central challenge in quantum information science, fundamentally limited by decoherence in noisy communication channels. In this work, we investigate the asymptotic limits of entanglement distribution across quantum networks utilizing intermediate repeater stations and local operations and classical communication (LOCC). We establish a strict dichotomy: the asymptotic preservation of entanglement over arbitrarily long distances is possible if and only if the underlying quantum channel admits a correctable subspace. For channels lacking such a subspace, we prove that the transmitted state converges exponentially fast to the set of separable states, rendering standard LOCC filtering insufficient. To counteract this exponential degradation, we analyze networks employing parallel channel uses per link. We derive a fundamental lower bound on the physical resource requirements, proving that for channels without a correctable subspace, the number of parallel channels per link must scale at least logarithmically with the number of intermediate stations to sustain a non-zero amount of entanglement. This theoretical limit serves as a stringent benchmark for quantum repeater architectures and underscores the necessity of advanced quantum error-correcting codes, such as qLDPC codes, which show promise in saturating this optimal resource scaling.