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Entropic uncertainty relations in the Heisenberg XXZ model and its controlling via filtering operations

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Abstract

The uncertainty principle is recognized as an elementary ingredient of quantum theory and sets up a significant bound to predict outcome of measurement for a couple of incompatible observables. In this work, we develop dynamical features of quantum memory-assisted entropic uncertainty relations (QMA-EUR) in a two-qubit Heisenberg XXZ spin chain with an inhomogeneous magnetic field. We specifically derive the dynamical evolutions of the entropic uncertainty with respect to the measurement in the Heisenberg XXZ model when spin A is initially correlated with quantum memory B. It has been found that the larger coupling strength \( J \) of the ferromagnetism (\( J < 0 \)) and the anti-ferromagnetism (\( J > 0 \)) chains can effectively degrade the measuring uncertainty. Besides, it turns out that the higher temperature can induce the inflation of the uncertainty because the thermal entanglement becomes relatively weak in this scenario, and there exists a distinct dynamical behavior of the uncertainty when an inhomogeneous magnetic field emerges. With the growing magnetic field \( \left| B \right| \), the variation of the entropic uncertainty will be non-monotonic. Meanwhile, we compare several different optimized bounds existing with the initial bound proposed by Berta et al. and consequently conclude Adabi et al.’s result is optimal. Moreover, we also investigate the mixedness of the system of interest, dramatically associated with the uncertainty. Remarkably, we put forward a possible physical interpretation to explain the evolutionary phenomenon of the uncertainty. Finally, we take advantage of a local filtering operation to steer the magnitude of the uncertainty. Therefore, our explorations may shed light on the entropic uncertainty under the Heisenberg XXZ model and hence be of importance to quantum precision measurement over solid state-based quantum information processing.

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References

  1. Heisenberg, W.: Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Z. Phys. 43, 172 (1927)

    Article  ADS  MATH  Google Scholar 

  2. Robertson, H.P.: Violation of Heisenberg’s uncertainty principle. Phys. Rev. 34, 163 (1929)

    Article  ADS  Google Scholar 

  3. Kennard, E.H.: Zur Quantenmechanik einfacher Bewegungstypen. Z. Phys. 44, 326 (1927)

    Article  ADS  MATH  Google Scholar 

  4. Deutsch, D.: Uncertainty in quantum measurements. Phys. Rev. Lett. 50, 631–633 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  5. Maassen, H., Uffink, J.B.M.: Generalized entropic uncertainty relations. Phys. Rev. Lett. 60, 1103 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  6. Kraus, K.: Complementary observables and uncertainty relations. Phys. Rev. D 35, 3070 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  7. Maccone, L., Pati, A.K.: Stronger uncertainty relations for the sum of variances. Phys. Rev. Lett. 113, 260401 (2014)

    Article  ADS  Google Scholar 

  8. Wang, K.K., Zhan, X., Bian, Z.H., Li, J., Zhang, Y.S., Xue, P.: Experimental investigation of the stronger uncertainty relations for all incompatible observables. Phys. Rev. A 93, 052108 (2016)

    Article  ADS  Google Scholar 

  9. Renes, J.M., Boileau, J.C.: Conjectured strong complementary information tradeoff. Phys. Rev. Lett. 103, 020402 (2009)

    Article  ADS  Google Scholar 

  10. Berta, M., Christandl, M., Colbeck, R., Renes, J.M., Renner, R.: The uncertainty principle in the presence of quantum memory. Nat. Phys. 6, 659–662 (2010)

    Article  Google Scholar 

  11. Prevedel, R., Hamel, D.R., Colbeck, R., Fisher, K., Resch, K.J.: Experimental investigation of the uncertainty principle in the presence of quantum memory and its application to witnessing entanglement. Nat. Phys. 7, 757–761 (2011)

    Article  Google Scholar 

  12. Li, C.F., Xu, J.S., Xu, X.Y., Li, K., Guo, G.C.: Experimental investigation of the entanglement-assisted entropic uncertainty principle. Nat. Phys. 7, 752–756 (2011)

    Article  Google Scholar 

  13. Nielson, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  14. Pati, A.K., Wilde, M.M., Devi, A.U., Rajagopal, A.K., Sudha, : Quantum discord and classical correlation can tighten the uncertainty principle in the presence of quantum memory. Phys. Rev. A 86, 042105 (2012)

    Article  ADS  Google Scholar 

  15. Coles, P.J., Piani, M.: Improved entropic uncertainty relations and information exclusion relations. Phys. Rev. A 89, 022112 (2014)

    Article  ADS  Google Scholar 

  16. Adabi, F., Salimi, S., Haseli, S.: Tightening the entropic uncertainty bound in the presence of quantum memory. Phys. Rev. A 93, 062123 (2016)

    Article  ADS  Google Scholar 

  17. Pramanik, T., Mal, S., Majumdar, A.S.: Lower bound of quantum uncertainty from extractable classical information. Quantum Inf. Process. 15, 981–999 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Ollivier, H., Zurek, W.H.: Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett. 88, 017901 (2001)

    Article  ADS  MATH  Google Scholar 

  19. Hu, M. L., Hu, X. Y., Wang, J. C., Peng, Y., Zhang, Y. R and Fan, H.: Quantum coherence and quantum correlations. arXiv: 1703.01852 (2017)

  20. Hu, M.L., Fan, H.: Upper bound and shareability of quantum discord based on entropic uncertainty relations. Phys. Rev. A 88, 014105 (2013)

    Article  ADS  Google Scholar 

  21. Dupuis, F., Fawzi, O., Wehner, S.: Entanglement sampling and applications. IEEE Trans. Inf. Theory 61, 1093 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Koenig, R., Wehner, S., Wullschleger, J.: Unconditional security from noisy quantum storage. IEEE Trans. Inf. Theory 58, 1962–1984 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Vallone, G., Marangon, D.G., Tomasin, M., Villoresi, P.: Quantum randomness certified by the uncertainty principle. Phys. Rev. A 90, 052327 (2014)

    Article  ADS  Google Scholar 

  24. Cerf, N.J., Bourennane, M., Karlsson, A., Gisin, N.: Security of quantum key distribution using d-level systems. Phys. Rev. Lett. 88, 127902 (2002)

    Article  ADS  Google Scholar 

  25. Grosshans, F., Cerf, N.J.: Continuous-variable quantum cryptography is secure against non-Gaussian attacks. Phys. Rev. Lett. 92, 047905 (2004)

    Article  ADS  Google Scholar 

  26. Jarzyna, M., Demkowicz-Dobrzański, R.: True precision limits in quantum metrology. New J. Phys. 17, 013010 (2015)

    Article  ADS  Google Scholar 

  27. Jia, L.J., Tian, Z.H., Jing, J.L.: Entropic uncertainty relation in de Sitter space. Ann. Phys. 353, 37–47 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Zou, H.M., Fang, M.F., Yang, B.Y., Guo, Y.N., He, W., Zhang, S.Y.: The quantum entropic uncertainty relation and entanglement witness in the two-atom system coupling with the non-Markovian environments. Phys. Scr. 89, 115101 (2014)

    Article  ADS  Google Scholar 

  29. Yao, C.M., Chen, Z.H., Ma, Z.H., Severini, S., Serafini, A.: Entanglement and discord assisted entropic uncertainty relations under decoherence. Sci. China Phys. Mech. Astron. 57, 1703–1711 (2014)

    Article  ADS  Google Scholar 

  30. Zhang, Y.L., Fang, M.F., Kang, G.D., Zhou, Q.P.: Reducing quantum-memory-assisted entropic uncertainty by weak measurement and weak measurement reversal. Int. J. Quantum Inf. 13, 1550037 (2015)

    Article  MATH  Google Scholar 

  31. Xu, Z.Y., Yang, W.L., Feng, M.: Quantum-memory-assisted entropic uncertainty relation under noise. Phys. Rev. A 86, 012113 (2012)

    Article  ADS  Google Scholar 

  32. Wang, D., Ming, F., Huang, A.J., Sun, W.Y., Shi, J.D., Ye, L.: Exploration of quantum-memory-assisted entropic uncertainty relations in a noninertial frame. Laser Phys. Lett. 14, 055205 (2017)

    Article  ADS  Google Scholar 

  33. Feng, J., Zhang, Y.Z., Gould, M.D., Fan, H.: Entropic uncertainty relations under the relativistic motion. Phys. Lett. B 726, 527–532 (2013)

    Article  ADS  MATH  Google Scholar 

  34. Huang, A.J., Shi, J.D., Wang, D., Ye, L.: Steering quantum-memory-assisted entropic uncertainty under unital and nonunital noises via filtering operations. Quantum Inf. Process. 16, 46 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Zheng, X., Zhang, G.F.: The effects of mixedness and entanglement on the properties of the entropic uncertainty in Heisenberg model with Dzyaloshinski–Moriya interaction. Quantum Inf. Process. 16, 1 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Huang, A.J., Wang, D., Wang, J.M., Shi, J.D., Sun, W.Y., Ye, L.: Exploring entropic uncertainty relation in the Heisenberg XX model with inhomogeneous magnetic field. Quantum Inf. Process. 16, 204 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Asoudeh, M., Karimipour, V.: Thermal entanglement of spins in an inhomogeneous magnetic field. Phys. Rev. A 71, 022308 (2005)

    Article  ADS  Google Scholar 

  38. Liang, Q.: Quantum correlations in a two-qubit Heisenberg XX model under intrinsic decoherence. Commun. Theor. Phys. 60, 391 (2013)

    Article  Google Scholar 

  39. Zhang, G.F., Li, S.S.: Thermal entanglement in a two-qubit Heisenberg XXZ spin chain under an inhomogeneous magnetic field. Phys. Rev. A 72, 034302 (2005)

    Article  ADS  Google Scholar 

  40. Wang, D., Ming, F., Huang, A.J., Sun, W.Y., Ye, L.: Entropic uncertainty for spin-1/2 XXX chains in the presence of inhomogeneous magnetic fields and its steering via weak measurement reversals. Laser Phys. Lett. 14, 095204 (2017)

    Article  ADS  Google Scholar 

  41. Sumana, K., Ajoy, S., Amit, B., Debasis, S.: Effect of local filtering on freezing phenomena of quantum correlation. Quantum Inf. Process. 14, 2517–2533 (2015)

    Article  MATH  Google Scholar 

  42. Michael, S., Ali, A.K.: Defeating entanglement sudden death by a single local filtering. Phys. Rev. A 86, 032304 (2012)

    Article  Google Scholar 

  43. Peters, N.A., Wei, T.C., Kwiat, P.G.: Mixed-state sensitivity of several quantum-information benchmarks. Phys. Rev. A 70, 052309 (2004)

    Article  ADS  Google Scholar 

  44. Singh, U., Bera, M.N., Dhar, H.S., Pati, A.K.: Maximally coherent mixed states: complementarity between maximal coherence and mixedness. Phys. Rev. A 91, 052115 (2015)

    Article  ADS  Google Scholar 

  45. Wang, S.C., Yu, Z.W., Wang, X.B.: Protecting quantum states from decoherence of finite temperature using weak measurement. Phys. Rev. A 89, 022318 (2014)

    Article  ADS  Google Scholar 

  46. Zhang, S.Y., Fang, M.F., Yu, M.: Controlling of entropic uncertainty in qubits system under the generalized amplitude damping channel via weak measurements. Int. J. Theor. Phys. 55, 1824–1832 (2016)

    Article  MATH  Google Scholar 

  47. Zhang, S.Y., Fang, M.F., Zhang, Y.L., Guo, Y.N., Zhao, Y.J., Tang, W.W.: Reduction of entropic uncertainty in entangled qubits system by local PT-symmetric operation. Chin. Phys. B 24, 090304 (2015)

    Article  ADS  Google Scholar 

  48. Flavien, H., Marco, T.Q., Joseph, B., Nicolas, B.: Genuine hidden quantum nonlocality. Phys. Rev. Lett. 111, 160402 (2013)

    Article  Google Scholar 

  49. Sun, Q.Q., Al-Amri, M., Davidovich, L., Suhail Zubairy, M.: Reversing entanglement change by a weak measurement. Phys. Rev. A 82, 052323 (2010)

    Article  ADS  Google Scholar 

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Acknowledgements

This work was supported by the National Science Foundation of China under Grant Nos. 61601002 and 11575001, Anhui Provincial Natural Science Foundation (Grant No. 1508085QF139), and the fund from CAS Key Laboratory of Quantum Information (Grant No. KQI201701).

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Authors and Affiliations

  1. School of Physics and Material Science, Anhui University, Hefei, 230601, China

    Fei Ming, Dong Wang, Wei-Nan Shi, Ai-Jun Huang, Wen-Yang Sun & Liu Ye

  2. CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, China

    Dong Wang

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  1. Fei Ming
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Correspondence to Dong Wang.

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Ming, F., Wang, D., Shi, WN. et al. Entropic uncertainty relations in the Heisenberg XXZ model and its controlling via filtering operations. Quantum Inf Process 17, 89 (2018). https://doi.org/10.1007/s11128-018-1857-2

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