time we had $$ \rho = \frac{1}{Z} e^{-\lambda^i X_i}$$ where $\lambda^i$ were some functions on our manifold and $$ Z = \mathrm{tr}(e^{-\lambda^i X_i})$$ was the partition function . Let's copy this idea. So, we'll start with our density matrix $\rho$, but then write it as $$ \rho = \frac{1}{Z} e^{-A}$$ where $A$