Entropy Accumulation

Abstract

We ask the question whether entropy accumulates, in the sense that the operationally relevant total uncertainty about an n-partite system A= (A1, … An) corresponds to the sum of the entropies of its parts Ai. The Asymptotic Equipartition Property implies that this is indeed the case to first order in n—under the assumption that the parts Ai are identical and independent of each other. Here we show that entropy accumulation occurs more generally, i.e., without an independence assumption, provided one quantifies the uncertainty about the individual systems Ai by the von Neumann entropy of suitably chosen conditional states. The analysis of a large system can hence be reduced to the study of its parts. This is relevant for applications. In device-independent cryptography, for instance, the approach yields essentially optimal security bounds valid for general attacks, as shown by Arnon-Friedman et al. (SIAM J Comput 48(1):181–225, 2019).