R\'enyi Entanglement of Purification Is Non-additive

Abstract

Entanglement of purification is a fundamental measure of total correlations whose additivity remains unresolved. We study its additivity for classical states on two qubits at different R\'enyi orders. For every $\alpha\in[0,1)$, we prove nonadditivity within this family, witnessed by two copies of a single state. We first solve the one-copy optimization exactly for the entire family at every R\'enyi order. We then restrict the two-copy optimization to a natural finite set of purifications and exhibit one whose entropy is strictly below the product value. In contrast, for $\alpha\in[2,\infty]$ we prove additivity under tensor products within this family. The interval $\alpha\in[1,2)$, including the von Neumann case $\alpha=1$, remains open, and we conjecture additivity there throughout the same family.