Symmetric extension in two-way quantum key distribution

TL;DR

This paper introduces symmetric extensions of bipartite quantum states as a tool to analyze quantum key distribution protocols, linking the success of key distillation to the ability to break symmetric extensions.

Contribution

It formulates the symmetric extension problem as a semidefinite program and applies it to analyze Bell-diagonal states in key distribution schemes.

Findings

States that cannot be distilled lack a symmetric extension

The semidefinite program efficiently solves the symmetric extension problem

Failure to break symmetric extension explains unsuccessful key distillation

Abstract

We introduce symmetric extensions of bipartite quantum states as a tool for analyzing protocols that distill secret key from quantum correlations. Whether the correlations are coming from a prepare-and-measure quantum key distribution scheme or from an entanglement-based scheme, the protocol has to produce effective states without a symmetric extension in order to succeed. By formulating the symmetric extension problem as a semidefinite program, we solve the problem for Bell-diagonal states. Applying this result to the six-state and BB84 schemes, we show that for the entangled states that cannot be distilled by current key distillation procedures, the failure can be understood in terms of a failure to break a symmetric extension.