Quantum counterpart of equipartition theorem: A Möbius inversion approach

The equipartition theorem is crucial in classical statistical physics, and recent studies have revealed its quantum counterpart for specific systems. This raises the question: does a quantum counterpart of the equipartition theorem exist for any given system, and if so, what is its concrete form? In this Letter, we employ the Möbius inversion approach to address these questions, providing a criterion to determine whether a system adheres to the quantum counterpart of the equipartition theorem. If it does, the corresponding distribution function can be readily derived. Furthermore, we construct the fermionic version of the criterion in a manner analogous to the bosonic case.