Quantum Capacities for Entanglement Networks

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We discuss quantum capacities for two types of entanglement networks: Q for the quantum repeater network with free classical communication, and R for the tensor network as the rank of the linear operation represented by the tensor network. We find that Q always equals R in the regularized case for the samenetwork graph. However, the relationships between the corresponding one-shot capacities Q₁ and R₁ 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 R₁ is a natural upper bound of Q₁. We analyze the possible gap between R₁ and Q₁ for certain networks, and compare them with the one-shot classical capacity of the corresponding classical network.