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Given a quantum Hamiltonian $H$ (e.g. the quantum Ising Hamiltonian $H= -h\sum_{i}X_i-\sum_{\langle i,j\rangle}Z_iZ_j$): we know that at temperature $T$, the system is in the state:

$$\rho(T) = e^{-H/T}.$$

It is clear $\rho(0)=|0\rangle\langle0|$, where $|0\rangle$ is the eigenvector of $H$ with smallest eigenvalue.

In other words, "$\rho(0)$ minimises $\text{tr}(H\rho)$".

Now, I'd like to make analogous statements to the above for $\rho(T)$ when $T>0$, i.e. I'd like to say that $\rho(T)$ minimises $\text{tr}(\tilde{H}\rho)$ for some $\tilde{H}$ which depends on $T$. My question is what is the $\tilde{H}$? (Or maybe first, does $\tilde{H}$ exist?) I think that the $\tilde{H}$ should be some "quantum version" of the free energy, but don't know what it should be exactly.

Thanks!

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I don't think this is a well-defined question. If you put $$\tilde{H} = H e^{H T}$$ then $$\text{tr} \, \tilde{H} \rho(T) = \text{tr}\, H \rho(0).$$ The operator $\tilde{H}$ is not really well-defined, in the sense that for $T>0$ its eigenvalues grow very rapidly at infinity.

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Okay, so I think this is correct, writing $D$ for the convex set of density operators and $S$ for the entropy: $$\rho(T) = \text{argmin}_{\rho \in \text{D}} [\text{tr}(\rho H)-TS(\rho)] = \text{argmin}_{\rho \in \text{D}} [\text{tr}(\rho (H+T\log \rho))].$$ The latter convex minimisation can be done via Lagrange multipliers giving $\rho(T)$ as desired. So if I were to invent some notation, I'd say something like: $$\tilde{H}=H+T\log ( \cdot ).$$

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