Smooth Entropy Calculus

Abstract

Smooth Rényi entropies are defined as optimizations (either minimizations or maximization) of Rényi entropies over a set of close states. For many applications it suffices to consider just two smooth Rényi entropies: the smooth min-entropy acts as a representative of all conditional Rényi entropies with $\alpha > 1$, whereas the smooth max-entropy acts as a representative for all Rényi entropies with $\alpha < 1$. These two entropies have particularly nice properties and can be expressed in various different ways, for example as semi-definite optimization problems. Most importantly, they give rise to an entropic (and fully quantum) version of the asymptotic equipartition property, which states that both the (regularized) smooth min- and max-entropies converge to the conditional von Neumann entropy for iid product states. This is because smoothing implicitly allows us to restrict our attention to a typical subspace where all conditional Rényi entropies coincide with the von Neumann entropy. Furthermore, we will see that the smooth entropies inherit many properties of the underlying Rényi entropies.