We consider the following task: how for a given quantum state $ρ$ to find a grounded Hamiltonian $H$ satisfying the condition $\mathrm{Tr} Hρ\leq E_0<+\infty$ in such a way that the von Neumann entropy of the Gibbs state $γ_H(E)$ corresponding to a given energy $E>0$ be as small as possible. We show that for any mixed state $ρ$ with finite entropy and any $E>0$ there exists a solution $H(ρ,E_0,E)$ of the above problem (unique in the non-degenerate case) which we call optimal Hamiltonian for the state $ρ$. Explicit expressions for $H(ρ,E_0,E)$, $γ_{H(ρ,E_0,E)}(E)$ and $S(γ_{H(ρ,E_0,E)}(E))$ are obtained. Analytical properties of the function $E\mapsto S(γ_{H(ρ,E_0,E)}(E))$ 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).

翻译：我们考虑以下任务：对于给定的量子态$ρ$，如何找到一个基态哈密顿量$H$，使其满足条件$\mathrm{Tr} Hρ\leq E_0<+\infty$，并且使得对应于给定能量$E>0$的吉布斯态$γ_H(E)$的冯·诺依曼熵尽可能小。我们证明，对于任意具有有限熵的混合态$ρ$和任意$E>0$，上述问题存在一个解$H(ρ,E_0,E)$（在非简并情况下唯一），我们称之为态$ρ$的最优哈密顿量。我们得到了$H(ρ,E_0,E)$、$γ_{H(ρ,E_0,E)}(E)$和$S(γ_{H(ρ,E_0,E)}(E))$的显式表达式。我们探讨了函数$E\mapsto S(γ_{H(ρ,E_0,E)}(E))$的解析性质。文中考虑了若干示例。我们还考虑了上述任务的一个修改版本，其中允许考虑任意哈密顿量（不限于基态）。文中描述了推动本研究的基本应用动机。作为示例，我们得到了冯·诺依曼熵和形成纠缠的新半连续性界，并进行了简要讨论（详细分析拟在另文中给出）。