working quantum repeater when using secret-key-agreement capacity upper bounds as a benchmark.'
author:
- Eneet Kaur
- 'Mark M. Wilde'
bibliography:
- 'Ref.bib'
title: Upper bounds on secret key agreement over lossy thermal bosonic channels
---

Introduction
============

One of the main goals of quantum information theory [@H13book; @H06; @W15book] is to establish bounds on communication rates for various information-processing tasks. An important application lies in the domain of secret communication, following the development of quantum key distribution [@bb84; @E91]. In recent years, there has been a growing interest in establishing bounds on the secret-key-agreement capacity of a quantum channel, which is the highest rate at which communicating parties can use the channel and public classical communication to distill a secret key [@TGW14IEEE; @TGW14Nat; @STW16; @PLOB15; @Goodenough2015; @TSW16; @AML16; @WTB16; @Christandl2017; @Wilde2016; @BA17; @RGRKVRHWE17; @TSW17]. Such bounds have been proven by exploiting the methods of quantum information theory and can be interpreted as setting the fundamental limitations of quantum key distribution whenever a quantum repeater is not available [@L15].

An important development occurred in [@TGW14Nat], in which it was established that there is a fundamental rate-loss trade-off that any repeaterless quantum key distribution protocol cannot overcome. That is, without a 