Uncertainty Relations of Non-Hermitian Operators: Theory and Experimental Scheme - Zhao, Xinzhi; Zhang, Chengjie (Frontiers in Physics, 2022)
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.