Dilation of stochastic matrices by coarse graining

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 × 4-matrix, which serves as a simplified model of the conditional action of Maxwell's demon.