AUTHOR=Zhao Xinzhi , Zhang Chengjie 

TITLE=Uncertainty Relations of Non-Hermitian Operators: Theory and Experimental Scheme

JOURNAL=Frontiers in Physics

VOLUME=Volume 10 - 2022

YEAR=2022

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2022.862868

DOI=10.3389/fphy.2022.862868

ISSN=2296-424X

ABSTRACT=The theoretical framework for the uncertainty relation of Hermitian operators is very perfect and has been applied in many fields. At the same time, non-Hermitian operators are also widely used in some other fields. However, the uncertainty relation of non-Hermitian operators remains to be explored. K.-W. Bong and his co-workers proposed the theory of unitary uncertainty relation and verified it in experiment [Phys. Rev. Lett. 120, 230402 (2018)]. In this work, we generalize this unitary uncertainty relation theory and propose uncertainty relations of non-Hermitian operators. Because of difficulties in the direct measurement of non-Hermitian operators in the uncertainty relations, we simplify the uncertainty relation of two non-Hermitian operators with pure states, and propose a realizable experimental measurement scheme by using Mach-Zehnder interferometer. When the two non-Hermitian operators are unitary, our result can reduce to Bong et al.’s result. Furthermore, for two non-Hermitian operators but not unitary, we obtain a generalized and
analogous result of theirs.