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Are observables necessarily Hermitian?

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Abstract

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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References

  1. Dirac, P.A.M.: The Principles of Quantum Mechanics. Clarendon, Oxford (1958)

    MATH  Google Scholar 

  2. Bender, C.M., Boettcher, S.: Real spectra in non-hermitian hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bender, C.M., Brody, D.C., Jones, H.F.: Complex extension of quantum mechanics. Phys. Rev. Lett. 89, 270401 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bender, C.M., Brody, D.C., Jones, H.F.: Erratum: Complex extension of quantum mechanics. Phys. Rev. Lett. 92, 119902 (2004)

    Article  Google Scholar 

  5. Bender, C.M.: Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947–1018 (2007)

    Article  MathSciNet  Google Scholar 

  6. Pati, A.K., Singh, U., Sinha, U.: Measuring non-Hermitian operators via weak values. Phys. Rev. A 92, 052120 (2015)

    Article  Google Scholar 

  7. Mitchison, G., Jozsa, R., Popescu, S.: Sequential weak measurement. Phys. Rev. A 76, 062105 (2007)

    Article  Google Scholar 

  8. Brodutch, A., Cohen, E.: Nonlocal measurements via quantum erasure. arXiv:1409.1575v2 [quant-ph] (2015)

  9. Aharonov, Y., Albert, D.Z., Vaidman, L.: How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett. 60, 1351 (1988)

    Article  Google Scholar 

  10. Aharonov, Y., Vaidman, L.: Properties of a quantum system during the time interval between two measurements. Phys. Rev. A 41, 11 (1990)

    Article  MathSciNet  Google Scholar 

  11. Aharonov, Y., Botero, A.: Quantum average of weak values. Phys. Rev. A 72, 052111 (2005)

    Article  Google Scholar 

  12. Zurek, W.H.: Relative state and the environment: einstein, envariance, quantum darwinism, and the existential interpretation. arXiv:0707.2832v1 [quant-ph] (2007)

  13. Sakurai, J.J., Tuan, S.F.: Modern Quantum Mechanics, Revised edn. Addison-Wesley, Boston (1994)

    Google Scholar 

  14. Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777 (1935)

    Article  MATH  Google Scholar 

  15. Feynman, R.P., Leighton, R.B., Stands, M.: The Feynman Lectures on Physics, vol. 3. Addison-Wesley, Boston (1964)

    Google Scholar 

  16. Busch, P., Grabowski, M., Lahti, P.J.: Operational Quantum Physics. Springer, Berlin (1995)

    MATH  Google Scholar 

  17. Busch, P., Lahti, P., Werner, R.F.: Quantum root-mean-square error and measurement uncertainty relations. Rev. Mod. Phys. 86, 1261 (2014)

    Article  Google Scholar 

  18. Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)

    MATH  Google Scholar 

  19. Lynch, R.: The quantum phase problem: a critical review. Phys. Rep. 256, 367–436 (1995)

    Article  MathSciNet  Google Scholar 

  20. Torgerson, J.R., Mandel, L.: Is there a unique operator for the phase difference of two quantum fields? Phys. Rev. Lett. 76, 3939 (1996)

    Article  Google Scholar 

  21. Yu, S.X.: Quantized phase difference. Phys. Rev. Lett. 79, 780 (1997)

    Article  Google Scholar 

  22. Bell, J.S.: On the Einstein Podolsky Rosen paradox. Physics (Long Island City, NY) 1, 195 (1964)

  23. Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A.: Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23, 880 (1969)

    Article  Google Scholar 

  24. Cirel’son, B.S.: Quantum generalization of Bell’s inequality. Lett. Math. Phys. 4, 93 (1980)

    Article  MathSciNet  Google Scholar 

  25. Cabello, A.: Violating Bell’s inequality Beyond Cirel’son’s bound. Phys. Rev. Lett. 88, 060403 (2002)

    Article  MathSciNet  Google Scholar 

  26. Werner, R.F., Wolf, M.M.: All-multipartite Bell-correlation inequalities for two dichotomic observables per site. Phys. Rev. A 64, 032112 (2001)

    Article  Google Scholar 

  27. Żukowski, M., Brukner, Č.: Bell’s theorem for general N-qubit states. Phys. Rev. Lett. 88, 210401 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Arnault, F.: A complete set of multidimensional Bell inequalities. J. Phys. A Math. Theor. 45, 255304 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

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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Authors and Affiliations

  1. Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, China

    Meng-Jun Hu, Xiao-Min Hu & Yong-Sheng Zhang

  2. Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, 230026, China

    Meng-Jun Hu, Xiao-Min Hu & Yong-Sheng Zhang

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  1. Meng-Jun Hu
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  2. Xiao-Min Hu
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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). https://doi.org/10.1007/s40509-016-0098-2

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