On quantum Renyi entropies: a new generalization and some properties

Abstract

The Renyi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies or mutual information, and have found many applications in information theory and beyond. Various generalizations of Renyi entropies to the quantum setting have been proposed, most notably Petz's quasi-entropies and Renner's conditional min-, max- and collision entropy. Here, we argue that previous quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of Renyi entropies that contains the von Neumann entropy, min-entropy, collision entropy and the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities, a duality relation, and an entropic uncertainty relation.

v1: contains several conjectures; v2: conjectures are resolved - see also arXiv:1306.5358 and arXiv:1306.5920; v3: published version

References in corpus (6)

• Sandwiched Rényi Divergence Satisfies Data Processing Inequality
• Monotonicity of a relative Rényi entropy
• Quantum hypothesis testing and the operational interpretation of the quantum Renyi relative entropies
• Generalized relative entropies and the capacity of classical-quantum channels
• Error exponents in hypothesis testing for correlated states on a spin chain
• Quantum state discrimination bounds for finite sample size

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