The Rényi 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 Rényi entropies to the quantum setting have been proposed, most prominently Petz's quasi-entropies and Renner's conditional min-, max-, and collision entropy. However, these quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of Rényi 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.
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In contrast, the mean
$H(\rho \oplus \sigma ) = \protect \qopname{}{m}{max}\protect \lbrace H(\rho ), H(\sigma ) \protect \rbrace$
would lead to a quantity that is not continuous.46.
A function f(X, Y) is jointly convex if, for any λ ∈ [0, 1] and normalized X1, X2, Y1, Y2 ⩾ 0, we have f(λX1 + (1 − λ)X2, λY1 + (1 − λ)Y2) ⩽ λf(X1, Y1) + (1 − λ)f(X2, Y2).
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