Quantum Capacities for Entanglement Networks

TL;DR

This paper explores the relationship between quantum capacities in entanglement networks, showing that regularized capacities are equal and analyzing the gap between one-shot capacities, with implications for quantum communication protocols.

Contribution

It establishes the equality of regularized quantum capacities and tensor network ranks, and analyzes the one-shot capacity relationships and bounds in entanglement networks.

Findings

Regularized quantum capacity equals tensor network rank in the same network.

One-shot capacities have a complex relationship, with min-cut bounds often not achievable.

Tensor networks can be viewed as stochastic protocols bounding quantum capacities.

Abstract

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.