Maximal R\'enyi Relative Entropy for $\alpha>2$

Quantum relative entropies play a fundamental role in quantum information theory. In the classical setting, R\'enyi relative entropies constitute, up to linear combinations, the most general class of relative entropies, naturally motivating the search for their minimal and maximal quantum extensions. The minimal extension is known to be the reverse sandwiched R\'enyi relative entropy for $\alpha\in[0,1/2)$ and the sandwiched R\'enyi relative entropy for $\alpha\geq 1/2$. In contrast, the maximal extension had previously been identified only for $\alpha\in[0,2]$, where it is given by the geometric R\'enyi relative entropy. In this work, we complete this characterization by proving that for $\alpha>2$, the maximal extension is given by the $\alpha$-$z$ R\'enyi relative entropy with $z=\alpha-1$. As an application, we determine when an energy-incoherent state can be transformed into an energy-coherent state by a Gibbs-preserving operation assisted by an uncorrelated catalyst, thereby fully characterizing the coherence-generating power of this class of operations in the catalytic setting.