# Unique Information and Secret Key Decompositions

## TL;DR

This paper investigates the properties of the unique information measure, proving a triangle inequality, and explores its implications as an upper bound on secret key rates, proposing a conjecture for its lower bound.

## Contribution

The paper proves a triangle inequality for the unique information measure and discusses its potential as a lower bound on two-way secret key rates.

## Key findings

- Proved a triangle inequality for the unique information measure
- Established that UI is an upper bound on the two-way secret key rate
- Conjectured that UI may also serve as a lower bound on the two-way rate

## Abstract

The unique information ($UI$) is an information measure that quantifies a deviation from the Blackwell order. We have recently shown that this quantity is an upper bound on the one-way secret key rate. In this paper, we prove a triangle inequality for the $UI$, which implies that the $UI$ is never greater than one of the best known upper bounds on the two-way secret key rate. We conjecture that the $UI$ lower bounds the two-way rate and discuss implications of the conjecture.