Every finite stochastic matrix is a marginal of a larger unistochastic matrix, so classical Markov transitions can be represented by unitary quantum evolution with a fixed ancilla.
Dilation of stochastic matrices by coarse graining
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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.
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The Born Representation Theorem and the Unistochastic Theorem
Every finite stochastic matrix is a marginal of a larger unistochastic matrix, so classical Markov transitions can be represented by unitary quantum evolution with a fixed ancilla.