Entropic Uncertainty Relations in Quantum Physics

1. Adesso G, Serafini A, Illuminati F (2004) Extremal entanglement and mixedness in continuous variable systems. Phys Rev A 70:022318

   Article  Google Scholar 

2. Bengtsson I, Życzkowski K (2006) Geometry of quantum states. Cambridge University Press, Cambridge

   Book  Google Scholar 

3. Bovino FA, Castagnoli G, Ekert A, Horodecki P, Alves CM, Sergienko AV (2005) Direct measurement of nonlinear properties of bipartite quantum states. Phys Rev Lett 95:240407

   Article  Google Scholar 

4. Gühne O, Lewenstein M (2004) Entropic uncertainty relations and entanglement. Phys Rev A 70:022316

   Article  Google Scholar 

5. Terhal BM (2002) Detecting quantum entanglement. J Theor Comput Sci 287:313

   Article  Google Scholar 

6. Giovannetti V, Lloyd S (2004) Additivity properties of a Gaussian channel. Phys Rev A 69:062307

   Article  Google Scholar 

7. Renner R, Gisin N, Kraus B (2005) Information-theoretic security proof for quantum-key-distribution protocols. Phys Rev A 72:012332

   Article  Google Scholar 

8. Lévay P, Nagy S, Pipek J (2005) Elementary formula for entanglement entropies of fermionic systems. Phys Rev A 72:022302

   Article  Google Scholar 

9. Beck C, Graudenz D (1992) Symbolic dynamics of successive quantum-mechanical measurements. Phys Rev A 46:6265

   Article  Google Scholar 

10. Kohler S, Hänggi P (2002) In: Leuchs G, Beth T (eds) Quantum information processing. Wiley-VCH, Berlin. arXiv:quant-ph/0206189

    Google Scholar 

11. Białas A, Czyż W (2000) Event by event analysis and entropy of multiparticle systems. Phys Rev D 61:074021

    Google Scholar 

12. Białas A, Czyż W, Zalewski K (2005) Moments of the particle phase-space density at freeze-out and coincidence probabilities. Acta Phys Pol B 36:3109

    Google Scholar 

13. Białas A, Czyż W, Zalewski K (2006) Moments of the Wigner function and Rényi entropies at freeze-out. Phys Rev C 73:034912

    Google Scholar 

14. Majka A, Wiślicki W (2003) Uniformity of the phase space and fluctuations in thermal equilibrium. Physica A 322C:313

    Article  Google Scholar 

15. Cybulski O, Matysiak D, Babin V, Hołyst R (2004) Pattern formation in nonextensive thermodynamics: selection criteria based on the Rényi entropy production. Phys Rev E 69:016110

    Google Scholar 

16. Cybulski O, Babin V, Hołyst R (2005) Minimization of the Rényi entropy production in the stationary states of the Brownian process with matched death and birth rates. J Chem Phys 122:174105

    Google Scholar 

17. Arbó DG, Reinhold CO, Burgdörfer J, Pattanayak AK, Stokely CL, Zhao W, Lancaster JC, Dunning FB (2003) Pulse-induced focusing of Rydberg wave packets. Phys Rev A 67:063401

    Article  Google Scholar 

18. Gnutzmann S, Życzkowski K (2001) Rényi-Wehrl entropies as measures of localization in phase space. J Phys A, Math Gen 34:10123

    Article  Google Scholar 

19. Verstraete F, Cirac JI (2006) Matrix product states represent ground states faithfully. Phys Rev B 73:094423

    Article  Google Scholar 

20. Dehesa JS, Martínez-Finkelshtein A, Sorokin VN (2002) Quantum-information entropies for highly excited states of single-particle systems with power-type potentials. Phys Rev A 66:062109

    Article  Google Scholar 

21. De Nicola S, Fedele R, Man‘ko MA, Man‘ko VI (2009) Entropic uncertainty relations for electromagnetic beams. Phys Scr 135:014053

    Article  Google Scholar 

22. Salcedo LL (2009) Phase space localization of antisymmetric functions. J Math Phys 50:012106

    Article  Google Scholar 

23. Varga I, Pipek J (2003) Rényi entropies characterizing the shape and the extension of the phase space representation of quantum wave functions in disordered systems. Phys Rev E 68:026202

    Article  Google Scholar 

24. Brukner C, Zeilinger A (2001) Conceptual inadequacy of the Shannon information in quantum measurements. Phys Rev A 63:022113

    Article  Google Scholar 

25. Timpson CG (2003) On a supposed Conceptual inadequacy of the Shannon information in quantum mechanics. Stud Hist Philos Mod Phys 33:441. arXiv:quant-ph/0112178

    Article  Google Scholar 

26. Shannon CE, Weaver W (1949) The mathematical theory of communication. University of Illinois Press, Urbana

    Google Scholar 

27. Heisenberg W (1927) Über den Anschaulichen Inhalt der quanten-theoretischen Kinematik und Mechanik. Z Phys 43:172

    Google Scholar 

28. Bialynicki-Birula I (2007) Rényi entropy and the uncertainty relations. In: Adenier G, Fuchs CA, Khrennikov AYu (eds) Foundations of probability and physics. AIP Conf Proc, vol 889. AIP, New York

    Google Scholar 

29. Peres A (1995) Quantum theory: concepts and methods. Kluwer, Dordrecht

    Google Scholar 

30. Partovi MH (1983) Entropic formulation of uncertainty for quantum measurements. Phys Rev Lett 50:1883

    Article  Google Scholar 

31. Heisenberg W (1930) The physical properties of the quantum theory. Dover, New York

    Google Scholar 

32. Bialynicki-Birula I (1984) Entropic uncertainty relations. Phys Lett 103 A:253

    Google Scholar 

33. Hardy G, Littlewood JL, Pólya G (1934) Inequalities. Cambridge University Press, Cambridge

    Google Scholar 

34. Jensen JLWV (1906) Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta Math 30:175

    Article  Google Scholar 

35. Hirschman II (1957) A note on entropy. Am J Math 79:152

    Article  Google Scholar 

36. Everett H III (1957) “Relative State” formulation of quantum mechanics. Rev Mod Phys 29:454

    Article  Google Scholar 

37. Everett H III (1973) The theory of the universal wave function. In: DeWitt BS, Graham N (eds) The many-world interpretation of quantum mechanics. Princeton University Press, Princeton. PhD thesis

    Google Scholar 

38. Bialynicki-Birula I, Mycielski J (1975) Uncertainty relations for information entropy in wave mechanics. Commun Math Phys 44:129

    Article  Google Scholar 

39. Beckner W (1975) Inequalities in Fourier analysis. Ann Math 102:159

    Article  Google Scholar 

40. Řehaček J, Bouchal Z, Čelechovský R, Hradil Z, Sánchez-Soto LL (2008) Experimental test of uncertainty relations for quantum mechanics on a circle. Phys Rev A 77:032110

    Article  Google Scholar 

41. Rényi A (1960) Some fundamental questions of information theory. MTA III Oszt Közl 251

    Google Scholar 

42. Rényi A (1960) On measures of information and entropy. In: Proceedings of the 4th Berkeley symposium on mathematics, statistics and probability, p 547

    Google Scholar 

43. Rényi A (1970) Probability theory. North-Holland, Amsterdam

    Google Scholar 

44. Bialynicki-Birula I (2006) Formulation of the uncertainty relations in terms of the Rényi entropies. Phys Rev A 74:052101

    Article  Google Scholar 

45. Babenko KI (1961) An inequality in the theory of Fourier integrals. Izv Akad Nauk SSSR, Ser Mat 25:531 (in Russian)

    Google Scholar 

46. Wilk G, Włodarczyk Z (2009) Uncertainty relations in terms of the Tsallis entropy. Phys Rev A 79:062108

    Article  Google Scholar 

47. Bialynicki-Birula I, Rudnicki Ł(2010) Comment on “Uncertainty relations in terms of the Tsallis entropy”. Phys Rev A 81:026101

    Article  Google Scholar 

48. Deutsch D (1983) Uncertainty in quantum measurements. Phys Rev Lett 50:631

    Article  Google Scholar 

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

    Google Scholar 

50. Maassen H, Uffink JBM (1988) Generalized entropic uncertainty relations. Phys Rev Lett 60:1103

    Article  Google Scholar 

51. Reed M, Simon B (1975) Methods of modern mathematical physics, vol II. Academic Press, New York

    Google Scholar 

52. Riesz M (1927) Sur les maxima des formes bilinéaires et sur les fonctionnelles linéaires. Acta Math 49:465

    Article  Google Scholar 

53. Schwinger J (1960) Unitary operator bases. Proc Natl Acad Sci USA 46:570

    Article  CAS  Google Scholar 

54. Bengtsson I (2007) Three ways to look at mutually unbiased bases. In: Adenier G, Fuchs CA, Khrennikov AYu (eds) Foundations of probability and physics. AIP Conf Proc, vol 889. AIP, New York. arXiv:quant-ph/0610216

    Google Scholar 

55. Zozor S, Portesi M, Vignat C (2008) Some extensions of the uncertainty principle. Physica A 387:4800–4808

    Article  CAS  Google Scholar 

56. Bialynicki-Birula I, Bialynicka-Birula Z (1976) Quantum electrodynamics of intense photon beams. New approximation method. Phys Rev A 14:1101

    Article  CAS  Google Scholar 

57. Bialynicki-Birula I, Madajczyk J (1985) Entropic uncertainty relations for angular distributions. Phys Lett 108 A:384

    Google Scholar 

58. Wehner S, Winter A (2009) Entropic uncertainty relations—a survey. arXiv:0907.3704v1 [quant-ph]

59. Ivonovic ID (1981) Geometrical description of quantal state determination. J Phys A 14:3241–3245

    Article  Google Scholar 

60. Azarchs A (2004) Entropic uncertainty relations for incomplete sets of mutually unbiased observables. arXiv:quant-ph/0412083v1

61. Sánchez J (1993) Entropic uncertainty and certainty relations for complementary observables. Phys Lett A 173:233–239

    Article  Google Scholar 

62. Wu S, Yu S, Mølmer K (2009) Entropic uncertainty relation for mutually unbiased bases. Phys Rev A 79:022104

    Article  Google Scholar 

63. Gross L (1975) Logarithmic Sobolev inequalities. Am J Math 97:1061–1083

    Article  Google Scholar 

64. Chafai D (2002) Gaussian maximum of entropy and reversed log-Sobolev inequality Séminaire de probabilitiés. Strasbourg 36:194–200

    Google Scholar 

65. Dodonov VV, Man‘ko VI (1989) Generalized uncertainty relations in quantum mechanics. In: Markov MA (ed) Invariants and evolution of nonstationary quantum systems. Proceedings of the Lebedev Physics Institute, vol 183. Nova Science, Commack

    Google Scholar 

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