Asymptotic Equipartition Theorems in von Neumann algebras

Journal article, 2026

The asymptotic equipartition property (AEP) in information theory shows that independent and identically distributed (i.i.d.) states behave as uniform states on its typical subspace. In particular, such a phenomenon can be expressed as that asymptotically, the min- and the max-relative entropy under appropriate smoothing coincide with the relative entropy. In this paper, we generalize several such equipartition properties to states on general von Neumann algebras. First, we show that the smooth max-relative entropy of i.i.d. states on a von Neumann algebra has an asymptotic rate given by the quantum relative entropy. In fact, our AEP not only applies to states, but also to quantum channels with appropriate restrictions. In addition, going beyond the i.i.d. assumption, we show that for states that are produced by a sequential process of quantum channels, the smooth max-relative entropy can be upper-bounded by the sum of appropriate channel relative entropies. Our main technical contributions are to extend to the context of general von Neumann algebras a chain rule for quantum channels, as well as an additivity result for the channel relative entropy with a replacer channel.