Quantum Proper Scoring Rules: Convex Duality, Bregman Risk, and the Coherence Gap

arXiv:2605.05268v2 Announce Type: replace Abstract: We generalize proper scoring rules to the quantum domain, replacing probability distributions with density operators. We define Quantum Value Functionals via convex generators and establish a duality theory yielding proper quantum scoring rules using McCarthy's tangent-plane construction. We derive local minimax optimal bounds for quantum state tomography under these incentives, proving a Local Holevo-McCarthy Bound that explicitly links asymptotic risk to the Bogoliubov-Kubo-Mori metric and the Holevo bound. Furthermore, we establish an exact coherence floor that quantifies the irreducible excess score of fixed-basis forecasting strategies when phases are unknown (minimax over the phase orbit), proving a worst-case minimax floor of $\log d$ over the full state space. Our results contextualize fundamental sample complexity separations compared to optimal quantum estimation strategies and guide the design of incentive-compatible quantum data markets and robust quantum machine learning protocols.