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arXiv · 1308.5961

A limit of the quantum Renyi divergence

Nilanjana Datta·Felix Leditzky
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Abstract

Recently, an interesting quantity called the quantum Renyi divergence (or "sandwiched" Renyi relative entropy) was defined for pairs of positive semi-definite operators $ρ$ and $σ$. It depends on a parameter $α$ and acts as a parent quantity for other relative entropies which have important operational significances in quantum information theory: the quantum relative entropy and the min- and max-relative entropies. There is, however, another relative entropy, called the 0-relative Renyi entropy, which plays a key role in the analysis of various quantum information-processing tasks in the one-shot setting. We prove that the 0-relative Renyi entropy is obtainable from the quantum Renyi divergence only if $ρ$ and $σ$ have equal supports. This, along with existing results in the literature, suggests that it suffices to consider two essential parent quantities from which operationally relevant entropic quantities can be derived - the quantum Renyi divergence with parameter $α\ge 1/2$, and the $α$-relative Rényi entropy with $α\in [0,1)$.

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Nilanjana Datta, Felix Leditzky. 2013-09-20. A limit of the quantum Renyi divergence. https://doi.org/10.1088/1751-8113%2F47%2F4%2F045304

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