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Start with the setting the open-quantum-systems literature calls a fine description: a Hilbert space $\mathcal H_f$ = $\mathcal H_S$ $\otimes$ $\mathcal H_B$ holding a system S and everything else, B , that a working physicist has decided not to track in detail — a resonant bath, an environment, a set of nuclear spins, whatever “everything else” means for the apparatus at hand. The fine Hamiltonian is $H_f = H_S \otimes I_B + I_S \otimes H_B + H_{SB}$. three Hermitian terms, each in units of energy: the system’s own generator, the bath’s own generator, and an interaction coupling the two. A fine state is a density operator $\rho$ on $\mathcal H_f$ — trace one, positive semidefinite, dimensionless as a probability object.

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