The von Neumann entropy of an n-partite system \(A_1^n\) given a system B can be written as the sum of the von Neumann entropies of the individual subsystems \(A_k\) given \(A_1^{k-1}\) and B. While it is known that such a chain rule does not hold for the smooth min-entropy, we prove a counterpart of this for a variant of the smooth min-entropy, which is equal to the conventional smooth min-entropy up to a constant. This enables us to lower bound the smooth min-entropy of an n-partite system in terms of, roughly speaking, equally strong entropies of the individual subsystems. We call this a universal chain rule for the smooth min-entropy, since it is applicable for all values of n. Using duality, we also derive a similar relation for the smooth max-entropy. Our proof utilises the entropic triangle inequality based technique developed in Marwah and Dupuis (Commun Math Phys 405(9):211, 2024. https://doi.org/10.1007/s00220-024-05074-8) for analysing approximation chains. Additionally, we also prove an approximate version of the entropy accumulation theorem, which significantly relaxes the conditions required on the state to bound its smooth min-entropy. In particular, it does not require the state to be produced through a sequential process like previous entropy accumulation type bounds. In our companion paper Marwah and Dupuis (Security proof for parallel DIQKD, 2025. https://arxiv.org/abs/2507.03991), we use it to prove the security of parallel device independent quantum key distribution.
