Classical Information-Theoretical View of Physical Measurements and Generalized Uncertainty Relations

TL;DR

This paper explores the classical information-theoretic approach to physical measurements, deriving generalized uncertainty relations for dynamic systems, analyzing boundary condition effects, and experimentally testing these relations with electron density data.

Contribution

It introduces a classical information-theoretic framework for generalized uncertainty relations and examines boundary condition effects on measurement uncertainties, supported by experimental data.

Findings

Boundary conditions can reduce measurement uncertainties quantitatively.

Generalized uncertainty relations are applicable to systems governed by Sturm--Liouville equations.

Experimental analysis confirms the reduction of uncertainty due to boundary conditions.

Abstract

General characterizations of physical measurements are discussed within the framework of the classical information theory. The uncertainty relation for simultaneous measurements of two physical observables is defined in this framework for generalized dynamic systems governed by a Sturm--Liouville type of equation of motion. In the first step, the reduction of Kennard--Robertson type uncertainties due to boundary conditions with a mean-square error is discussed quantitatively with reference to the information entropy. Several concrete examples of generalized uncertainty relations are given. Then, by considering disturbance effects, a universally valid uncertainty relation is investigated for the generalized equation of motion with a certain boundary condition. Necessary conditions for violation (reduction) of the Heisenberg-type uncertainty relation are discussed in detail. The reduction of the generalized uncertainty relation due to the boundary condition is then tested experimentally by re-analyzing data for measured electron densities in a hydrogen molecule encapsulated in a fullerene C60 cage.