Topos quantum theory provides representations of quantum states as direct generalizations of the probability distribution, namely probability valuation. In this article, we consider extensions of a known bijective correspondence between quantum states and probability valuations to composite systems and to state transformations. We show that multipartite probability valuations on composite systems have a bijective correspondence to positive over pure tensor states, according to a candidate definition of the composite systems in topos quantum theory. Among the multipartite probability valuations, a special attention is placed to Markov chains which are defined by generalizing classical Markov chains from probability theory. We find an incompatibility between the multipartite probability valuations and a monogamy property of quantum states, which trivializes the Markov chains to product probability valuations. Several observations on the transformations of probability valuations are deduc 查看全文>>