Asymptotic Equipartition Theorems in von Neumann algebras

TL;DR

This paper extends the asymptotic equipartition property to states and channels in von Neumann algebras, showing that smooth max relative entropy converges to quantum relative entropy and providing bounds for sequential processes.

Contribution

It generalizes AEP to von Neumann algebras and introduces new chain rule and additivity results for quantum channels.

Findings

Smooth max relative entropy converges to quantum relative entropy for i.i.d. states.

Upper bounds for max relative entropy in sequential channel processes.

Extended chain rule and additivity results for quantum channels.

Abstract

The Asymptotic Equipartition Property (AEP) in information theory establishes that independent and identically distributed (i.i.d.) states behave in a way that is similar to uniform states. In particular, with appropriate smoothing, for such states both the min and the max relative entropy asymptotically 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.