# Coarse graining Shannon and von Neumann entropies

## TL;DR

This paper develops explicit, mathematically well-defined models for coarse graining of classical and quantum entropies, analyzing how entropy flows and hidden information evolve in these processes, with potential implications for black hole information puzzles.

## Contribution

It introduces simple, calculable models for coarse graining of Shannon and von Neumann entropies, including quantum decoherence and super-scattering matrices, to clarify entropy flow and hidden correlations.

## Key findings

- Models effectively track entropy increase during coarse graining
- Quantum decoherence models reveal hidden correlations in von Neumann entropy
- Super-scattering matrices aid in analyzing quantum entropy flow

## Abstract

The nature of coarse graining is intuitively "obvious", but it is rather difficult to find explicit and calculable models of the coarse graining process (and the resulting entropy flow) discussed in the literature. What we would like to have at hand is some explicit and calculable process that takes an arbitrary system, with specified initial entropy S, and monotonically and controllably drives the entropy to its maximum value. This does not have to be a physical process, in fact for some purposes it is better to deal with a gedanken-process, since then it is more obvious how the "hidden information" is hiding in the fine-grain correlations that one is simply agreeing not to look at. We shall present several simple mathematically well-defined and easy to work with conceptual models for coarse graining. We shall consider both the classical Shannon and quantum von Neumann entropies, including models based on quantum decoherence, and analyze the entropy flow in some detail. When coarse-graining the quantum von Neumann entropy, we find it extremely useful to introduce an adaptation of Hawking's super-scattering matrix. These explicit models that we shall construct allow us to quantify and keep clear track of the entropy that appears when coarse-graining the system, and the information that can be hidden in unobserved correlations. (While not the focus of the current article, in the longer run these considerations are of interest when addressing black hole information puzzle.)