Measurement uncertainty relation for three observables

Sixia Yu, Ya-Li Mao, Chang Niu, Hu Chen, Zheng-Da Li, Jingyun Fan

Published 2022-11-17Version 1

Measurement uncertainty relations (MUR) are essential for quantum foundations and quantum information science. MURs for two observables have been studied extensively and tested in experiments there is only a few known MUR for three or more observables. In this work we shall establish rigorously a MUR for three unbiased qubit observables, which was previously shown to hold true under some presumptions. The uncertainty, which is quantified by the total statistic distance between the target observables and the jointly implemented observables, is lower bounded by an incompatibility measure that reflects the joint measurement conditions. We derive a necessary and sufficient condition for the MUR to be saturated. The exact values of incompatibility measure are analytically calculated for some symmetric triplets. To facilitate experimental tests of MURs we propose a straightforward implementation of the optimal measurements. We anticipate that this work may enrich the understanding of quantum mechanics and atract more interest in quantum fundamental physics. This work presents a complete theory relevant to a parallel work [Y.L. Mao, et al.}, Testing Heisenberg's measurement uncertainty relation of three observables, arXiv 2022] on experimental tests.