  Consider a sequential process in which each step outputs a system $A_i$ and
updates a side information register $E$. We prove that if this process
satisfies a natural "non-signalling" condition between past outputs and future
side information, the min-entropy of the outputs $A_1, \dots, A_n$ conditioned
on the side information $E$ at the end of the process can be bounded from below
by a sum of von Neumann entropies associated with the individual steps. This is
a generalisation of the entropy accumulation theorem (EAT), which deals with a
more restrictive model of side information: there, past side information cannot
be updated in subsequent rounds, and newly generated side information has to
satisfy a Markov condition. Due to its more general model of side-information,
our generalised EAT can be applied more easily and to a broader range of
cryptographic protocols. As examples, we give the first multi-round security
proof for blind randomness expansion and a simplified analysis of the E91 QKD
protocol. The proof of our generalised EAT relies on a new variant of Uhlmann's
theorem and new chain rules for the Renyi divergence and entropy, which might
be of independent interest.
