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arXiv:2007.06644 · Journal of Mathematical Physics

Equality conditions of data processing inequality forα-zRényi relative entropies

Haonan Zhang

Published Oct 1, 2020• 12 citations
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Abstract

The α–z Rényi relative entropies are a two-parameter family of Rényi relative entropies that are quantum generalizations of the classical α-Rényi relative entropies. In the work [Adv. Math. 365, 107053 (2020)], we decided the full range of (α, z) for which the data processing inequality (DPI) is valid. In this paper, we give algebraic conditions for the equality in DPI. For the full range of parameters (α, z), we give necessary conditions and sufficient conditions. For most parameters, we give equivalent conditions. This generalizes and strengthens the results of Leditzky et al. [Lett. Math. Phys. 107, 61–80 (2017)].

DOI
10.1063/5.0022787

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We study the three-parameter family of quantum conditional Rényi entropies Hα,zλH_{α,z}^λHα,zλ​ introduced in \textit{Commun. Math. Phys.} 407, 180 (2026). We prove that the previously established data-processing region is sharp. We then characterize the club-sandwiched family through a classical coarse-graining principle: within the full data-processing region, universal monotonicity under deterministic coarse-graining of the classical conditioned register holds if and only if z=αz = αz=α. Finally, we establish rigidity of entropic duality by showing that any universal duality relation for finite-dimensional tripartite pure states must satisfy the known parameter relations. Thus, the data-processing region, the club-sandwiched subfamily, and the duality transformation are each uniquely characterized by their respective universal properties.