# Incompatible Multiple Consistent Sets of Histories and Measures of Quantumness

## TL;DR

This paper proposes a classification of multiple consistent sets in the consistent histories approach to quantum theory based on the existence of unifying probabilities, linking it to measures of quantumness and classical-quantum boundaries.

## Contribution

It introduces a new classification scheme for consistent histories using unifying probabilities, connecting quantum measures with classicality and extending the single framework rule.

## Key findings

- Existence of unifying probabilities correlates with classical regimes.
- Non-existence indicates higher quantumness and incompatible sets.
- Examples include Bell, CHSH, and Leggett-Garg inequalities.

## Abstract

In the consistent histories (CH) approach to quantum theory probabilities are assigned to histories subject to a consistency condition of negligible interference. The approach has the feature that a given physical situation admits multiple sets of consistent histories that cannot in general be united into a single consistent set, leading to a number of counter-intuitive or contrary properties if propositions from different consistent sets are combined indiscriminately. An alternative viewpoint is proposed in which multiple consistent sets are classified according to whether or not there exists any unifying probability for combinations of incompatible sets which replicates the consistent histories result when restricted to a single consistent set. A number of examples are exhibited in which this classification can be made, in some cases with the assistance of the Bell, CHSH or Leggett-Garg inequalities together with Fine's theorem. When a unifying probability exists logical deductions in different consistent sets can in fact be combined, an extension of the "single framework rule". It is argued that this classification coincides with intuitive notions of the boundary between classical and quantum regimes and in particular, the absence of a unifying probability for certain combinations of consistent sets is regarded as a measure of the "quantumness" of the system. The proposed approach and results are closely related to recent work on the classification of quasi-probabilities and this connection is discussed.