We show that the hierarchical decomposition of quantum entropy is determined by the Wedderburn structure of the causally accessible observable algebra. Given a finite-dimensional von Neumann subalgebra AC ⊆ B (H) encoding the observables accessible under a causal constraint C, we prove that any continuous functional on the state space of AC satisfying a block independence condition decomposes uniquely into a classical contribution over the center and quantum contributions within each block. The branching structure of this decomposition — the grouping tree — is not postulated but uniquely determined by the block decomposition of AC. The framework is illustrated in a bipartite system with three levels of causal accessibility, demonstrating that the grouping tree grows, restructures, or collapses as the accessible algebra changes. This identifies the decomposition structure of quantum entropy as an algebraic consequence of accessible observables, rather than an independent axiom on entropy functionals.
Kaibin Liu (Sun,) studied this question.