Linear programs for entanglement and key distribution in the quantum internet

TL;DR

This paper introduces a linear programming approach to efficiently estimate the maximum rates of entanglement and key distribution in quantum networks, covering bipartite, multipartite, and multi-user scenarios.

Contribution

It develops a unified linear programming framework for bounding quantum network communication rates, extending classical max-flow min-cut theorems to quantum settings.

Findings

Provides linear programs for bipartite and multipartite quantum networks.

Derives bounds for parallel entanglement distribution among multiple user pairs.

Introduces Steiner tree concepts for multipartite entanglement in quantum networks.

Abstract

Quantum networks will allow to implement communication tasks beyond the reach of their classical counterparts. A pressing and necessary issue for the design of quantum network protocols is the quantification of the rates at which these tasks can be performed. Here, we propose a simple recipe that yields efficiently computable lower and upper bounds on the maximum achievable rates. For this we make use of the max-flow min-cut theorem and its generalization to multi-commodity flows to obtain linear programs. We exemplify our recipe deriving the linear programs for bipartite settings, settings where multiple pairs of users obtain entanglement in parallel as well as multipartite settings, covering almost all known situations. We also make use of a generalization of the concept of paths between user pairs in a network to Steiner trees spanning a group of users wishing to establish Greenberger-Horne-Zeilinger states.