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22
Classical representations of quantum mechanics related to statistically complete observables, Wissenschaft und Technik
, 1997
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Reconstruction of Quantum Theory
"... What belongs to quantum theory is no more than what is needed for its derivation. Keeping to this maxim, we record a paradigmatic shift in the foundations of quantum mechanics, where the focus has recently moved from interpreting to reconstructing quantum theory. Several historic and contemporary re ..."
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What belongs to quantum theory is no more than what is needed for its derivation. Keeping to this maxim, we record a paradigmatic shift in the foundations of quantum mechanics, where the focus has recently moved from interpreting to reconstructing quantum theory. Several historic and contemporary reconstructions are analyzed, including the work of Hardy, Rovelli, and Clifton, Bub and Halvorson. We conclude by discussing the importance of a novel concept of intentionally incomplete reconstruction.
Quantum information processing, operational quantum logic, convexity, and th foundations of physics
 Studies in the History and Philosophy of Modern Physics, 34:343–379
, 2003
"... Quantum information science is a source of taskrelated axioms whose consequences can be explored in general settings encompassing quantum mechanics, classical theory, and more. Quantum states are compendia of probabilities for the outcomes of possible operations we may perform on a system: ‘‘operat ..."
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Quantum information science is a source of taskrelated axioms whose consequences can be explored in general settings encompassing quantum mechanics, classical theory, and more. Quantum states are compendia of probabilities for the outcomes of possible operations we may perform on a system: ‘‘operational states.’ ’ I discuss general frameworks for ‘‘operational theories’ ’ (sets of possible operational states of a system), in which convexity plays key role. The main technical content of the paper is in a theorem that any such theory naturally gives rise to a ‘‘weak effect algebra’ ’ when outcomes having the same probability in all states are identified and in the introduction of a notion of ‘‘operation algebra’ ’ that also takes account of sequential and conditional operations. Such frameworks are appropriate for investigating what things look like from an ‘‘inside view,’ ’ i.e., for describing perspectival information that one subsystem of the world can have about another. Understandinghow such views can combine, and whether an overall ‘‘geometric’ ’ picture (‘‘outside view’’) coordinating them all can be had, even if this picture is very different in structure from the perspectives within it, is the key to whether we may be able to achieve a unified, ‘‘objective’ ’ physical view in which quantum mechanics is the appropriate description for certain perspectives, or whether quantum mechanics is truly telling us we must go beyond this ‘‘geometric’ ’ conception of physics.
Ordered linear spaces and categories as frameworks for informationprocessing characterizations of quantum and classical theory. Available as arXiv:0908.2354
"... The advent of quantum computation and quantum information science has been accompanied by a revival of the project of characterizing quantum and classical theory within a setting significantly more general than both. Part of the motivation is to obtain a clear conceptual understanding of the sources ..."
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The advent of quantum computation and quantum information science has been accompanied by a revival of the project of characterizing quantum and classical theory within a setting significantly more general than both. Part of the motivation is to obtain a clear conceptual understanding of the sources of quantum theory’s greaterthanclassical power in areas like cryptography and computation, as well as of the limits it appears to share with classical theory. This line of work suggests supplementing traditional approaches to the axiomatic characterization of quantum mechanics within broader classes of theories, with an approach in which some or all of the axioms concern the informationprocessing power of the theory. In this paper, we review some of our recent results (with collaborators) on information processing in an ordered linear spaces framework for probabilistic theories. These include demonstrations that many “inherently quantum” phenomena are in reality quite general characteristics of nonclassical theories, quantum or otherwise. As an example, a set of states in such a theory is broadcastable if, and only if, it is contained in a simplex whose vertices are cloneable, and therefore distinguishable by a single measurement. As another example, information that can be obtained about a system in this framework without causing disturbance to the system state, must be inherently classical. We also review results on teleportation protocols in the framework, and the fact that any nonclassical theory without entanglement allows exponentially secure bit commitment in this framework. Finally, we sketch some ways of formulating our framework in terms of categories, and in this light consider the relation of our work to that of Abramsky, Coecke, Selinger, Baez and others on information processing and other aspects of theories formulated categorically. 1
Early Greek Thought and perspectives for the Interpretation of Quantum Mechanics: Preliminaries to an Ontological Approach
 In The Blue Book of Einstein Meets Magritte
, 1999
"... It will be shown in this article that an ontological approach for some problems related to the interpretation of Quantum Mechanics (QM) could emerge from a reevaluation of the main paradox of early Greek thought: the paradox of Being and nonBeing, and the solutions presented to it by Plato and Ari ..."
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It will be shown in this article that an ontological approach for some problems related to the interpretation of Quantum Mechanics (QM) could emerge from a reevaluation of the main paradox of early Greek thought: the paradox of Being and nonBeing, and the solutions presented to it by Plato and Aristotle. More well known are the derivative paradoxes of Zeno: the paradox of motion and the paradox of the One and the Many. They stem from what was perceived by classical philosophy to be the fundamental enigma for thinking about the world: the seemingly contradictory results that followed from the coincidence of being and nonbeing in the world of change and motion as we experience it, and the experience of absolute existence here and now. The most clear expression of both stances can be found, again following classical thought, in the thinking of Heraclitus of Ephesus and Parmenides of Elea. The problem put forward by these paradoxes reduces for both Plato and Aristotle to the possibility of the existence of stable objects as a necessary condition for knowledge. Hence the primarily ontological nature of the solutions they proposed: Plato’s Theory of Forms and Aristotle’s metaphysics and logic. Plato’s and Aristotle’s systems are argued here to do on the ontological level essentially the same: to introduce stability in the world by introducing the notion of a separable, stable object,
Extended quantum mechanics
 Acta Physica Slovaca
, 2000
"... The work can be considered as an essay on mathematical and conceptual structure of nonrelativistic quantum mechanics (QM) which is related here to some other (more general, but also to more special – “approximative”) theories. QM is here primarily equivalently reformulated in the form of a Poisson s ..."
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The work can be considered as an essay on mathematical and conceptual structure of nonrelativistic quantum mechanics (QM) which is related here to some other (more general, but also to more special – “approximative”) theories. QM is here primarily equivalently reformulated in the form of a Poisson system on the phase space consisting of density matrices, where the “observables”, as well as “symmetry generators ” are represented by a specific type of real valued (densely defined) functions, namely the usual quantum expectations of corresponding selfadjoint operators. It is shown in this work that inclusion of additional (“nonlinear”) symmetry generators (i.e. “Hamiltonians”) into this reformulation of (linear) QM leads to a considerable extension of the theory: two kinds of quantum “mixed states ” should be distinguished, and operator – valued functions of density matrices should be used in the rôle of “nonlinear observables”. A general framework for physical theories is obtained in this way: By different choices of the sets of “nonlinear observables ” we obtain, as special cases, e.g. classical mechanics on homogeneous spaces of kinematical symmetry groups, standard (linear) QM, or nonlinear extensions of QM; also various “quasiclassical approximations ” to QM are
Physicallyrelativized ChurchTuring Hypotheses. Applied Mathematics and Computation 215, 4
 in the School of Mathematics at the University of Leeds, U.K. © 2012 ACM 00010782/12/03 $10.00 march 2012  vol. 55  no. 3  communications of the acm 83
"... Abstract. We turn ‘the ’ ChurchTuring Hypothesis from an ambiguous source of sensational speculations into a (collection of) sound and welldefined scientific problem(s): Examining recent controversies, and causes for misunderstanding, concerning the state of the ChurchTuring Hypothesis (CTH), sug ..."
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Abstract. We turn ‘the ’ ChurchTuring Hypothesis from an ambiguous source of sensational speculations into a (collection of) sound and welldefined scientific problem(s): Examining recent controversies, and causes for misunderstanding, concerning the state of the ChurchTuring Hypothesis (CTH), suggests to study the CTH relative to an arbitrary but specific physical theory—rather than vaguely referring to “nature ” in general. To this end we combine (and compare) physical structuralism with (models of computation in) complexity theory. The benefit of this formal framework is illustrated by reporting on some previous, and giving one new, example result(s) of computability
Bell's Inequalities And Algebraic Structure
, 1996
"... . We provide an overview of the connections between Bell's inequalities and algebraic structure. 1. Introduction Motivated by the desire to bring into the realm of testable hypotheses at least some of the important matters concerning the interpretation of quantum mechanics evoked in the controversy ..."
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. We provide an overview of the connections between Bell's inequalities and algebraic structure. 1. Introduction Motivated by the desire to bring into the realm of testable hypotheses at least some of the important matters concerning the interpretation of quantum mechanics evoked in the controversy surrounding the EinsteinPodolsky Rosen paradox [18][5], Bell discovered the first example [3][4] of a family of inequalities which are now generally called Bell's inequalities. These inequalities provide an upper bound on the strength of correlations between systems which are no longer interacting but have interacted in the past. Stated briefly, Bell showed that if the correlation experiments can be modelled by a single classical probability measure, then the strength of the correlations must satisfy a bound which is violated by certain quantum mechanical predictions (and, as has been verified experimentally, by nature  for a review of this original application of Bell's inequalities and ...
Series Preproceedings of the Workshop “Physics and Computation ” 2008
, 2008
"... In the 1940s, two different views of the brain and the computer were equally important. One was the analog technology and theory that had emerged before the war. The other was the digital technology and theory that was to become the main paradigm of computation. 1 The outcome of the contest between ..."
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In the 1940s, two different views of the brain and the computer were equally important. One was the analog technology and theory that had emerged before the war. The other was the digital technology and theory that was to become the main paradigm of computation. 1 The outcome of the contest between these two competing views derived from technological and epistemological arguments. While digital technology was improving dramatically, the technology of analog machines had already reached a significant level of development. In particular, digital technology offered a more effective way to control the precision of calculations. But the epistemological discussion was, at the time, equally relevant. For the supporters of the analog computer, the digital model — which can only process information transformed and coded in binary — wouldn’t be suitable to represent certain kinds of continuous variation that help determine brain functions. With analog machines, on the contrary, there would be few or no layers between natural objects and the work and structure of computation (cf. [4, 1]). The 1942–52 Macy Conferences in cybernetics helped to validate digital theory and logic as legitimate ways to think about the brain and the machine [4]. In particular, those conferences helped made McCullochPitts ’ digital model
Consistency  What's Logic Got to Do with It?
, 1996
"... this paper, I want to explore the origin of the modern conception of the idea of consistency in logic in the work of German mathematician David Hilbert. My interest in the development of the modern idea of consistency arises from my belief that an overriding concern with a strict requirement of cons ..."
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this paper, I want to explore the origin of the modern conception of the idea of consistency in logic in the work of German mathematician David Hilbert. My interest in the development of the modern idea of consistency arises from my belief that an overriding concern with a strict requirement of consistency, borrowed primarily from the rigors of modern developments in logic, has prevented latter day twentieth century philosophers from producing philosophical systems of the type produced in earlier times.