Intermediate determinism in general probabilistic theories

TL;DR

This paper explores the concept of intermediate determinism in quantum and general probabilistic theories, extending from properties to measurements and identifying conditions for its presence beyond quantum mechanics.

Contribution

It generalizes intermediate determinism to measurements in probabilistic theories and characterizes the conditions under which it holds, beyond quantum theory.

Findings

Intermediate determinism is guaranteed by the structure of effects in quantum theories.

Necessary and sufficient conditions for intermediate determinism in general probabilistic theories are identified.

Neither the no-restriction hypothesis nor Gleason-type theorems are necessary or sufficient for intermediate determinism.

Abstract

Quantum theory is indeterministic, but not completely so. When a system is in a pure state there are properties it possesses with certainty, known as actual properties. The actual properties of a quantum system (in a pure state) fully determine the probability of finding the system to have any other property. We call this feature intermediate determinism. In dimensions of at least three, the intermediate determinism of quantum theory is guaranteed by the structure of its lattice of properties. This observation follows from Gleason's theorem, which is why it fails to hold in dimension two. In this work we extend the idea of intermediate determinism from properties to measurements. Under this extension intermediate determinism follows from the structure of quantum effects for separable Hilbert spaces of any dimension, including dimension two. Then, we find necessary and sufficient conditions for a general probabilistic theory to obey intermediate determinism. We show that, although related, both the no-restriction hypothesis and a Gleason-type theorem are neither necessary nor sufficient for intermediate determinism.