Fundamental Limit on Angular Measurements and Rotations from Quantum Mechanics and General Relativity

TL;DR

This paper establishes a fundamental limit on the precision of angular measurements and rotations, dictated by quantum mechanics and general relativity, implying a potential discreteness of quantum state space.

Contribution

It derives a universal precision bound for angular measurements based on physical principles, linking quantum mechanics and gravity, and explores implications for the nature of quantum state space.

Findings

Precision limit is proportional to inverse of device size in Planck units

Small rotations cannot be experimentally distinguished beyond this limit

Supports the possibility of a discrete quantum state space

Abstract

We show that the precision of an angular measurement or rotation (e.g., on the orientation of a qubit or spin state) is limited by fundamental constraints arising from quantum mechanics and general relativity (gravitational collapse). The limiting precision is $r^{-1}$ in Planck units, where $r$ is the physical extent of the (possibly macroscopic) device used to manipulate the spin state. This fundamental limitation means that spin states $S_1$ and $S_2$ cannot be experimentally distinguished from each other if they differ by a sufficiently small rotation. Experiments cannot exclude the possibility that the space of quantum state vectors (i.e., Hilbert space) is fundamentally discrete, rather than continuous. We discuss the implications for finitism: does physics require infinity or a continuum?