Markov operators on cones and non-commutative consensus

The analysis of classical consensus algorithms relies on contraction properties of Markov matrices with respect to the Hilbert semi-norm (infinitesimal version of Hilbert's projective metric) and to the total variation norm. We generalize these properties to the case of operators on cones. This is motivated by the study of “non-commutative consensus”, i.e., of the dynamics of linear maps leaving invariant cones of positive semi-definite matrices. Such maps appear in quantum information (Kraus maps), and in the study of matrix means. We give a characterization of the contraction rate of an abstract Markov operator on a cone, which extends classical formulæ obtained by Dœblin and Dobrushin in the case of Markov matrices. In the special case of Kraus maps, we relate the absence of contraction to the positivity of the “zero-error capacity” of a quantum channel. We finally show that a number of decision problems concerning the contraction rate of Kraus maps reduce to finding a rank one matrix in linear spaces satisfying certain conditions and discuss complexity issues.