Optimal Hamiltonian for a quantum state with finite entropy

We consider the following task: how for a given quantum state $\rho$ to find a grounded Hamiltonian $H$ satisfying the condition $\mathrm{Tr} H\rho\leq E_0<+\infty$ in such a way that the von Neumann entropy of the Gibbs state $\gamma_H(E)$ corresponding to a given energy $E>0$ be as small as possible. We show that for any mixed state $\rho$ with finite entropy and any $E>0$ there exists a solution $H(\rho,E_0,E)$ of the above problem (unique in the non-degenerate case) which we call optimal Hamiltonian for the state $\rho$. Explicit expressions for $H(\rho,E_0,E)$, $\gamma_{H(\rho,E_0,E)}(E)$ and $S(\gamma_{H(\rho,E_0,E)}(E))$ are obtained. Analytical properties of the function $E\mapsto S(\gamma_{H(\rho,E_0,E)}(E))$ are explored. Several examples are considered. We also consider a modification of the above task in which arbitrary Hamiltonians (not necessarily grounded) are considered. The basic application motivated this research is described. As examples, new semicontinuity bounds for the von Neumann entropy and for the entanglement of formation are obtained and briefly discussed (with the intention to give a detailed analysis in a separate article).