Discretization of the Bloch sphere, fractal invariant sets and Bell's theorem.

T N Palmer

doi:10.1098/rspa.2019.0350

Abstract

An arbitrarily dense discretization 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 discretized representation incorporates many of the defining characteristics of quantum systems: completementarity, uncertainty relationships and (with a simple Cartesian product of discretized spheres) entanglement. Unlike Meyer’s earlier discretization of the Bloch Sphere, there are no orthonormal triples, hence the Kocken–Specker theorem is not nullified. A physical interpretation of points on the discretized 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 factorization, 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.

Highlights

The fields R and C are deeply embedded in the formalism of both classical and quantum theories of physics
We argue that if the latter are based on processes occurring in space–time, the latter are plausible definitions of free choice and local causality, when the invariant set model is locally causal
Many scientists will be sympathetic to the implications of Hilbert’s observation—that if the infinite is nowhere to be found in reality, it should neither be found in our descriptions of reality. This is problematic for quantum theory where, even for finite-dimensional systems, the notion of the infinitesimal plays a vital role [4]

Summary

Introduction

The fields R and C are deeply embedded in the formalism of both classical and quantum theories of physics. It is shown how two of the most important defining properties of quantum theory: complementarity and the uncertainty relationships, derive from geometric properties of spherical triangles and number-theoretic properties of trigonometric functions. Such a limit may conceivably be a manifestation in the invariant set model of the inherently decoherent nature of gravity.

With this define

Then cos φ

Hence b

We now have

Discussion