A Secret Common Information Duality for Tripartite Noisy Correlations

TL;DR

This paper investigates the duality between secret correlation simulation and extraction in tripartite noisy systems, connecting it with classical notions of common information, and provides bounds and conjectures on key rates.

Contribution

It introduces a duality framework linking secret correlation simulation and extraction, and establishes bounds on secret key rates using common information concepts.

Findings

Reexpression of Winter's key cost result via bipartite protocol monotone.

Construction of distributions where Gács and Körner common information bounds the secret key rate.

Conjecture that the bound holds generally for non-communicative key agreement.

Abstract

We explore the duality between the simulation and extraction of secret correlations in light of a similar well-known operational duality between the two notions of common information due to Wyner, and G\'acs and K\"orner. For the inverse problem of simulating a tripartite noisy correlation from noiseless secret key and unlimited public communication, we show that Winter's (2005) result for the key cost in terms of a conditional version of Wyner's common information can be simply reexpressed in terms of the existence of a bipartite protocol monotone. For the forward problem of key distillation from noisy correlations, we construct simple distributions for which the conditional G\'acs and K\"orner common information achieves a tight bound on the secret key rate. We conjecture that this holds in general for non-communicative key agreement models. We also comment on the interconvertibility of secret correlations under local operations and public communication.