Entropic Uncertainty Relations in Quantum Physics

Uncertainty relations have become the trademark of quantum theory since they were formulated by Bohr and Heisenberg. This review covers various generalizations and extensions of the uncertainty relations in quantum theory that involve the Renyi and the Shannon entropies. The advantages of these entropic uncertainty relations are pointed out and their more direct connection to the observed phenomena is emphasized. Several remaining open problems are mentioned.

[1]  M. Lewenstein,et al.  Entropic uncertainty relations and entanglement , 2004, quant-ph/0403219.

[2]  I. Bialynicki-Birula,et al.  Entropic uncertainty relations , 1984 .

[3]  Szilvia Nagy,et al.  Elementary formula for entanglement entropies of fermionic systems , 2005 .

[4]  R. Renner,et al.  An information-theoretic security proof for QKD protocols , 2005, quant-ph/0502064.

[5]  Hans Maassen.  Quantum probability applied to the damped harmonic oscillator , 2004 .

[6]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[7]  Karol Zyczkowski,et al.  Rényi-Wehrl entropies as measures of localization in phase space , 2001 .

[8]  J. Schwinger.  UNITARY OPERATOR BASES. , 1960, Proceedings of the National Academy of Sciences of the United States of America.

[9]  K R Parthasarathy,et al.  An Entropic Uncertainty Principle for Quantum Measurements , 2001 .

[10]  I. Bialynicki-Birula,et al.  Quantum electrodynamics of intense photon beams. New approximation method , 1976 .

[11]  M. Reed.  Methods of Modern Mathematical Physics. I: Functional Analysis , 1972 .

[12]  Rudolf Ahlswede,et al.  Quantum data processing , 1999, IEEE Trans. Inf. Theory.

[13]  Barbara M. Terhal.  Detecting quantum entanglement , 2002, Theor. Comput. Sci..

[14]  I. Bialynicki-Birula,et al.  Uncertainty relations for information entropy in wave mechanics , 1975 .

[15]  Steeve Zozor,et al.  Some extensions of the uncertainty principle , 2007, 0709.3011.

[16]  Kraus.  Complementary observables and uncertainty relations. , 1987, Physical review. D, Particles and fields.

[17]  I. Bialynicki-Birula.  Rényi Entropy and the Uncertainty Relations , 2007 .

[18]  F. Illuminati,et al.  Extremal entanglement and mixedness in continuous variable systems , 2004, quant-ph/0402124.

[19]  Ingemar Bengtsson.  Three Ways to Look at Mutually Unbiased Bases , 2007 .

[20]  On a supposed conceptual inadequacy of the Shannon information in quantum mechanics , 2001, quant-ph/0112178.

[21]  Event by event analysis and entropy of multiparticle systems , 2000 .

[22]  C. Beck,et al.  Symbolic dynamics of successive quantum-mechanical measurements. , 1992, Physical Review A. Atomic, Molecular, and Optical Physics.

[23]  M. Takesaki.  Theory of Operator Algebras II , 1979 .

[24]  A. Winter,et al.  Entropic uncertainty relations—a survey , 2009, 0907.3704.

[25]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[26]  Djalil CHAFAÏ.  Gaussian maximum of entropy and reversed log-Sobolev inequality , 2001 .

[27]  H. Everett.  "Relative State" Formulation of Quantum Mechanics , 1957 .

[28]  A. Martínez-Finkelshtein,et al.  Quantum-information entropies for highly excited states of single-particle systems with power-type potentials , 2002 .

[29]  J. Jensen.  Sur les fonctions convexes et les inégalités entre les valeurs moyennes , 1906 .

[30]  M. Riesz.  Sur les maxima des formes bilinéaires et sur les fonctionnelles linéaires , 1926 .

[31]  R. Renner,et al.  Information-theoretic security proof for quantum-key-distribution protocols , 2005, quant-ph/0502064.

[32]  D. Petz,et al.  Quantum Entropy and Its Use , 1993 .

[33]  W. Heisenberg.  Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik , 1927 .

[34]  J. Schwinger.  THE GEOMETRY OF QUANTUM STATES. , 1960, Proceedings of the National Academy of Sciences of the United States of America.

[35]  Entropic uncertainty relations for electromagnetic beams , 2009, 0902.4362.

[36]  Quantum Electrodynamics of Intense Photon Beams , 1973 .

[37]  Phase space localization of antisymmetric functions , 2002, quant-ph/0211017.

[38]  Seth Lloyd,et al.  Additivity properties of a Gaussian channel , 2004, quant-ph/0403075.

[39]  Uniformity of the phase space and fluctuations in thermal equilibrium , 2002, cond-mat/0211061.

[40]  Gaussian maximum of entropy and reversed log-Sobolev inequality , 2001, math/0102227.

[41]  Asher Peres,et al.  Quantum Theory: Concepts and Methods , 1994 .

[42]  A. Sergienko,et al.  Direct Measurement of Nonlinear Properties of Bipartite Quantum States , 2005, Open systems & information dynamics.

[43]  F. Dunning,et al.  Pulse-induced focusing of Rydberg wave packets , 2003 .

[44]  Z. Bouchal,et al.  Experimental test of uncertainty relations for quantum mechanics on a circle , 2007, 0712.0230.

[45]  A. Zeilinger,et al.  Conceptual inadequacy of the Shannon information in quantum measurements , 2000, quant-ph/0006087.

[46]  Adam Azarchs.  Entropic uncertainty relations for incomplete sets of mutually unbiased observables , 2004 .

[47]  Lukasz Rudnicki,et al.  Comment on Uncertainty relations in terms of the Tsallis entropy , 2010 .

[48]  Grzegorz Wilk,et al.  Uncertainty relations in terms of Tsallis entropy , 2008, 0806.1660.

[49]  F. A. Bovino,et al.  Direct Measurement of Nonlinear Properties of Bipartite Quantum States , 2006, Open Syst. Inf. Dyn..

[50]  A. Arbab,et al.  A New Formulation of Quantum Mechanics , 2010, 1002.4709.

[51]  I. D. Ivonovic.  Geometrical description of quantal state determination , 1981 .

[52]  M. Partovi.  Entropic formulation of uncertainty for quantum measurements , 1983 .

[53]  R. Jackson.  Inequalities , 2007, Algebra for Parents.

[54]  K. Babenko,et al.  An inequality in the theory of Fourier integrals , 1961 .

[55]  W. Beckner.  Inequalities in Fourier analysis , 1975 .

[56]  Moments of the Wigner function and Renyi entropies at freeze-out , 2005, hep-ph/0512293.

[57]  Eliahu Levy,et al.  A note about entropy , 2006 .

[58]  Shengjun Wu,et al.  Entropic uncertainty relation for mutually unbiased bases , 2008, 0811.2298.

[59]  M. Hall.  Almost-periodic time observables for bound quantum systems , 2008, 0803.3721.

[60]  V. Babin,et al.  Pattern formation in nonextensive thermodynamics: selection criterion based on the Renyi entropy production. , 2005, The Journal of chemical physics.

[61]  F. Verstraete,et al.  Matrix product states represent ground states faithfully , 2005, cond-mat/0505140.

[62]  Andrei Khrennikov,et al.  Foundations of Probability and Physics , 2002 .

[63]  H. L. Le Roy,et al.  Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability; Vol. IV , 1969 .

[64]  K. Parthasarathy.  An Introduction to Quantum Stochastic Calculus , 1992 .

[65]  Jorge Sánchez,et al.  Entropic uncertainty and certainty relations for complementary observables , 1993 .

[66]  Entropic uncertainty relations for angular distributions , 1985 .

[67]  János Pipek,et al.  Rényi entropies characterizing the shape and the extension of the phase space representation of quantum wave functions in disordered systems. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[68]  Iwo Bialynicki-Birula.  Formulation of the uncertainty relations in terms of the Rényi entropies , 2006 .

[69]  Maassen,et al.  Generalized entropic uncertainty relations. , 1988, Physical review letters.

[70]  D. Deutsch.  Uncertainty in Quantum Measurements , 1983 .

[71]  I. Chuang,et al.  Quantum Computation and Quantum Information: Bibliography , 2010 .

[72]  Alfréd Rényi,et al.  Probability Theory , 1970 .