Tighter Uncertainty and Reverse Uncertainty Relations

TL;DR

This paper introduces new, tighter state-dependent uncertainty relations for quantum observables, establishing both lower and upper bounds on variances, thus deepening understanding of quantum measurement limitations.

Contribution

It presents novel state-dependent uncertainty and reverse uncertainty relations that are tighter than existing ones, using inequalities like reverse Cauchy-Schwarz and Dunkl-Williams.

Findings

Derived tighter uncertainty relations than Robertson-Schrödinger.

Established state-dependent upper bounds on variances.

Showed fundamental limits on sharpness of incompatible observables.

Abstract

We prove a few novel state-dependent uncertainty relations for product as well the sum of variances of two incompatible observables. These uncertainty relations are shown to be tighter than the Roberson-Schr\"odinger uncertainty relation and other ones existing in the current literature. Also, we derive state dependent upper bound to the sum and the product of variances using the reverse Cauchy-Schwarz inequality and the Dunkl-Williams inequality. Our results suggest that not only we cannot prepare quantum states for which two incompatible observables can have sharp values, but also we have both, lower and upper limits on the variances of quantum mechanical observables at a fundamental level.