Information Causality is a Special Point in the Dual of the Gray-Wyner Region

TL;DR

This paper explores the relationship between Information Causality and the Gray-Wyner region, proposing a new postulate to extend the connection to correlated inputs, but finds limitations within quantum theory.

Contribution

It identifies Information Causality as a point in the dual of the Gray-Wyner region for independent inputs and introduces the Accessibility of Mutual Information postulate to study correlated inputs.

Findings

Information Causality aligns with the dual of the Gray-Wyner region for independent inputs.

The proposed Accessibility of Mutual Information postulate is not satisfied by quantum theory.

The approach highlights limitations in extending classical information principles to quantum correlations.

Abstract

Information Causality contributes to the program of deriving fundamentals of quantum theory from information theoretic principles. It puts restrictions on the amount of information learned by a party (Bob) from the other party (Alice) in a one-way communication scenario: Bob receives an index b, and after a one-way communication from Alice, tries to recover the b-th bit of Alice's input. In this manuscript we note that Information Causality is indeed a point in the dual of the Gray-Wyner region in the case where Alice's input bits are chosen independently at random. The main motivation of this work was to study this connection in the case of correlated input bits. Our approach to forge this connection was to propose a new postulate called Accessibility of Mutual Information (AMI) on the underling physical theory. This approach, explained in the first version of this paper, fails since AMI is not satisfied by the quantum theory.