Optimal Hamiltonian for a quantum state with finite entropy

TL;DR

This paper investigates how to find an optimal Hamiltonian for a given quantum state that minimizes the entropy of the associated Gibbs state at a fixed energy, providing explicit solutions and exploring their properties.

Contribution

It introduces a method to explicitly construct the optimal Hamiltonian for a quantum state with finite entropy, including analytical properties and examples, advancing understanding of quantum thermodynamics.

Findings

Existence and uniqueness of the optimal Hamiltonian for mixed states.

Explicit formulas for the optimal Hamiltonian and Gibbs state.

New bounds for von Neumann entropy and entanglement of formation.

Abstract

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).