The Born rule is derived from operational assumptions, together with assumptions of quantum mechanics that concern only the deterministic development of the state. Unlike Gleason’s theorem, the argument applies even if probabilities are de…ned for only a single resolution of the identity, so it applies to a variety of foundational approaches to quantum mechanics. It also provides a probability rule for state spaces that are not Hilbert spaces. 1

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.

Abstract. Let A be a von Neumann algebra with no direct summand of Type I2, and let P(A) be its lattice of projections. Let X be a Banach space. Let m: P(A) → X be a bounded function such that m(p + q) = m(p) + m(q) whenever p and q are orthogonal projections. The main theorem states that m has a unique extension to a bounded linear operator from A to X. In particular, each bounded complex-valued finitely additive quantum measure on P(A) has a unique extension to a bounded linear functional on A. Physical background In von Neumann’s approach to the mathematical foundations of quantum mechanics, the bounded observables of a physical system are identified with a real linear space, L, of bounded selfadjoint operators on a Hilbert space H. It is reasonable to assume that L is closed in the weak operator topology and that whenever x ∈ L then x 2 ∈ L. (Thus L is a Jordan algebra and contains spectral projections.) Then the projections in L form a complete orthomodular lattice, P, otherwise known as the lattice of “questions ” or the quantum logic of the physical system. A quantum measure is a map µ: P → R such that whenever p and q are orthogonal projections µ(p + q) = µ(p) + µ(q). In Mackey’s formulation of quantum mechanics [11] his Axiom VII makes the assumption that L = L(H)sa. Mackey states, that in contrast to his other axioms, Axiom VII has no physical justification; it is adopted for mathematical convenience. One of the technical advantages of this axiom was that, by Gleason’s Theorem, a completely additive positive quantum measure on the projections of L(H) is the restriction of a bounded linear functional (provided H is not two-dimensional). In order to weaken Axiom VII it was desirable to strengthen Gleason’s Theorem.

Quantum theory has provoked intense discussions about its interpretation since its pioneer days, beginning with Bohr’s view of quantum theory as a theory of knowledge. We show that such an epistemic perspective can be consistently complemented by Einstein’s ontically oriented position.

Gleason's theorem states that any totally additive measure on the closed subspaces, or projections, of a Hilbert space of dimension greater than two is given by a positive operator of trace class. In this paper we give a constructive proof of that theorem. A measure on the projections of a real or complex Hilbert space assigns to

ABSTRACT. A variety of ideas arisiüg in decoherence theory, and in the ongoing debate over Everett’s relative-state theory, can be linked to issues in relativity theory and the philosophy of time, speci…cally the relational theory of tense and of identity over time. These have been systematically presented in companion papers (Saunders 1995, 1996a); in what follows we shall consider the same circle of ideas, but speci…cally in relation to the interpretation of probability, and its identi…cation with relations in the Hilbert space norm. The familiar objection that Everett’s approach yields probabilities di¤erent from quantum mechanics is easily dealt with. The more fundamental question is how to interpret these probabilities consistent with the relational theory of change, and the relational theory of identity over time. I shall show that the relational theory needs nothing more than the physical, minimal criterion of identity as de…ned by Everett’s theory, and that this can be transparently interpreted in terms of the ordinary notion of the chance occurrence of an event, as witnessed in the present. It is in this sense that the theory has empirical content. 1

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The ABL rule is derived and shown to be a tool of standard quantum mechanics. Objections by Kastner [quant-ph/0003098v3] and others to the counterfactual use of the ABL rule are shown to be groundless. In particular, this use is not restricted in the way Kastner has claimed. A variant of the three-box experiment due to Vaidman is discussed. It is argued that Born probabilities (and hence state vectors or density operators) are not the right basis for drawing ontological inferences. What quantum mechanics is trying to tell us about the world must be inferred from the objective ABL probabilities that are assigned to counterfactuals. The correct inferences, however, will remain incomprensible until a prevalent but inconsistent way of thinking about the temporal aspect of the world is rejected. 1 OBJECTIVE PROBABILITIES AND THE ABL RULE Following Mermin [1], I characterized some of the probabilities that quantum mechanics allows us to calculate as being objective in the sense that they have nothing to do with ignorance—there is nothing for us to be ignorant of [2]. In order to be objective, they

When you write papers, how many times do you want to make some citations at a place but you are not sure which papers to cite? Do you wish to have a recommendation system which can recommend a small number of good candidates for every place that you want to make some citations? In this paper, we present our initiative of building a context-aware citation recommendation system. High quality citation recommendation is challenging: not only should the citations recommended be relevant to the paper under composition, but also should match the local contexts of the places citations are made. Moreover, it is far from trivial to model how the topic of the whole paper and the contexts of the citation places should affect the selection and ranking of citations. To tackle the problem, we develop a context-aware approach. The core idea is to design a novel non-parametric probabilistic model which can measure the context-based relevance between a citation context and a document. Our approach can recommend citations for a context effectively. Moreover, it can recommend a set of citations for a paper with high quality. We implement a prototype system in CiteSeerX. An extensive empirical evaluation in the Cite-SeerX digital library against many baselines demonstrates the effectiveness and the scalability of our approach.

Contexts are maximal collections of co-measurable observables “bundled together ” to form a “quasi-classical mini-universe. ” Different notions of contexts are discussed for classical, quantum and generalized urn–automaton systems. PACS numbers: 02.10.-v,02.50.Cw,02.10.Ud