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Multiparty pseudotelepathy
 Proceedings of the 8th International Workshop on Algorithms and Data Structures, Volume 2748 of Lecture Notes in Computer Science
, 2003
"... Quantum information processing is at the crossroads of physics, mathematics and computer science. It is concerned with that we can and cannot do with quantum information that goes beyond the abilities of classical information processing devices. Communication complexity is an area of classical compu ..."
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Cited by 18 (6 self)
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Quantum information processing is at the crossroads of physics, mathematics and computer science. It is concerned with that we can and cannot do with quantum information that goes beyond the abilities of classical information processing devices. Communication complexity is an area of classical computer science that aims at quantifying the amount of communication necessary to solve distributed computational problems. Quantum communication complexity uses quantum mechanics to reduce the amount of communication that would be classically required. Pseudotelepathy is a surprising application of quantum information processing to communication complexity. Thanks to entanglement, perhaps the most nonclassical manifestation of quantum mechanics, two or more quantum players can accomplish a distributed task with no need for communication whatsoever, which would be an impossible feat for classical players. After a detailed overview of the principle and purpose of pseudotelepathy, we present a survey of recent and nosorecent work on the subject. In particular, we describe and analyse all the pseudotelepathy games currently known to the authors.
Physical traces: Quantum vs. classical information processing
 In Proceedings of Category Theory and Computer Science 2002 (CTCS’02), volume 69 of Electronic Notes in Theoretical Computer Science. Elsevier Science
, 2003
"... a setting that enables qualitative differences between classical and quantum processes to be explored. The key construction is the physical interpretation/realization of the traced monoidal categories of finitedimensional vector spaces with tensor product as monoidal structure and of finite sets an ..."
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Cited by 17 (5 self)
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a setting that enables qualitative differences between classical and quantum processes to be explored. The key construction is the physical interpretation/realization of the traced monoidal categories of finitedimensional vector spaces with tensor product as monoidal structure and of finite sets and relations with Cartesian product as monoidal structure, both of them providing a socalled wavestyle GoI. The developments in this paper reveal that envisioning state update due to quantum measurement as a process provides a powerful tool for developing highlevel approaches to quantum information processing.
Quantum probability and decision theory, revisited
 IN THE EVERETT INTERPRETATION”, STUDIES IN THE HISTORY AND PHILOSOPHY OF MODERN PHYSICS
, 2002
"... An extended analysis is given of the program, originally suggested by Deutsch, of solving the probability problem in the Everett interpretation by means of decision theory. Deutsch’s own proof is discussed, and alternatives are presented which are based upon different decision theories and upon Glea ..."
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Cited by 16 (2 self)
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An extended analysis is given of the program, originally suggested by Deutsch, of solving the probability problem in the Everett interpretation by means of decision theory. Deutsch’s own proof is discussed, and alternatives are presented which are based upon different decision theories and upon Gleason’s Theorem. It is argued that decision theory gives Everettians most or all of what they need from ‘probability’. Contact is made with Lewis’s Principal Principle linking subjective credence with objective chance: an Everettian Principal Principle is formulated, and shown to be at least as defensible as the usual Principle. Some consequences of (Everettian) quantum mechanics for decision theory itself are also discussed.
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.
2002), Derivation of the Born Rule from Operational Assumptions
 In Proceedings of the Royal Society A
, 2004
"... 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 appl ..."
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Cited by 15 (2 self)
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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
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.
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 or c ..."
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Cited by 12 (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
Time, quantum mechanics, and probability
 Synthese
, 1998
"... ABSTRACT. A variety of ideas arisiüg in decoherence theory, and in the ongoing debate over Everett’s relativestate 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 pr ..."
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Cited by 11 (0 self)
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ABSTRACT. A variety of ideas arisiüg in decoherence theory, and in the ongoing debate over Everett’s relativestate 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
Quantum Probability Theory
, 2006
"... The mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933. Quantum theory as nonclassical probability theory was incorporated into the beginnings of noncommutative measure theory by von Neumann in the early thirties, as well. To precisely this e ..."
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Cited by 10 (3 self)
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The mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933. Quantum theory as nonclassical probability theory was incorporated into the beginnings of noncommutative measure theory by von Neumann in the early thirties, as well. To precisely this end, von Neumann initiated the study of what are now called von Neumann algebras and, with Murray, made a first classification of such algebras into three types. The nonrelativistic quantum theory of systems with finitely many degrees of freedom deals exclusively with type I algebras. However, for the description of further quantum systems, the other types of von Neumann algebras are indispensable. The paper reviews quantum probability theory in terms of general von Neumann algebras, stressing the similarity of the conceptual structure of classical and noncommutative probability theories and emphasizing the correspondence between the classical and quantum concepts, though also indicating the nonclassical nature of quantum probabilistic predictions. In addition, differences between the probability theories in the type I, II and III settings are explained. A brief description is given of quantum systems