**What happens to entropy production when conserved quantities fail to commute with each other** - Billy Braasch

A fundamental challenge is to define quantum thermodynamic
quantities-for example, heat, work, and entropy production. We extend the definition of entropy production to a deeply quantum regime involving noncommuting observables. Consider two systems prepared in different thermal states. A unitary transports observables ("charges") between the systems. Three common formulae model the entropy produced. They cast entropy as an extensive thermodynamic variable, as an informationtheoretic uncertainty measure, and as a quantifier of irreversibility. Often, the charges are assumed to commute with each other (e.g., energy and particle number), and the entropy-production formulae equal each other. Yet quantum charges can fail to commute, inviting generalizations of the
three formulae. Chargesâ€™ noncommutation, we find, breaks the formulae's
equivalence. Furthermore, different formulae quantify different physical
effects of charges' noncommutation on entropy production. For instance,
entropy production can signal contextuality-true nonclassicality-by
becoming nonreal. This work opens the door of stochastic
thermodynamics to charges that are peculiarly quantum by failing to
commute with each other.