Canonical quantum thermal preparation

Adopt canonical preparation as follows. Fix the positive Boltzmann constant $k_B>0$ in J/K. Let $H=H^{\ast }$ act on a finite-dimensional complex Hilbert space of dimension $d\ge 1$, or be self-adjoint on a nonzero separable Hilbert space, bounded below and satisfying $e^{-\beta H}\in {\mathcal{S}}_1$ for $\beta$ in an open interval about the chosen value. For temperature $T>0$, put $\beta =(k_{B}T{)}^{-1}$ and adopt ${\rho }_{\beta }=\frac{e^{-\beta H}}{Z_{\beta }},Z_{\beta }=\mathrm{Tr}(e^{-\beta H})\mathrm{.}$ In the infinite-dimensional case all scalar energy moments used in a conclusion must also be finite on the actual spectral domain, as required by Self-adjoint trace-Gibbs normalization and form-energy optimization. With $H$ in joules, $k_B$ in J/K, $T$ in K, and $\beta$ in J${}^{-1}$, the exponent is dimensionless and $Z_{\beta }$ is a dimensionless trace.

This is an adopted preparation rule in the normal-state framework. It is not a theorem that unitary dynamics equilibrates, does not specify a bath or equilibration time, and makes no empirical claim. The finite and trace-Gibbs variational results describe the mathematical optimization associated with this state but do not establish that a physical system is prepared in it.