General mathematical properties of Boltzmann’s quantum entropy

Establish the general mathematical properties of Boltzmann’s quantum entropy, defined as the logarithm of the regularized Hilbert-space volume of pure states compatible with a quantum preparation, including additivity, subadditivity, continuity, and its behavior under composition and coarse graining.

Background

The paper introduces Boltzmann’s quantum entropy as a preparation-dependent quantity obtained from the Hilbert-space volume of pure states satisfying regularized quantum constraints. Unlike von Neumann entropy, which is defined from a density matrix, this quantity measures the size of the compatible microscopic-state set associated directly with a preparation.

The authors derive explicit volume formulas for several classes of preparations, including subspace restrictions, fixed expectation values, partial-trace descriptions, and an imperfect-detector coarse-graining map. However, they do not establish whether the proposed entropy satisfies standard structural properties expected of an entropy, such as additivity, subadditivity, and continuity, nor how it behaves when systems are composed or subjected to coarse graining. These properties are therefore explicitly identified as unresolved mathematical questions.