Betting on Quantum Objects

TL;DR

This paper proves a quantum Dutch book theorem linking beliefs about measurement outcomes and quantum objects, with implications for interpretations of quantum mechanics and the logic of Hilbert spaces.

Contribution

It introduces a quantum Dutch book theorem applicable to beliefs about quantum objects prior to measurement, extending classical probabilistic coherence to quantum contexts.

Findings

Vector states correspond to Dutch book-avoiding beliefs

The theorem applies to both measurement outcomes and quantum objects

Defenders of the eigenstate-value orthodoxy face a trilemma

Abstract

Dutch book arguments have been applied to beliefs about the outcomes of measurements of quantum systems, but not to beliefs about quantum objects prior to measurement. In this paper, we prove a quantum version of the probabilists' Dutch book theorem that applies to both sorts of beliefs: roughly, if ideal beliefs are given by vector states, all and only Born-rule probabilities avoid Dutch books. This theorem and associated results have implications for operational and realist interpretations of the logic of a Hilbert lattice. In the latter case, we show that the defenders of the eigenstate-value orthodoxy face a trilemma. Those who favor vague properties avoid the trilemma, admitting all and only those beliefs about quantum objects that avoid Dutch books.