Quantum Information Elicitation

TL;DR

This paper extends classical scoring rules to quantum settings, allowing agents to report quantum states and designing mechanisms to incentivize truthful reporting, with connections to quantum information theory.

Contribution

It introduces quantum scoring rules, characterizes their structure, and explores the elicitation complexity of quantum properties, advancing the intersection of quantum information and mechanism design.

Findings

Spectral scores have elegant structure and relate to quantum information theory.

Eigenvectors of quantum beliefs are elicitable, eigenvalues are not.

Quantum entropy has maximal elicitation complexity.

Abstract

In the classic scoring rule setting, a principal incentivizes an agent to truthfully report their probabilistic belief about some future outcome. This paper addresses the situation when this private belief, rather than a classical probability distribution, is instead a quantum mixed state. In the resulting quantum scoring rule setting, the principal chooses both a scoring function and a measurement function, and the agent responds with their reported density matrix. Several characterizations of quantum scoring rules are presented, which reveal a familiar structure based on convex analysis. Spectral scores, where the measurement function is given by the spectral decomposition of the reported density matrix, have particularly elegant structure and connect to quantum information theory. Turning to property elicitation, eigenvectors of the belief are elicitable, whereas eigenvalues and entropy have maximal elicitation complexity. The paper concludes with a discussion of other quantum information elicitation settings and connections to the literature.