Classical information theoretic view of physical measurements and generalized uncertainty relations

Uncertainty relations are discussed in detail not only for free particles but also for bound states within the framework of classical information theory. Uncertainty relation for simultaneous measurements of two physical observables is defined in this framework for generalized dynamic systems governed by a Sturm-Liouville-type equation of motion. In the first step, the reduction of Kennard-Robertson type uncertainties because of 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. The necessary conditions for violation (reduction) of the Heisenberg-type uncertainty relation are discussed in detail. The reduction of the generalized uncertainty relation because of the boundary condition is discussed by reanalyzing experimental data for measured electron densities in a hydrogen molecule encapsulated in a fullerene C-60 cage.