  We extend entropy production to a deeply quantum regime involving
noncommuting conserved quantities. Consider a unitary transporting conserved
quantities ("charges") between two systems initialized in thermal states. Three
common formulae model the entropy produced. They respectively cast entropy as
an extensive thermodynamic variable, as an information-theoretic uncertainty
measure, and as a quantifier of irreversibility. Often, the charges are assumed
to commute with each other (e.g., energy and particle number). Yet quantum
charges can fail to commute. Noncommutation invites generalizations, which we
posit and justify, of the three formulae. Charges' 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 up stochastic
thermodynamics to noncommuting - and so particularly quantum - charges.
