Compatibility for Probabilistic Theories

TL;DR

This paper introduces a compatibility index for probabilistic theories, positioning quantum mechanics and classical probability at opposite ends, and explores the structure of observables via positive vector measures.

Contribution

It defines a new compatibility index for PTs, analyzes its implications for quantum and classical theories, and proposes a representation of observables through positive vector measures.

Findings

Quantum mechanics has compatibility index 0.

Classical probability theory has compatibility index 1.

Potential existence of intermediate compatibility indices in certain PTs.

Abstract

We define an index of compatibility for a probabilistic theory (PT). Quantum mechanics with index 0 and classical probability theory with index 1 are at the two extremes. In this way, quantum mechanics is at least as incompatible as any PT. We consider a PT called a concrete quantum logic that may have compatibility index strictly between 0 and 1, but we have not been able to show this yet. Finally, we show that observables in a PT can be represented by positive, vector-valued measures.