Maximal Rényi Relative Entropy for $α>2$

Abstract

Quantum relative entropies play a fundamental role in quantum information theory. In the classical setting, Rényi 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 Rényi relative entropy for $α\in[0,1/2)$ and the sandwiched Rényi relative entropy for $α\geq 1/2$. In contrast, the maximal extension had previously been identified only for $α\in[0,2]$, where it is given by the geometric Rényi relative entropy. In this work, we complete this characterization by proving that for $α>2$, the maximal extension is given by the $α$-$z$ Rényi relative entropy with $z=α-1$. 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.

The paper determines the maximal additive, data-processing-monotone quantum extension of classical Rényi relative entropy in the previously unresolved range $\alpha>2$. It proves that this extension is $D_{\alpha,\alpha-1}$, the $\alpha$–$z$ Rényi divergence with $z=\alpha-1$. Together with the known geometric extension for $\alpha\in[0,2]$, this completes the characterization of the minimal and maximal quantum Rényi extensions, without classifying all intermediate extensions. The main step identifies the regularized prepared Rényi divergence—the minimum classical Rényi cost of jointly preparing a pair of quantum states—with $D_{\alpha,\alpha-1}$. Data processing supplies the lower bound. For the upper bound, an explicit preparation channel, constructed by pinching $\sigma^{-1}$ in the eigenspaces of $\rho$, establishes $D_\alpha^{\mathbb P}(\rho\|\sigma)\leq D_{\alpha,\alpha-1}(\rho\|\sigma)+\log|\operatorname{spec}(\rho)|$. On $n$ copies in dimension $d$, the correction is at most $(d-1)\log(n+1)$ and therefore vanishes per copy. Using comparison theorems for preordered semirings, the paper derives the optimal exact asymptotic conversion rate from a classical state pair to a quantum pair as the minimum ratio of input to output maximal divergences, including both argument orderings. It also characterizes catalytic Gibbs-preserving conversion from full-rank energy-incoherent inputs to full-rank quantum targets: all corresponding divergence inequalities are necessary and sufficient when arbitrarily small target error is allowed, with the catalyst returned exactly and uncorrelated. These applications assume coinciding supports; exact catalytic conversion has strict sufficient and non-strict necessary inequalities. General quantum-to-quantum conversion remains uncharacterized.