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Betting on the outcomes of measurements: a Bayesian theory of quantum probability
, 2003
"... We develop a systematic approach to quantum probability as a theory of rational bettingin 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 ..."
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Cited by 12 (4 self)
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We develop a systematic approach to quantum probability as a theory of rational bettingin 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 amongothers. Quantum gambles are closely related to quantum logic and provide a new semantics for it. We conclude with a philosophical discussion on the interpretation of quantum mechanics.
An Unentangled Gleason's Theorem
 CONTEMPORARY MATHEMATICS
, 2002
"... The purpose of this note is to give a generalization of Gleason's theorem inspired by recent work in quantum information theory. For multipartite quantum systems, each of dimension three or greater, the only nonnegative frame functions over the set of unentangled states are those given by the stand ..."
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The purpose of this note is to give a generalization of Gleason's theorem inspired by recent work in quantum information theory. For multipartite quantum systems, each of dimension three or greater, the only nonnegative frame functions over the set of unentangled states are those given by the standard Born probability rule. However, if one system is of dimension 2 this is not necessarily the case.
Generalizations of Kochen and Specker’s theorem and the effectiveness of Gleason’s theorem
 Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 35, 177194
, 2004
"... Abstract. Kochen and Specker’s theorem can be seen as a consequence of Gleason’s theorem and logical compactness. Similar compactness arguments lead to stronger results about finite sets of rays in Hilbert space, which we also prove by a direct construction. Finally, we demonstrate that Gleason’s th ..."
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Cited by 3 (1 self)
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Abstract. Kochen and Specker’s theorem can be seen as a consequence of Gleason’s theorem and logical compactness. Similar compactness arguments lead to stronger results about finite sets of rays in Hilbert space, which we also prove by a direct construction. Finally, we demonstrate that Gleason’s theorem itself has a constructive proof, based on a generic, finite, effectively generated set of rays, on which every quantum state can be approximated. 1. Gleason’s Theorem and Logical Compactness Kochen and Specker’s (1967) theorem (KS) puts a severe constraint on possible hiddenvariable interpretations of quantum mechanics. Often it is considered an improvement on a similar argument derived from Gleason (1957) theorem (see, for example, Held. 2000). This is true in the sense that KS provide an explicit construction of a finite set of rays on which no twovalued homomorphism exists. However, the fact that there is such a finite set follows from Gleason’s theorem using a simple logical compactness argument (Pitowsky 1998, a similar point is made in Bell 1996). The existence of finite sets of rays with other interesting features