Convex Split Lemma without Inequalities

TL;DR

This paper refines the convex split lemma by replacing max mutual information with collision mutual information, leading to tighter bounds in quantum source coding and a universal upper bound on smoothed max mutual information.

Contribution

It introduces a new equality-based refinement of the convex split lemma and establishes a universal bound on smoothed max mutual information independent of system dimensions.

Findings

Tighter achievability bounds for quantum source coding tasks.

A universal upper bound on smoothed max mutual information.

Implications for the reverse quantum Shannon theorem.

Abstract

We introduce a refinement to the convex split lemma by replacing the max mutual information with the collision mutual information, transforming the inequality into an equality. This refinement yields tighter achievability bounds for quantum source coding tasks, including state merging and state splitting. Furthermore, we derive a universal upper bound on the smoothed max mutual information, where "universal" signifies that the bound depends exclusively on R\'enyi entropies and is independent of the system's dimensions. This result has significant implications for quantum information processing, particularly in applications such as the reverse quantum Shannon theorem.