  Modern quantum experiments provide examples of transport with non-commuting
quantities, offering a tool to understand the interplay between thermal and
quantum effects. Here we set forth a theory for non-Abelian transport in the
linear response regime. Our key insight is to use generalized Gibbs ensembles
with non-commuting charges as the basic building blocks and strict
charge-preserving unitaries in a collisional setup. The linear response
framework is then built using a collisional model between two reservoirs. We
show that the transport coefficients obey Onsager reciprocity. Moreover, we
find that quantum coherence, associated to the non-commutativity, acts so as to
reduce the net entropy production, when compared to the case of commuting
transport. This therefore provides a clear connection between quantum coherent
transport and dissipation. As an example, we study heat and squeezing fluxes in
bosonic systems, characterizing a set of thermosqueezing coefficients with
potential applications in metrology and heat-to-work conversion in the quantum
regime.
