A mechanistic macroscopic physical entity with a three-dimensional Hilbert space description

TL;DR

This paper constructs a classical macroscopic system modeled in a three-dimensional real Hilbert space, challenging the notion that Gleason's theorem prohibits hidden-variable models in higher dimensions.

Contribution

It provides an explicit classical system representation in three-dimensional Hilbert space, offering a new perspective on Gleason's theorem and hidden-variable models.

Findings

Classical system modeled in 3D Hilbert space.

Probability structure linked to measurement context ignorance.

Discussion on Gleason's theorem implications.

Abstract

It is sometimes stated that Gleason's theorem prevents the construction of hidden-variable models for quantum entities described in a more than two-dimensional Hilbert space. In this paper however we explicitly construct a classical (macroscopical) system that can be represented in a three-dimensional real Hilbert space, the probability structure appearing as the result of a lack of knowledge about the measurement context. We briefly discuss Gleason's theorem from this point of view.