Entropy Accumulation

We ask the question whether entropy accumulates, in the sense that the operationally relevant total uncertainty about an n-partite system $$A = (A_1, \ldots A_n)$$A = ( A 1 , … A n ) corresponds to the sum of the entropies of its parts $$A_i$$A i . The Asymptotic Equipartition Property implies that this is indeed the case to first order in n—under the assumption that the parts $$A_i$$A i 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 $$A_i$$A i 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).