Quantum communication capacity transition of complex quantum networks

Quantum network is the key to enable distributed quantum information processing. As the single-link communication rate decays exponentially with the distance, to enable reliable end-to-end quantum communication, the number of nodes need to grow with the network scale. For highly connected networks, we identify a threshold transition in the capacity as the density of network nodes increases -- below a critical density, the rate is almost zero, while above the threshold the rate increases linearly with the density. Notably, our prediction of the critical density is one-order-of-magnitude higher than that in the recent work [Phys. Rev. Lett.124, 210501 (2020)]. Surprisingly, above the threshold the typical communication capacity between two nodes is independent of the distance between them, due to multi-path routing enabled by the quantum network. In contrast, for less connected networks such as scale-free networks, the end-to-end capacity saturates to constants as the number of nodes increase, and always decays with the distance; Our results are based on capacity evaluations, therefore is universal for general architectures and protocols of quantum networks.