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Quantum mechanics as quantum information (and only a little more), Quantum Theory: Reconsideration of Foundations
, 2002
"... In this paper, I try once again to cause some goodnatured trouble. The issue remains, when will we ever stop burdening the taxpayer with conferences devoted to the quantum foundations? The suspicion is expressed that no end will be in sight until a means is found to reduce quantum theory to two or ..."
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Cited by 66 (6 self)
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In this paper, I try once again to cause some goodnatured trouble. The issue remains, when will we ever stop burdening the taxpayer with conferences devoted to the quantum foundations? The suspicion is expressed that no end will be in sight until a means is found to reduce quantum theory to two or three statements of crisp physical (rather than abstract, axiomatic) significance. In this regard, no tool appears better calibrated for a direct assault than quantum information theory. Far from a strained application of the latest fad to a timehonored problem, this method holds promise precisely because a large part—but not all—of the structure of quantum theory has always concerned information. It is just that the physics community needs reminding. This paper, though takingquantph/0106166 as its core, corrects one mistake and offers several observations beyond the previous version. In particular, I identify one element of quantum mechanics that I would not label a subjective term in the theory—it is the integer parameter D traditionally ascribed to a quantum system via its Hilbertspace dimension. 1
Quantum Foundations in the Light of Quantum Information
 PROCEEDINGS OF THE NATO ADVANCED RESEARCH WORKSHOP, MYKONOS GREECE
, 2001
"... In this paper, I try to cause some goodnatured trouble. The issue at stake is when will we ever stop burdening the taxpayer with conferences and workshops devoted— explicitly or implicitly—to the quantum foundations? The suspicion is expressed that no end will be in sight until a means is found to ..."
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Cited by 17 (2 self)
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In this paper, I try to cause some goodnatured trouble. The issue at stake is when will we ever stop burdening the taxpayer with conferences and workshops devoted— explicitly or implicitly—to the quantum foundations? The suspicion is expressed that no end will be in sight until a means is found to reduce quantum theory to two or three statements of crisp physical (rather than abstract, axiomatic) significance. In this regard, no tool appears to be better calibrated for a direct assault than quantum information theory. Far from being a strained application of the latest fad to a deepseated problem, this method holds promise precisely because a large part (but not all) of the structure of quantum theory has always concerned information. It is just that the physics community has somehow forgotten this.
A Constructive Proof of Gleason’s Theorem
 J. Func. Anal
, 1999
"... 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 ..."
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Cited by 16 (2 self)
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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
The changemaking problem
 J. Assoc. Comput. Mach
, 1975
"... 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 u ..."
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Cited by 15 (0 self)
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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 complexvalued 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 twodimensional). In order to weaken Axiom VII it was desirable to strengthen Gleason’s Theorem.
Contexts in quantum, classical and partition logic
 In Handbook of Quantum Logic
, 2006
"... Contexts are maximal collections of comeasurable observables “bundled together ” to form a “quasiclassical miniuniverse. ” 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 ..."
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Cited by 8 (7 self)
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Contexts are maximal collections of comeasurable observables “bundled together ” to form a “quasiclassical miniuniverse. ” 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
Embedding Quantum Universes in Classical Ones
, 1999
"... this paper; the propositional structure encountered in the quantum mechanics of spin  state measurements of a spin onehalf particle along two directions ( mod p) , that is, the modular, orthocomplemented lattice MO 2 drawn in Fig. 1 ( where p 2 = ( p + ) and q 2 = ( q + ) ) ..."
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Cited by 4 (1 self)
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this paper; the propositional structure encountered in the quantum mechanics of spin  state measurements of a spin onehalf particle along two directions ( mod p) , that is, the modular, orthocomplemented lattice MO 2 drawn in Fig. 1 ( where p 2 = ( p + ) and q 2 = ( q + ) )
An extension of gleason’s theorem for quantum computation
 International Journal of Theoretical Physics
"... We develop a synthesis of Turing’s paradigm of computation and von Neumann’s quantum logic to serve as a model for quantum computation with recursion, such that potentially nonterminating computation can take place, as in a quantum Turing machine. This model is based on the extension of von Neumann ..."
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We develop a synthesis of Turing’s paradigm of computation and von Neumann’s quantum logic to serve as a model for quantum computation with recursion, such that potentially nonterminating computation can take place, as in a quantum Turing machine. This model is based on the extension of von Neumann’s quantum logic to partial states, defined here as subprobability measures on the Hilbert space, equipped with the natural pointwise partial ordering. The subprobability measures allow a certain probability for the nontermination of the computation. We then derive an extension of Gleason’s theorem and show that, for Hilbert spaces of dimension greater than two, the partial order of subprobability measures is order isomorphic with the collection of partial density operators, i.e. trace class positive operators with trace between zero and one, equipped with the usual partial ordering induced from positive operators. We show that the expected value of a bounded observable with respect to a partial state can be defined as a closed bounded interval, which extends the classical definition of expected value. 1
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
Piron's and Bell's Geometric Lemmas and Gleason's Theorem
 Found. Physics
, 2000
"... INTRODUCTION The Gleason theorem (1) is the cornerstone of measurement theory in quantum mechanics. It says, that if the quantum mechanical system can be described by a Hilbert space of dimension at least three, then any state of the physical system corresponds to von Neumann operator. The origi ..."
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INTRODUCTION The Gleason theorem (1) is the cornerstone of measurement theory in quantum mechanics. It says, that if the quantum mechanical system can be described by a Hilbert space of dimension at least three, then any state of the physical system corresponds to von Neumann operator. The original proof of this theorem is highly nontrivial, and only after 30 years later a simpler proof using also Piron's geometrical lemma [Ref. 2, pp. 75#78] was present by Cooke et al. (3) Today Gleason's theorem is used in quantum measurement as well as in mathematics. Dvurec# enskij is the author of a monograph, (4) where there are described plenty of applications of Gleason's theorem to different areas of mathematics. 1737 00159018#00#10001737#18.00#0 # 2000 Plenum Publishing Corporation 1