A classification of classical representations for quantum-like systems

TL;DR

This paper classifies classical representations of quantum-like systems, revealing their structural properties and providing explicit examples, thereby deepening understanding of the classical-quantum interface.

Contribution

It introduces a poset structure for non-equivalent classical representations and constructs explicit finite-dimensional quantum mechanics representations.

Findings

The collection of non-equivalent representations forms a poset.

No smallest representation exists due to the poset not being a semi-lattice.

An explicit finite-dimensional classical representation of quantum mechanics is constructed.

Abstract

For general non-classical systems, we study the different classical representations that fulfill the specific context dependence imposed by the hidden measurement system formalism introduced in quant-ph/0008061. We show that the collection of non-equivalent representations has a poset structure. We also show that in general, there exists no 'smallest' representation, since this poset is not a semi-lattice. Then we study the possible representations of quantum-like measurement systems. For example, we show that there exists a classical representation of finite dimensional quantum mechanics with ${\Bbb N}$ as a set of states for the measurement context, and we build an explicit example of such a representation.