Discretisation of the Bloch Sphere, Fractal Invariant Sets and Bell's Theorem

TL;DR

This paper constructs a dense discretisation of the Bloch sphere using number theory, revealing quantum-like properties and offering a new deterministic interpretation of Bell's theorem through fractal invariant sets.

Contribution

It introduces a novel finite, discretised model of the Bloch sphere that captures key quantum features and provides a deterministic framework for understanding Bell inequality violations.

Findings

Discretised Bloch sphere models quantum properties like complementarity and entanglement.

The model offers a deterministic explanation for Bell inequality violations.

There is a maximum number of qubits that can be entangled in this representation.

Abstract

An arbitrarily dense discretisation of the Bloch sphere of complex Hilbert states is constructed, where points correspond to bit strings of fixed finite length. Number-theoretic properties of trigonometric functions (not part of the quantum-theoretic canon) are used to show that this constructive discretised representation incorporates many of the defining characteristics of quantum systems: completementarity, uncertainty relationships and (with a simple Cartesian product of discretised spheres) entanglement. Unlike Meyer's earlier discretisation of the Bloch Sphere, there are no orthonormal triples, hence the Kocken-Specker theorem is not nullified. A physical interpretation of points on the discretised Bloch sphere is given in terms of ensembles of trajectories on a dynamically invariant fractal set in state space, where states of physical reality correspond to points on the invariant set. This deterministic construction provides a new way to understand the violation of the Bell inequality without violating statistical independence or factorisation, where these conditions are defined solely from states on the invariant set. In this finite representation there is an upper limit to the number of qubits that can be entangled, a property with potential experimental consequences.