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Phase estimation Under energy conservation

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Abstract

We discuss the accuracy of phase estimation for single-qubit and multi-qubit systems under energy conservation constraints. For the case where both the target and the auxiliary system are qubits, we analytically maximize the FI over all possible energy-conserving joint measurements and auxiliary states, and derive the form of optimal measurement and state. Then, we generalize the results to the case where the auxiliary system is d-dimensional and discover that the achievable FI can approximate quantum Fisher information for large d. Finally, we discuss the difference between global and local measurements when considering multi-qubit target systems. Our results can contribute to the study of restrictions of conservation laws on quantum measurements.

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References

  1. Helstrom, C.W.: Quantum detection and estimation theory. J. Stat. Phys. 1(2), 231–252 (1969). https://doi.org/10.1007/BF01007479

    Article  ADS  MathSciNet  Google Scholar 

  2. PARIS, M.G.A,: Quantum estimation for quantum technology. J. Quantum Inf, Int (2009). https://doi.org/10.1142/S0219749909004839

  3. Giovannetti, V., Lloyd, S., Maccone, L.: Quantum metrology. Phys. Rev. Lett. 96(1), 010401 (2006). https://doi.org/10.1103/PhysRevLett.96.010401

    Article  ADS  MathSciNet  Google Scholar 

  4. Braunstein, S.L.: Quantum limits on precision measurements of phase. Phys. Rev. Lett. 69(25), 3598 (1992). https://doi.org/10.1103/PhysRevLett.69.3598

    Article  ADS  Google Scholar 

  5. Liu, J., Yuan, H., Lu, X.-M., Wang, X.: Quantum fisher information matrix and multiparameter estimation. J. Phys. A: Math. Theoret. 53(2), 023001 (2020). https://doi.org/10.1088/1751-8121/ab5d4d

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Wasilewski, W., Jensen, K., Krauter, H., Renema, J.J., Balabas, M., Polzik, E.S.: Quantum noise limited and entanglement-assisted magnetometry. Phys. Rev. Lett. 104(13), 133601 (2010). https://doi.org/10.1103/PhysRevLett.104.133601

    Article  ADS  Google Scholar 

  7. Dowling, J.P.: Correlated input-port, matter-wave interferometer: Quantum-noise limits to the atom-laser gyroscope. Phys. Rev. A 57(6), 4736 (1998). https://doi.org/10.1103/PhysRevA.57.4736

    Article  ADS  Google Scholar 

  8. Abbott, B.P., Abbott, R., Abbott, T., Abernathy, M., Acernese, F., Ackley, K., Adams, C., Adams, T., Addesso, P., Adhikari, R.: Gw150914: Implications for the stochastic gravitational-wave background from binary black holes. Phys. Rev. Lett. 116(13), 131102 (2016). https://doi.org/10.1103/PhysRevLett.116.131102

    Article  ADS  MathSciNet  Google Scholar 

  9. Lasky, P.D., Thrane, E., Levin, Y., Blackman, J., Chen, Y.: Detecting gravitational-wave memory with ligo: implications of gw150914. Phys. Rev. Lett. 117(6), 061102 (2016). https://doi.org/10.1103/PhysRevLett.117.061102

    Article  ADS  Google Scholar 

  10. Leibfried, D., Barrett, M.D., Schaetz, T., Britton, J., Chiaverini, J., Itano, W.M., Jost, J.D., Langer, C., Wineland, D.J.: Toward heisenberg-limited spectroscopy with multiparticle entangled states. Science 304(5676), 1476–1478 (2004). https://doi.org/10.1126/science.1097576

    Article  ADS  Google Scholar 

  11. Giovannetti, V., Lloyd, S., Maccone, L.: Quantum-enhanced measurements: beating the standard quantum limit. Science 306(5700), 1330–1336 (2004). https://doi.org/10.1126/science.1104149

    Article  ADS  Google Scholar 

  12. Caves, C.M., Thorne, K.S., Drever, R.W., Sandberg, V.D., Zimmermann, M.: On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. i. issues of principle. Rev. Mod. Phys 52(2), 341 (1980). https://doi.org/10.1103/RevModPhys.52.341

  13. Liu, J., Yuan, H.: Quantum parameter estimation with optimal control. Phys. Rev. A 96, 012117 (2017). https://doi.org/10.1103/PhysRevA.96.012117

    Article  ADS  Google Scholar 

  14. Liu, J., Yuan, H.: Control-enhanced multiparameter quantum estimation. Phys. Rev. A 96, 042114 (2017). https://doi.org/10.1103/PhysRevA.96.042114

    Article  ADS  Google Scholar 

  15. Nielsen, M.A., Chuang, I.: Quantum computation and quantum information. AM J. Phys. 70(5), 558–559 (2002). https://doi.org/10.1119/1.1463744

    Article  ADS  Google Scholar 

  16. Navascués, M., Popescu, S.: How energy conservation limits our measurements. Phys. Rev. Lett. 112(14), 140502 (2014). https://doi.org/10.1103/PhysRevLett.112.140502

    Article  ADS  Google Scholar 

  17. Wigner, E.P.: Die messung quantenmechanischer operatoren. Philosophical Reflections and Syntheses, 147–154 (1995)

  18. Araki, H., Yanase, M.M.: Measurement of quantum mechanical operators. Phys. Rev. 120, 622–626 (1960). https://doi.org/10.1103/PhysRev.120.622

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Yanase, M.M.: Optimal measuring apparatus. Phys. Rev. 123, 666–668 (1961). https://doi.org/10.1103/PhysRev.123.666

    Article  ADS  Google Scholar 

  20. Ozawa, M.: Conservation laws, uncertainty relations, and quantum limits of measurements. Phys. Rev. Lett. 88, 050402 (2002). https://doi.org/10.1103/PhysRevLett.88.050402

    Article  ADS  MathSciNet  Google Scholar 

  21. Gea-Banacloche, J., Ozawa, M.: Constraints for quantum logic arising from conservation laws and field fluctuations. J. opt. B. Quantum. Semiclass Opt. 7(10), 326 (2005). https://doi.org/10.1088/1464-4266/7/10/017

    Article  ADS  Google Scholar 

  22. Ahmadi, M., Jennings, D., Rudolph, T.: The wigner-araki-yanase theorem and the quantum resource theory of asymmetry. New J. Phys 15(1), 013057 (2013). https://doi.org/10.1088/1367-2630/15/1/013057

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Greenberger, D.M., Horne, M.A., Shimony, A., Zeilinger, A.: Bell’s theorem without inequalities. AM J. Phys. 58(12), 1131–1143 (1990). https://doi.org/10.1119/1.16243

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Braunstein, S.L., Caves, C.M.: Statistical distance and the geometry of quantum states. Phys. Rev. Lett. 72(22), 3439 (1994). https://doi.org/10.1103/PhysRevLett.72.3439

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Zhang, M., Yu, H.-M., Yuan, H., Wang, X., Demkowicz-Dobrza ński, R., Liu, J.: Quanestimation: An open-source toolkit for quantum parameter estimation. Phys. Rev. Res. 4, 043057 (2022). https://doi.org/10.1103/PhysRevResearch.4.043057

  26. Tóth, G., Apellaniz, I.: Quantum metrology from a quantum information science perspective. J. Phys. A Math. Theor 47(42), 424006 (2014). https://doi.org/10.1088/1751-8113/47/42/424006

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Liu, J., Jing, X.-X., Zhong, W., Wang, X.-G.: Quantum fisher information for density matrices with arbitrary ranks. Commun. Theoretical Phys. 61(1), 45 (2014). https://doi.org/10.1088/0253-6102/61/1/08

    Article  ADS  MATH  Google Scholar 

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Acknowledgements

This work was supported by National Natural Science Foundation of China under Grant No. 11774205 and the Young Scholars Program of Shandong University.

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  1. School of Information Science and Engineering, Shandong University, Qingdao, 266237, Shandong, China

    Sen Han & Xueyuan Hu

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  1. Sen Han
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  2. Xueyuan Hu
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XH contributed to the study conception. All authors contributed to the main conclusions of the article. The first draft of the manuscript was written by SH and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Xueyuan Hu.

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Han, S., Hu, X. Phase estimation Under energy conservation. Quantum Inf Process 22, 178 (2023). https://doi.org/10.1007/s11128-023-03948-9

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