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The analysis of classical consensus algorithms relies on contractionproperties of adjoints of Markov operators, with respect to Hilbert'sprojective metric or to a related family of seminorms (Hopf's oscillation orHilbert's seminorm). We generalize these properties to abstract consensusoperators over normal cones, which include the unital completely positive maps(Kraus operators) arising in quantum information theory. In particular, we showthat the contraction rate of such operators, with respect to the Hopfoscillation seminorm, is given by an analogue of Dobrushin's ergodicitycoefficient. We derive from this result a characterization of the contractionrate of a non-linear flow, with respect to Hopf's oscillation seminorm and toHilbert's projective metric.

Dobrushin ergodicity coefficient for Markov operators on cones, and beyond