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We extend entropy production to a deeply quantum regime involvingnoncommuting conserved quantities. Consider a unitary transporting conservedquantities ("charges") between two systems initialized in thermal states. Threecommon formulae model the entropy produced. They respectively cast entropy asan extensive thermodynamic variable, as an information-theoretic uncertaintymeasure, and as a quantifier of irreversibility. Often, the charges are assumedto commute with each other (e.g., energy and particle number). Yet quantumcharges can fail to commute. Noncommutation invites generalizations, which weposit and justify, of the three formulae. Charges' noncommutation, we find,breaks the formulae's equivalence. Furthermore, different formulae quantifydifferent physical effects of charges' noncommutation on entropy production.For instance, entropy production can signal contextuality - truenonclassicality - by becoming nonreal. This work opens up stochasticthermodynamics to noncommuting - and so particularly quantum - charges.

What happens to entropy production when conserved quantities fail to commute with each other