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

References in corpus (5)

• A nonequilibrium equality for free energy differences
• A Quantum Fluctuation Theorem
• From quantum mechanics to classical statistical physics: generalized Rokhsar-Kivelson Hamiltonians and the "Stochastic Matrix Form" decomposition
• Birkhoff's polytope and unistochastic matrices, N=3 and N=4
• Sequential measurements and entropy