  The grand canonical ensemble lies at the core of quantum and classical
statistical mechanics. A small system thermalizes to this ensemble while
exchanging heat and particles with a bath. A quantum system may exchange
quantities represented by operators that fail to commute. Whether such a system
thermalizes and what form the thermal state has are questions about truly
quantum thermodynamics. Here we investigate this thermal state from three
perspectives. First, we introduce an approximate microcanonical ensemble. If
this ensemble characterizes the system-and-bath composite, tracing out the bath
yields the system's thermal state. This state is expected to be the equilibrium
point, we argue, of typical dynamics. Finally, we define a resource-theory
model for thermodynamic exchanges of noncommuting observables. Complete
passivity---the inability to extract work from equilibrium states---implies the
thermal state's form, too. Our work opens new avenues into equilibrium in the
presence of quantum noncommutation.
