  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.
