Quantum Capacities for Entanglement Networks

We discuss quantum capacities for two types of entanglement networks: $mathcal{Q}$ for the quantum repeater network with free classical communication, and $mathcal{R}$ for the tensor network as the rank of the linear operation represented by the tensor network. We find that $mathcal{Q}$ always equals $mathcal{R}$ in the regularized case for the samenetwork graph. However, the relationships between the corresponding one-shot capacities $mathcal{Q}_1$ and $mathcal{R}_1$ are more complicated, and the min-cut upper bound is in general not achievable. We show that the tensor network can be viewed as a stochastic protocol with the quantum repeater network, such that $mathcal{R}_1$ is a natural upper bound of $mathcal{Q}_1$. We analyze the possible gap between $mathcal{R}_1$ and $mathcal{Q}_1$ for certain networks, and compare them with the one-shot classical capacity of the corresponding classical network.