Probability density functions of quantum mechanical observable uncertainties

TL;DR

This paper derives the probability density functions of uncertainties for quantum observables in random pure states, providing new insights into uncertainty regions and state-independent relations in quantum mechanics.

Contribution

It analytically determines the PDFs of uncertainties for arbitrary qubit observables, enabling a novel approach to studying uncertainty relations.

Findings

Derived explicit PDFs for uncertainties of qubit observables.

Characterized uncertainty regions via the supports of these PDFs.

Proposed a new method to analyze state-independent uncertainty relations.

Abstract

We study the uncertainties of quantum mechanical observables, quantified by the standard deviation (square root of variance) in Haar-distributed random pure states. We derive analytically the probability density functions (PDFs) of the uncertainties of arbitrary qubit observables. Based on these PDFs, the uncertainty regions of the observables are characterized by the supports of the PDFs. The state-independent uncertainty relations are then transformed into the optimization problems over uncertainty regions, which opens a new vista for studying state independent uncertainty relations. Our results may be generalized to multiple observable case in higher dimensional spaces.