For α>2, every quantum relative entropy extending the classical Rényi relative entropy is upper bounded by D_{α,α-1}, which is therefore the maximal quantum Rényi relative entropy.
The following lemma provides an explicit characterization of such elements
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Maximal R\'enyi Relative Entropy for $\alpha>2$
For α>2, every quantum relative entropy extending the classical Rényi relative entropy is upper bounded by D_{α,α-1}, which is therefore the maximal quantum Rényi relative entropy.