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Dilation of stochastic matrices by coarse graining

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arxiv 2106.03513 v1 pith:QQ2MT25W submitted 2021-06-07 math-ph math.MP

classification math-phmath.MP
keywords dilationcoarsegrainingstochasticmatricesactingactionanalogy
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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.

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  1. The Born Representation Theorem and the Unistochastic Theorem

    quant-ph 2026-08 conditional novelty 5.0 of 10

    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.

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