Uncertainty Relations for Joint Localizability and Joint Measurability in Finite-Dimensional Systems

TL;DR

This paper explores the relationship between joint localizability and joint measurability in finite-dimensional quantum systems, establishing bounds and uncertainty relations for projection-valued measures.

Contribution

It demonstrates a simple relationship between localizability and measurability bounds, extending uncertainty relations to finite-dimensional systems and general projection-valued measures.

Findings

Bound on joint localizability implies a bound on joint measurability.

Uncertainty relations for pairs of projection-valued measures are derived.

The results generalize previous work to finite-dimensional quantum systems.

Abstract

Two quantities quantifying uncertainty relations are examined. In J.Math.Phys. 48, 082103 (2007), Busch and Pearson investigated the limitation on joint localizability and joint measurement of position and momentum by introducing overall width and error bar width. In this paper, we show a simple relationship between these quantities for finite-dimensional systems. Our result indicates that if there is a bound on joint localizability, it is possible to obtain a similar bound on joint measurability. For finite-dimensional systems, uncertainty relations for a pair of general projection-valued measures are obtained as by-products.