Dilation of stochastic matrices by coarse graining

TL;DR

This paper explores methods to represent stochastic matrices as bi-stochastic matrices through dilation techniques, inspired by quantum theory, and discusses entropy implications with illustrative examples.

Contribution

It introduces and compares two dilation methods for stochastic matrices, broadening the understanding of their structure and entropy behavior in probabilistic systems.

Findings

Two dilation methods for stochastic matrices are proposed and analyzed.

Entropy balance is discussed in the context of these dilations.

An example illustrates the application to a Maxwell's demon model.

Abstract

We consider two different ways of representing stochastic matrices by bi-stochastic ones acting on a larger probability space, referred to as ``dilation by uniform coarse graining" and ``environmental dilation". The latter is motivated by analogy to the dilation of operations in quantum theory. Both types of dilation can be viewed as special cases of a general ``dilation by coarse graining". We also discuss the entropy balance and illustrate our results, among others, by an example of a stochastic $4\times 4$-matrix, which serves as a simplified model of the conditional action of Maxwell's demon.