Title: On Quantum Equipartition Theorem for General Systems

Abstract: Does the quantum equipartition theorem truly exist for any given system? If so, what is the concrete form of such a theorem? The extension of the equipartition theorem, a fundamental principle in classical statistical physics, to the quantum regime raises these two crucial questions. In the present Letter, we focus on how to answer them for arbitray systems. For this propose, the inverse problem of the quantum equipartition theorem has been successfully solved. This result, termed as the inverse equipartition theorem, toghther with nonnegativity and normalizability of the distribution function $\mathbb{P}(\omega)$ serves as a criterion for determining whether a given system adheres to quantum equipartition theorem. If yes, the concrete form of the theorem can be readily obtained. Fermionic version of them is also discussed.Our results can be viewed as a general solution to the topics of quantum equipartition theorem.