Convex Split Lemma without Inequalities

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 Rényi 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.