Geometric and algebraic origins of additive uncertainty relations

TL;DR

This paper introduces constructive, state-independent uncertainty relations for the sum of variances of two observables, utilizing geometric and algebraic methods to derive tight bounds applicable to quantum systems.

Contribution

It presents a novel approach based on joint numerical range and uncertainty range to derive semianalytical, tight bounds for the sum of variances, improving previous numerical methods.

Findings

Derived tight bounds for the sum of variances of angular momentum components.

Applicable to detecting quantum entanglement and other problems.

Boundaries are semianalytical and can be made arbitrarily tight.

Abstract

Constructive techniques to establish state-independent uncertainty relations for the sum of variances of arbitrary two observables are presented. We investigate the range of simultaneously attainable pairs of variances, which can be applied to a wide variety of problems including finding exact bound for the sum of variances of two components of angular momentum operator for any total angular momentum quantum number $j$ and detection of quantum entanglement. Resulting uncertainty relations are state-independent, semianalytical, bounded-error and can be made arbitrarily tight. The advocated approach, based on the notion of joint numerical range of a number of observables and uncertainty range, allows us to improve earlier numerical works and to derive semianalytical tight bounds for the uncertainty relation for the sum of variances expressed as roots of a polynomial of a single real variable.