Entropy as a bound for expectation values and variances of a general quantum mechanical observable

TL;DR

This paper establishes bounds on the expectation values and variances of quantum observables based on von Neumann entropy, linking quantum information theory with measurement uncertainties.

Contribution

It introduces a novel relationship between von Neumann entropy and observable expectation values and uncertainties, including a reverse uncertainty relation for multiple observables.

Findings

Expectation values and variances are bounded by entropy.

Von Neumann entropy encompasses information about measurement uncertainties.

A reverse uncertainty relation for multiple observables is derived.

Abstract

Quantum information-theoretic approach has been identified as a way to understand the foundations of quantum mechanics as early as 1950 due to Shannon. However there hasn't been enough advancement or rigorous development of the subject. In the following paper we try to find relationship between a general quantum mechanical observable and it von Neumann entropy. We find that the expectation values and the uncertainties of the observables have bounds which depend on the entropy. The results also show that von Neumann entropy is not just the uncertainty of the state but also encompasses the information about expectation values and uncertainties of any observable which depends on the observers choice for a particular measurement. Also a reverse uncertainty relation is derived for n quantum mechanical observables.