Natural information measures for contextual probabilistic models

My greatest concern was what to call it. I thought of calling it an ‘information’Â’, but the word was overly used, so I decided to call it an “uncertainty’. When I discussed it with John von Neumann, he had a better idea. Von Neumann told me, “You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, nobody knows what entropy really is, so in a debate you will always have an advantage”.[1] (see also [2], page 35).

[1]  J. Yngvason The role of type III factors in quantum field theory , 2004, math-ph/0411058.

[2]  A. J. Short,et al.  Entropy in general physical theories , 2009, 0909.4801.

[3]  Benjamin Schumacher,et al.  A new proof of the quantum noiseless coding theorem , 1994 .

[4]  U. von Toussaint,et al.  Bayesian inference and maximum entropy methods in science and engineering , 2004 .

[5]  Carl A. Hein,et al.  Entropy in operational statistics and quantum logic , 1979 .

[6]  K. Knuth Measuring Questions: Relevance and its Relation to Entropy , 2004, physics/0409084.

[7]  Gerard J. Milburn,et al.  Geometry of quantum states: an introduction to quantum entanglement by Ingemar Bengtsson and Karol Zyczkowski , 2006, Quantum Inf. Comput..

[8]  Stefan Arnborg,et al.  On the foundations of Bayesianism , 2001 .

[9]  Serge Fehr,et al.  On quantum Rényi entropies: A new generalization and some properties , 2013, 1306.3142.

[10]  R. T. Cox The Algebra of Probable Inference , 1962 .

[11]  David P. DiVincenzo,et al.  Quantum information and computation , 2000, Nature.

[12]  J. Aczél,et al.  Lectures on Functional Equations and Their Applications , 1968 .

[13]  W. Ochs A new axiomatic characterization of the von Neumann entropy , 1975 .

[14]  S. Zozor,et al.  General entropy-like uncertainty relations in finite dimensions , 2013, 1311.5602.

[15]  Kevin S. Van Horn,et al.  Constructing a logic of plausible inference: a guide to Cox's theorem , 2003, Int. J. Approx. Reason..

[16]  Schumacher,et al.  Quantum coding. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[17]  H. Barnum,et al.  Entropy and information causality in general probabilistic theories , 2009, 0909.5075.

[18]  H. Maassen Quantum Probability Theory , 2022 .

[19]  C. Tsallis Possible generalization of Boltzmann-Gibbs statistics , 1988 .

[20]  Mihai Sanduleac,et al.  Energy and Information , 2010 .

[21]  Gary Oas,et al.  Negative probabilities and counter-factual reasoning in quantum cognition , 2014, 1404.3921.

[22]  Ruediger Schack,et al.  Quantum-Bayesian Coherence , 2009, 1301.3274.

[23]  Adán Cabello,et al.  Proposal for revealing quantum nonlocality via local contextuality. , 2009, Physical review letters.

[24]  J. Neumann Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren , 1930 .

[25]  H. Dishkant,et al.  Logic of Quantum Mechanics , 1976 .

[26]  H. Geiringer On the Foundations of Probability Theory , 1967 .

[27]  A. Rényi On Measures of Entropy and Information , 1961 .

[28]  Alexey E. Rastegin,et al.  Tests for quantum contextuality in terms of Q-entropies , 2012, Quantum Inf. Comput..

[29]  R. T. Cox,et al.  The Algebra of Probable Inference , 1962 .

[30]  Kevin H. Knuth,et al.  Lattice duality: The origin of probability and entropy , 2013, Neurocomputing.

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

[32]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[33]  J. Acacio de Barros,et al.  Exploring non-signalling polytopes with negative probability , 2014, 1404.3831.

[34]  R. Rossignoli,et al.  Generalized entropic measures of quantum correlations , 2010, 1104.5678.

[35]  J. Neyman,et al.  Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability , 1963 .

[36]  J. Aczel,et al.  WHY THE SHANNON AND HARTLEY ENTROPIES ARE 'NATURAL' , 1974 .

[37]  Mirko Navara,et al.  The Pasting Constructions for Orthomodular Posets , 1991 .

[38]  A unique characterization of the generalized Boltzmann-Gibbs-Shannon entropy , 1976 .

[39]  Matt Farr,et al.  Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics , 2015 .

[40]  J. Neumann,et al.  On Rings of Operators. III , 1940 .

[41]  Jeffrey Bub,et al.  Quantum computation from a quantum logical perspective , 2006, Quantum Inf. Comput..

[42]  Federico Holik,et al.  On the Lattice Structure of Probability Spaces in Quantum Mechanics , 2011, 1112.4616.

[43]  Kevin H. Knuth,et al.  Quantum Theory and Probability Theory: Their Relationship and Origin in Symmetry , 2011, Symmetry.

[44]  Kevin Knuth,et al.  Toward Question-Asking Machines: The Logic of Questions and the Inquiry Calculus , 2005, AISTATS.

[45]  H. Barnum,et al.  Cloning and Broadcasting in Generic Probabilistic Theories , 2006, quant-ph/0611295.

[46]  Karl Svozil,et al.  Quantum Logic in Algebraic Approach , 2001 .

[47]  L. Ballentine,et al.  Probabilistic and Statistical Aspects of Quantum Theory , 1982 .

[48]  P. Porcelli,et al.  On rings of operators , 1967 .

[49]  Richard Phillips Feynman,et al.  The Concept of Probability in Quantum Mechanics , 1951 .

[50]  A. Plastino,et al.  Generalized Probabilities in Statistical Theories , 2014, Quantum Reports.

[51]  C. F. Roos,et al.  Compatibility and noncontextuality for sequential measurements , 2009, 0912.4846.

[52]  A. Gleason Measures on the Closed Subspaces of a Hilbert Space , 1957 .

[53]  Federico Holik,et al.  A discussion on the origin of quantum probabilities , 2012, 1211.4952.

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

[55]  J. von Neumann,et al.  On rings of operators. II , 1937 .

[56]  A. Plastino,et al.  Convex polytopes and quantum separability , 2011, 1109.3034.

[57]  R. T. Cox Probability, frequency and reasonable expectation , 1990 .

[58]  Cliff Hooker,et al.  The Logico-Algebraic Approach to Quantum Mechanics , 1975 .