Lower bound of quantum uncertainty from extractable classical information

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• 作者：T. Pramanik ; S. Mal ; A. S. Majumdar
• 关键词：Uncertainty relation ; Classical information ; Quantum discord
• 刊名：Quantum Information Processing
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：15
• 期：2
• 页码：981-999
• 全文大小：693 KB
• 参考文献：1.Heisenberg, W.: Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Z. Phys. 43, 172–198 (1927)CrossRef ADS MATH
  2.Kennard, E.H.: Zur Quantenmechanik einfacher Bewegungstypen. Z. Phys. 44, 326–352 (1927)CrossRef ADS MATH
  3.Robertson, H.P.: The uncertainty principle. Phys. Rev. 34, 163–164 (1929)CrossRef ADS
  4.Hirschmann, I.I.: A note on entropy. Am. J. Math. 79, 152–156 (1957)CrossRef
  5.Beckner, W.: Inequalities in Fourier analysis. Ann. Math. 102, 159–182 (1975)CrossRef MathSciNet MATH
  6.Bialynicki-Birula, I., Mycielski, J.: Uncertainty relations for information entropy in wave mechanics. Commun. Math. Phys. 44, 129–132 (1975)CrossRef ADS MathSciNet
  7.Deutsch, D.: Uncertainty in quantum measurements. Phys. Rev. Lett. 50, 631–633 (1983)CrossRef ADS MathSciNet
  8.Kraus, K.: Complementary observables and uncertainty relations. Phys. Rev. D 35, 3070–3075 (1987)CrossRef ADS MathSciNet
  9.Maassen, H., Uffink, J.B.M.: Generalized entropic uncertainty relations. Phys. Rev. Lett. 60, 1103–1106 (1988)CrossRef ADS MathSciNet
  10.Wehner, S., Winter, A.: Entropic uncertainty relations—a survey. New J. Phys. 12, 025009 (2010)CrossRef ADS MathSciNet
  11.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)CrossRef
  12.Coles, P.J., Piani, M.: Improved entropic uncertainty relations and information exclusion relations. Phys. Rev. A 89, 022112 (2014)CrossRef ADS
  13.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)CrossRef
  14.Li, C., Xu, J., Xu, X., Li, K., Guo, G.-C.: Experimental investigation of the entanglement-assisted entropic uncertainty principle. Nat. Phys. 7, 752–756 (2011)CrossRef
  15.Pramanik, T., Chowdhury, P., Majumdar, A.S.: Fine-grained lower limit of entropic uncertainty in the presence of quantum memory. Phys. Rev. Lett. 110, 020402 (2013)CrossRef ADS
  16.Oppenheim, J., Wehner, S.: The uncertainty principle determines the nonlocality of quantum mechanics. Science 330, 1072–1074 (2010)CrossRef ADS MathSciNet MATH
  17.Pati, A.K., Wilde, M.M., Usha Devi, A.R., 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)CrossRef ADS
  18.Henderson, L., Vedral, V.: Classical, quantum and total correlations. J. Phys. A Math. Gen. 34, 6899–6905 (2001)CrossRef ADS MathSciNet MATH
  19.Ollivier, H., Zurek, W.H.: Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett. 88, 017901 (2001)CrossRef ADS
  20.Devetak, I., Winter, A.: Distillation of secret key and entanglement from quantum states. Proc. R. Soc. A 461, 207–235 (2005)CrossRef ADS MathSciNet MATH
  21.Pramanik, T., Majumdar, A.S.: Fine-grained uncertainty relation and nonlocality of tripartite systems. Phys. Rev. A 85, 024103 (2012)CrossRef ADS
  22.Pramanik, T., Kaplan, M., Majumdar, A.S.: Fine-grained Einstein–Podolsky–Rosen-steering inequalities. Phys. Rev. A 90, 050305(R) (2014)CrossRef ADS
  23.Coles, Patrick J., Colbeck, Roger, Yu, Li, Zwolak, Michael: Uncertainty relations from simple entropic properties. Phys. Rev. Lett. 108, 210405 (2012)CrossRef ADS
  24.Dakić, B., Vedral, V., Brukner, C̆.: Necessary and sufficient condition for nonzero quantum discord. Phys. Rev. Lett. 105, 190502 (2010)CrossRef ADS
  25.Werner, R.F.: Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40, 4277–4281 (1989)CrossRef ADS
  26.Luo, S.: Quantum discord for two-qubit systems. Phys. Rev. A 77, 042303 (2008)CrossRef ADS
  27.Puchala, Z., Rudnicki, L., Zyczkowski, K.: Majorization entropic uncertainty relations. J. Phys. A Math. Theor. 46, 272002 (2013)CrossRef ADS MathSciNet
  28.Friedland, S., Gheorghiu, V., Gour, G.: Universal uncertainty relations. Phys. Rev. Lett. 111, 230401 (2013)CrossRef ADS
  29.Rudnicki, L., Puchala, Z., Zyczkowski, K.: Strong majorization entropic uncertainty relations. Phys. Rev. A 89, 052115 (2014)CrossRef ADS
  
• 作者单位：T. Pramanik (1)
  S. Mal (2)
  A. S. Majumdar (2)
  
  1. LTCI, Telecom ParisTech, 23 Avenue dItalie, 75214, Paris Cedex 13, France
  2. S. N. Bose National Centre for Basic Sciences, Salt Lake, Kolkata, 700 098, India
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Physics
  Mathematics
  Engineering, general
  Computer Science, general
  Characterization and Evaluation Materials
  
• 出版者：Springer Netherlands
• ISSN：1573-1332

文摘

The sum of entropic uncertainties for the measurement of two non-commuting observables is not always reduced by the amount of entanglement (quantum memory) between two parties, and in certain cases may be impacted by quantum correlations beyond entanglement (discord). An optimal lower bound of entropic uncertainty in the presence of any correlations may be determined by fine-graining. Here we express the uncertainty relation in a new form where the maximum possible reduction in uncertainty is shown to be given by the extractable classical information. We show that the lower bound of uncertainty matches with that using fine-graining for several examples of two-qubit pure and mixed entangled states, and also separable states with non-vanishing discord. Using our uncertainty relation, we further show that even in the absence of any quantum correlations between the two parties, the sum of uncertainties may be reduced with the help of classical correlations. Keywords Uncertainty relation Classical information Quantum discord