This work investigates the structural origin of quantum entropy from the perspective of opera- tional symmetry. Rather than postulating symmetry properties directly on entropy functionals, the analysis begins with the geometry of quantum state space under operational indistinguishability. We show that invariance under representational redundancy—mathematically expressed as uni- tary conjugation—naturally induces a quotient structure on the space of density operators 1. The resulting orbit space is isomorphic to the classical probability simplex formed by unordered spectra. Consequently, any admissible state functional must factor through this spectral quotient. Upon imposing strong compositional consistency and continuity conditions on the reduced space, classical structural results apply and uniquely select the Shannon entropy 2, 3. Lifting this result back to operators yields the von Neumann entropy as a direct corollary. Entropy uniqueness thus appears as a structural consequence of quotient geometry combined with classical compositional principles, rather than as an independently postulated axiom.
Kaibin Liu (Fri,) studied this question.