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Are observables necessarily Hermitian?


Observables are believed that they must be Hermitian in quantum theory. Based on the obviously physical fact that only eigenstates of observable and its corresponding probabilities, i.e., spectrum distribution of observable are actually observed, we argue that observables need not necessarily be Hermitian. More generally, observables should be reformulated as normal operators including Hermitian operators as a subclass. This reformulation is consistent with the quantum theory currently used and does not change any physical results. The Clauser–Horne–Shimony–Holt (CHSH) inequality is taken as an example to show that our opinion does not conflict with conventional quantum theory and gives the same physical results. Reformulation of observables as normal operators not only coincides with the physical facts, but also will deepen our understanding of measurement in quantum theory.

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This work is supported by National Natural Science Foundation of China (No. 61275122, No. 11674306, No. 61590932), National Fundamental Research Program of China and Strategic Priority Research Program (B) of CAS (No. XDB01030200).

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Correspondence to Meng-Jun Hu.

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Hu, MJ., Hu, XM. & Zhang, YS. Are observables necessarily Hermitian?. Quantum Stud.: Math. Found. 4, 243–249 (2017).

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  • Quantum measurement
  • Non-Hermitian observable
  • Normal operator