Generalised Entropy Accumulation

Consider a sequential process in which each step outputs a system <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>A</mml:mi> <mml:mi>i</mml:mi></mml:msub> </mml:math> and updates a side information register <i>E</i>. 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>A</mml:mi> <mml:mn>1</mml:mn></mml:msub> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:msub><mml:mi>A</mml:mi> <mml:mi>n</mml:mi></mml:msub> </mml:mrow> </mml:math> conditioned on the side information <i>E</i> 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) (Dupuis et al. in Commun Math Phys 379: 867-913, 2020), 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 Rényi divergence and entropy, which might be of independent interest.