Quantum Proper Scoring Rules: Minimax Estimation and Resource-Theoretic Advantages

TL;DR

This paper extends proper scoring rules to quantum systems, establishing a duality theory, optimal bounds for quantum state estimation, and quantifying quantum resource advantages in forecasting.

Contribution

It introduces a comprehensive framework for quantum proper scoring rules, linking quantum resources to estimation performance and guiding quantum sensor and machine learning design.

Findings

Derived a Quantum Cramér-Rao-McCarthy Bound linking risk to quantum Fisher information.

Quantified the economic value of quantum coherence, entanglement, and adaptivity.

Established scaling separations between classical and quantum estimation strategies.

Abstract

We generalize proper scoring rules to the quantum domain, replacing probability distributions with density operators. We define Quantum Value Functionals via operator convex generators and establish a complete duality theory yielding proper quantum scoring rules. We derive minimax optimal bounds for quantum state tomography under McCarthy-type incentives, proving a Quantum Cram\'er-Rao-McCarthy Bound that explicitly links minimax risk to the curvature of the generating function and the Quantum Fisher Information. We quantify the economic value of quantum resources (coherence, entanglement, adaptivity) in forecasting tasks, establishing scaling separations between classical and quantum estimation strategies. Our results guide the design of quantum sensors, incentive-compatible quantum data markets, and robust quantum machine learning protocols.