Probability density functions of quantum mechanical observable uncertainties
标题：Probability density functions of quantum mechanical observable uncertainties
简介：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 support
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 stateindependent
uncertainty relations. Our results may be generalized to multiple observable cases in
higher dimensional spaces.