Linear programs for entanglement and key distribution in the quantum internet

Stefan Bäuml,Go Kato,Koji Azuma,David Elkouss

doi:10.1038/s42005-020-0318-2

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.

Highlights

Quantum networks will allow to implement communication tasks beyond the reach of their classical counterparts
Even in the case of point-to-point links, entanglement makes the characterization of capacities notably more complicated than its classical counterpart, with phenomena such as superactivation[9], it was unclear how much it would be possible to borrow from the theory of classical networks
In this paper, using the approach taken in refs. 13–15,24, i.e., defining a network capacity as a rate per the total number of channel uses or per time, we introduce or generalize the capacities for private or quantum communication in the following scenarios: bipartite communication, concurrent communication between multiple user pairs with the objective of (1) maximizing the sum of rates achieved by the user pairs or (2) maximizing the worst-case rate that can be achieved by any pair, as well as multipartite state sharing where the goal is either to distribute Greenberger–Horne–Zeilinger (GHZ) or multipartite private states[5] for a group of network users

Summary

Introduction

Quantum networks will allow to implement communication tasks beyond the reach of their classical counterparts. 10,11 introduced the quantum problem and successfully established upper and lower bounds on a capacity of a quantum network which quantifies the maximum size of bipartite maximally entangled states (for quantum teleportation) or private states (for quantum key distribution) per network use, as a generalization of the fundamental/established notion of classical network capacity[12]. Protocols involving distillation of Bell pairs across all edges of a network, and entanglement swapping along paths have been used to provide lower bounds on the network capacities[10,11,14] Using such simple routing methods, it was shown in refs. We do so by considering edge-disjoint Steiner trees spanning the set of users

Objectives

Methods

Results

Conclusion