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. We show how transport coefficients obey Onsager reciprocity and identify non-commutativity-induced reductions in the entropy production. 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.
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