Probabilistic logic of quantum observations

A probabilistic propositional logic, endowed with an epistemic component for asserting (non-)compatibility of diagonizable and bounded observables, is presented and illustrated for reasoning about the random results of projective measurements made on a given quantum state. Simultaneous measurements are assumed to imply that the underlying observables are compatible. A sound and weakly complete axiomatization is provided relying on the decidable first-order theory of real closed ordered fields. The proposed logic is proved to be a conservative extension of classical propositional logic.

[1]  W. Browder,et al.  Annals of Mathematics , 1889 .

[2]  P. Dirac.  Principles of Quantum Mechanics , 1982 .

[3]  Paul Adrien Maurice Dirac,et al.  A new notation for quantum mechanics , 1939, Mathematical Proceedings of the Cambridge Philosophical Society.

[4]  A. Tarski.  A Decision Method for Elementary Algebra and Geometry , 2023 .

[5]  W. Rudin.  Real and complex analysis , 1968 .

[6]  C. Piron,et al.  On the Foundations of Quantum Physics , 1976 .

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

[8]  Gaisi Takeuti,et al.  Quantum Set Theory , 1981 .

[9]  Gary M. Hardegree.  An axiom system for orthomodular quantum logic , 1981 .

[10]  Theodore Hailperin,et al.  Probability logic , 1984, Notre Dame J. Formal Log..

[11]  E. Panarella.  Heisenberg uncertainty principle , 1987 .

[12]  Charles H. Bennett,et al.  Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. , 1993, Physical review letters.

[13]  Pavel Exner,et al.  Hilbert Space Operators in Quantum Physics , 1994 .

[14]  W. van der Hoek,et al.  Epistemic logic for AI and computer science , 1995, Cambridge tracts in theoretical computer science.

[15]  Christopher Isham,et al.  Lectures On Quantum Theory: Mathematical And Structural Foundations , 1995 .

[16]  Ernest W. Adams,et al.  A primer of probability logic , 1996 .

[17]  Karl Svozil,et al.  Quantum Logic , 1998, Discrete mathematics and theoretical computer science.

[18]  Bob Coecke,et al.  The Sasaki Hook Is Not a [Static] Implicative Connective but Induces a Backward [in Time] Dynamic One That Assigns Causes , 2001 .

[19]  D. Marker.  Model theory : an introduction , 2002 .

[20]  Joseph Y. Halpern.  Reasoning about uncertainty , 2003 .

[21]  Manas K. Patra,et al.  A Logic for Probability in Quantum Systems , 2003, CSL.

[22]  Constance de Koning,et al.  Editors , 2003, Annals of Emergency Medicine.

[23]  Constantin A. Drossos.  Probability and Logic , 2004 .

[24]  Alexandru Baltag,et al.  Complete Axiomatizations for Quantum Actions , 2005 .

[25]  Alexandru Baltag,et al.  LQP: the dynamic logic of quantum information , 2006, Mathematical Structures in Computer Science.

[26]  Rohit Chadha,et al.  Reasoning About States of Probabilistic Sequential Programs , 2006, CSL.

[27]  Amílcar Sernadas,et al.  Weakly complete axiomatization of exogenous quantum propositional logic , 2005, Inf. Comput..

[28]  W. Carnielli,et al.  Logics of Formal Inconsistency , 2007 .

[29]  Rohit Chadha,et al.  Reasoning about probabilistic sequential programs , 2007, Theor. Comput. Sci..

[30]  Alexandru Baltag,et al.  A Dynamic-Logical Perspective on Quantum Behavior , 2008, Stud Logica.

[31]  Mladen Pavicic,et al.  Standard Logics Are Valuation-Nonmonotonic , 2008, J. Log. Comput..

[32]  C. Holbrow,et al.  The Heisenberg Uncertainty Principle , 2009 .

[33]  Alexandru Baltag,et al.  Correlated Knowledge: an Epistemic-Logic View on Quantum Entanglement , 2010 .

[34]  Walter Alexandre Carnielli,et al.  Paraconsistent Machines and their Relation to Quantum Computing , 2008, J. Log. Comput..

[35]  B. Hall.  Quantum Theory for Mathematicians , 2013 .

[36]  Robert B. Griffiths,et al.  The New Quantum Logic , 2013, 1311.2619.

[37]  J. D. Trimmer,et al.  THE PRESENT SITUATION IN QUANTUM MECHANICS: A TRANSLATION OF SCHR6DINGER'S "CAT PARADOX" PAPER , 2014 .

[38]  Jort Bergfeld,et al.  Duality for the Logic of Quantum Actions , 2015, Stud Logica.

[39]  A. B. Henriques,et al.  Epistemic nature of quantum reasoning , 2015, 1504.06460.

[40]  Cristina Sernadas,et al.  On probability and logic , 2016, 1601.02915.

[41]  Chrysafis Hartonas,et al.  First-order frames for orthomodular quantum logic , 2016, J. Appl. Non Class. Logics.

[42]  M. Pavicic.  Classical Logic and Quantum Logic with Multiple and Common Lattice Models , 2016, 1609.00822.

[43]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[44]  Niki Pfeifer,et al.  Probability Logic , 2019, ArXiv.