Betting on the Outcomes of Measurements: A Bayesian Theory of Quantum Probability

We develop a systematic approach to quantum probability as a theory of rational betting in quantum gambles. In these games of chance the agent is betting in advance on the outcomes of several (finitely many) incompatible measurements. One of the measurements is subsequently chosen and performed and the money placed on the other measurements is returned to the agent. We show how the rules of rational betting imply all the interesting features of quantum probability, even in such finite gambles. These include the uncertainty principle and the violation of Bell's inequality among others. Quantum gambles are closely related to quantum logic and provide a new semantics to it. We conclude with a philosophical discussion on the interpretation of quantum mechanics.

[1]  Allen Stairs,et al.  Incompleteness, Nonlocality and Realism: A Prolegomenon to the Philosophy of Quantum Mechanics. , 1990 .

[2]  I. Pitowsky,et al.  George Boole's ‘Conditions of Possible Experience’ and the Quantum Puzzle , 1994, The British Journal for the Philosophy of Science.

[3]  R. Schack,et al.  Quantum probability from decision theory? , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[4]  William Demopoulos.  Elementary propositions and essentially incomplete knowledge: A framework for the interpretation of quantum mechanics , 2004 .

[5]  Jeffrey Bub,et al.  A uniqueness theorem for ‘no collapse’ interpretations of quantum mechanics , 1996 .

[6]  The Interpretation of Quantum Mechanics , 1974 .

[7]  J. Neumann,et al.  The Logic of Quantum Mechanics , 1936 .

[8]  Rob Clifton,et al.  Unremarkable contextualism: Dispositions in the Bohm theory , 1995 .

[9]  William Demopoulos,et al.  The Possibility Structure of Physical Systems , 1976 .

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

[11]  Jeffrey Bub,et al.  Interpreting the Quantum World , 1997 .

[12]  Nguyen Ba An.  Quantum dialogue , 2004 .

[13]  I. Pitowsky.  Quantum Probability ― Quantum Logic , 1989 .

[14]  Cliff Hooker,et al.  Foundations and philosophy of statistical theories in the physical sciences , 1976 .

[15]  M. Kernaghan.  Bell-Kochen-Specker theorem for 20 vectors , 1994 .

[16]  G. Gregory,et al.  Probability, Induction and Statistics , 1974 .

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

[18]  C. Fuchs,et al.  Quantum probabilities as Bayesian probabilities , 2001, quant-ph/0106133.

[19]  Bruno de Finetti,et al.  Probability, induction and statistics , 1972 .

[20]  J. Neumann.  Mathematical Foundations of Quantum Mechanics , 1955 .

[21]  L. Hardy.  Quantum Theory From Five Reasonable Axioms , 2001, quant-ph/0101012.

[22]  S. BellJ,et al.  Einstein‐Podolsky‐Rosen逆理 量子力学での遠隔作用か , 1987 .

[23]  K. Svozil,et al.  Optimal tests of quantum nonlocality , 2000, quant-ph/0011060.

[24]  I. Pitowsky,et al.  Relativity, Quantum Mechanics and EPR , 1992, PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association.

[25]  R. Mcweeny.  On the Einstein-Podolsky-Rosen Paradox , 2000 .

[26]  C. Fuchs.  Quantum Mechanics as Quantum Information (and only a little more) , 2002, quant-ph/0205039.

[27]  Mermin,et al.  Simple unified form for the major no-hidden-variables theorems. , 1990, Physical review letters.

[28]  Asher Peres,et al.  Quantum Theory Needs No ‘Interpretation’ , 2000 .

[29]  Stanley Gudder,et al.  Quantum probability—Quantum logic (Lecture notes in physics 321). By Itamar Pitowsky , 1989 .

[30]  Albert Einstein,et al.  Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? , 1935 .

[31]  Mario Bunge,et al.  Philosophy of Physics , 1972 .

[32]  M. Kafatos.  Bell's theorem, quantum theory and conceptions of the universe , 1989 .

[33]  Hilary Putnam,et al.  Mathematics, Matter and Method: Philosophical Papers , 1979 .

[34]  C. Caves,et al.  Quantum Bayes rule , 2000, quant-ph/0008113.

[35]  A. Zeilinger,et al.  Going Beyond Bell’s Theorem , 2007, 0712.0921.

[36]  I. Pitowsky.  Range Theorems for Quantum Probability and Entanglement , 2001 .

[37]  E. Specker,et al.  The Problem of Hidden Variables in Quantum Mechanics , 1967 .