Optimal Hamiltonian for a quantum state with finite entropy

TL;DR

The paper addresses how to construct a grounded Hamiltonian $H$ that minimizes the Gibbs-state entropy $S(\gamma_H(E))$ at a fixed energy for a given quantum state $\rho$ with finite entropy, under a linear energy constraint $\mathrm{Tr} H\rho \le E_0$. It proves existence of an optimal $H(\rho,E_0,E)$ and provides explicit constructions in two regimes (A and B); the Gibbs state satisfies $\gamma_{H(\rho,E_0,E)}(E)=\langle\theta\rho\rangle_{\rm q}$ with $\theta=E/E_0$, and $S(\gamma_{H(\rho,E_0,E)}(E))$ is expressed via a Shannon-entropy functional of $\langle\theta p_1,\theta p_2,\dots\rangle_{\rm c}$, depending only on the eigenvalues of $\rho$. The authors show that the energy-to-entropy map $G_{E_0}^{\rho}(E)$ is nondecreasing, concave, and differentiable, with precise finite/infinite-rank behaviors and breakpoints governed by the spectrum of $\rho$. They illustrate the theory with several examples, highlighting piecewise-constant regimes and the role of the null subspace dimension $m$, and then apply the framework to obtain faithful semicontinuity bounds for entropic quantities, notably a new bound for Entanglement of Formation that tightens prior results. The work provides a practical method to bound informational measures for infinite-dimensional systems by optimizing over Hamiltonians under energy constraints, with potential impact on stability analyses and quantum information protocols in continuous-variable settings.

Abstract

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. A basic application of the above task is briefly described (with the intention to give a detailed description in a separate article). As an example, a new semicontinuity bound for the entanglement of formation is obtained.

Figures (5)

• Figure 1: The values of $S(\gamma_{H(\bar{\rho}_n,E_0,E)}(E))$ with $n=10$ along with the values of $m(E)\doteq\dim\ker H(\rho_n,E_0,E)$ with factor $1/10$.
• Figure 2: The values of $S(\gamma_{H(\rho_n,E_0,E)}(E))$ with $n=10$ along with the values of $m(E)\doteq\dim\ker H(\rho_n,E_0,E)$ with factor $1/10$.
• Figure 3: The values of $S(\gamma_{H(\rho_0,E_0,E)}(E))$ and $S(\gamma_N(E))$ with $E_0=1$ along with the values of $m(E)\doteq\dim\ker H(\rho_0,E_0,E)$ with factor $1/10$.
• Figure 4: The values of $S(\gamma_{H(\rho_0,E_0,E)}(E))$ and $S(\gamma_N(E))$ with $E_0=10$ along with the values of $m(E)\doteq\dim\ker H(\rho_0,E_0,E)$ with factor $1/10$.
• Figure 5: The values of $S(\gamma_{H(\rho_0,E_0,E)}(E))$ and $S(\gamma_N(E))$ with $E_0=100$ along with the values of $m(E)\doteq\dim\ker H(\rho_0,E_0,E)$ with factor $1/50$.