Entropy accumulation with improved second-order

The entropy accumulation theorem states that the smooth min-entropy of an n-partite system A = (A_1 ,. .. , A_n) is lower-bounded by the sum of the von Neumann entropies of suitably chosen conditional states up to corrections that are sublinear in n. This theorem is particularly suited to proving the security of quantum cryptographic protocols, and in particular so-called device-independent protocols for randomness expansion and key distribution, where the devices can be built and preprogrammed by a malicious supplier. However, while the bounds provided by this theorem are optimal in the first order, the second-order term is bounded more crudely, in such a way that the bounds deteriorate significantly when the theorem is applied directly to protocols where parameter estimation is done by sampling a small fraction of the positions, as is done in most QKD protocols. The objective of this paper is to improve this second-order sublinear term and remedy this problem. On the way, we prove various bounds on the divergence variance, which might be of independent interest