Smooth Entropy Calculus

Abstract

Smooth Renyi entropies are defined as optimizations (either minimizations or maximization) of Renyi entropies over a set of close states. For many applications it suffices to consider just two smooth Renyi entropies: the smooth min-entropy acts as a representative of all conditional Renyi entropies with \(\alpha > 1\), whereas the smooth max-entropy acts as a representative for all Renyi 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 Renyi entropies coincide with the von Neumann entropy. Furthermore, we will see that the smooth entropies inherit many properties of the underlying Renyi entropies.