Quantum uncertainty and the spectra of symmetric operators

TL;DR

This paper investigates the relationship between the spectral properties of symmetric operators in quantum mechanics and the bounds on measurement uncertainty, with implications for quantum information theory.

Contribution

It establishes new results linking eigenvalue spacing of self-adjoint extensions to uncertainty bounds in symmetric operators.

Findings

Bounded uncertainty implies symmetric but not self-adjoint operators.

Eigenvalue spacing relates to the lower bounds of uncertainty.

Applications discussed in quantum and classical information theory.

Abstract

In certain circumstances, the uncertainty, $\Delta S [\phi]$, of a quantum observable, $S$, can be bounded from below by a finite overall constant $\Delta S>0$, \emph{i.e.}, $\Delta S [\phi] \geq \Delta S$, for all physical states $\phi$. For example, a finite lower bound to the resolution of distances has been used to model a natural ultraviolet cutoff at the Planck or string scale. In general, the minimum uncertainty of an observable can depend on the expectation value, $t=\langle \phi, S \phi\rangle$, through a function $\Delta S_t$ of $t$, \emph{i.e.}, $\Delta S [\phi]\ge \Delta S_t$, for all physical states $\phi$ with $\langle \phi, S \phi\rangle=t$. An observable whose uncertainty is finitely bounded from below is necessarily described by an operator that is merely symmetric rather than self-adjoint on the physical domain. Nevertheless, on larger domains, the operator possesses a family of self-adjoint extensions. Here, we prove results on the relationship between the spacing of the eigenvalues of these self-adjoint extensions and the function $\Delta S_t$. We also discuss potential applications in quantum and classical information theory.