We show that three fundamental information-theoretic constraints -- the impossibility of superluminal information transfer between two physical systems by performing measurements on one of them, the impossibility of broadcasting the information contained in an unknown physical state, and the impossibility of unconditionally secure bit commitment -- suffice to entail that the observables and state space of a physical theory are quantum-mechanical. We demonstrate the converse derivation in part, and consider the implications of alternative answers to a remaining open question about (...) nonlocality and bit commitment. (shrink)

We consider probabilistic theories in which the most elementary system, a two-dimensional system, contains one bit of information. The bit is assumed to be contained in any complete set of mutually complementary measurements. The requirement of invariance of the information under a continuous change of the set of mutually complementary measurements uniquely singles out a measure of information, which is quadratic in probabilities. The assumption which gives the same scaling of the number of degrees of freedom with the dimension as (...) in quantum theory follows essentially from the assumption that all physical states of a higher dimensional system are those and only those from which one can post-select physical states of two-dimensional systems. The requirement that no more than one bit of information (as quantified by the quadratic measure) is contained in all possible post-selected two-dimensional systems is equivalent to the positivity of density operator in quantum theory. (shrink)

Carlo Rovelli's relational interpretation of quantum mechanics holds that a system's states or the values of its physical quantities as normally conceived only exist relative to a cut between a system and an observer or measuring instrument. Furthermore, on Rovelli's account, the appearance of determinate observations from pure quantum superpositions happens only relative to the interaction of the system and observer. Jeffrey Barrett ([1999]) has pointed out that certain relational interpretations suffer from what we might call the ‘determinacy problem', but (...) Barrett misclassifies Rovelli's interpretation by lumping it in with Mermin's view, as Rovelli's view is quite different and has resources to escape the particular criticisms that Barrett makes of Mermin's view. Rovelli's interpretation still leaves us with a paradox having to do with the determinacy of measurement outcomes, which can be accepted only if we are willing to give up on certain elements of the ‘absolute’ view of the world. (shrink)

Quantum mechanics and probability theory share one peculiarity. Both have well established mathematical formalisms, yet both are subject to controversy about the meaning and interpretation of their basic concepts. Since probability plays a fundamental role in QM, the conceptual problems of one theory can affect the other. We first classify the interpretations of probability into three major classes: inferential probability, ensemble probability, and propensity. Class is the basis of inductive logic; deals with the frequencies of events in repeatable experiments; describes (...) a form of causality that is weaker than determinism. An important, but neglected, paper by P. Humphreys demonstrated that propensity must differ mathematically, as well as conceptually, from probability, but he did not develop a theory of propensity. Such a theory is developed in this paper. Propensity theory shares many, but not all, of the axioms of probability theory. As a consequence, propensity supports the Law of Large Numbers from probability theory, but does not support Bayes theorem. Although there are particular problems within QM to which any of the classes of probability may be applied, it is argued that the intrinsic quantum probabilities are most naturally interpreted as quantum propensities. This does not alter the familiar statistical interpretation of QM. But the interpretation of quantum states as representing knowledge is untenable. Examples show that a density matrix fails to represent knowledge. (shrink)

In this article we present a possible way to make usual quantum mechanics fully compatible with physical realism, defined as the statement that the goal of physics is to study entities of the natural world, existing independently from any particular observer’s perception, and obeying universal and intelligible rules. Rather than elaborating on the quantum formalism itself, we propose a new quantum ontology, where physical properties are attributed jointly to the system, and to the context in which it is embedded. In (...) combination with a quantization principle, this non-classical definition of physical reality sheds new light on counter-intuitive features of quantum mechanics such as the origin of probabilities, non-locality, and the quantum-classical boundary. (shrink)

Quantum mechanics is not about 'quantum states': it is about values of physical variables. I give a short fresh presentation and update on the *relational* perspective on the theory, and a comment on its philosophical implications.

Carlo Rovelli’s inspiring “Relational Quantum Mechanics” serves several aims at once: it provides a new vision of what the world of quantum mechanics is like, and it offers a program to derive the theory’s formalism from a set of simple postulates pertaining to information processing. I propose here to concentrate entirely on the former, to explore the world of quantum mechanics as Rovelli depicts it. It is a fascinating world in part because of Rovelli’s reliance on the information-theory approach to (...) the foundations of quantum mechanics, and in part because its presentation involves taking sides on a fundamental divide within philosophy itself. (shrink)

We study the EPR-type correlations from the perspective of the relational interpretation of quantum mechanics. We argue that these correlations do not entail any form of “non-locality”, when viewed in the context of this interpretation. The abandonment of strict Einstein realism implied by the relational stance permits to reconcile quantum mechanics, completeness, (operationally defined) separability, and locality.

The abstract description of a physical system is developed, along lines originally suggested by Birkhoff and von Neumann, in terms of the complete lattice of propositions associated with that system, and the distinction between classical and quantum systems is made precise. With the help of the notion of state, a propositional system is defined: it is remarked that every irreducible propositional system (of more than three dimensions) is isomorphic to the lattice of all closed subspaces of a Hilbert space constructed (...) on some division ring with involution. The propositional system consisting of a family of separable complex Hilbert spaces is treated as a particular case which is sufficiently general to include both classical and quantum mechanics. The theory of the Galilean particle without spin is given as an illustration. Finally, the basis for the statistical interpretation of wave mechanics is developed with the help of Gleason's theorem. In an appendix, a proof of essentially the first part of Gleason's theorem is given which is a little different (perhaps more geometric) from that originally given by Gleason. (shrink)

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.

A paradigmatic shift in the foundations of quantum mechanics is recorded, from interpreting to reconstructing quantum theory. Examples of reconstruction are analyzed, and conceptual foundations of the information-theoretic reconstruction developed. A concept of intentionally incomplete reconstruction is introduced to mark the novel content of research in the foundation of quantum theory. ‡Many thanks to Lucien Hardy, Jeff Bub and Bill Demopoulos for their comments. This research was supported through the ANR grant ANR-06-BLAN-0348-01. Part of this research was held at the (...) Perimeter Institute for Theoretical Physics. Research at Perimeter Institute for Theoretical Physics is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI. †To contact the author, please write to: CEA-Saclay, 91191 Gif-sur-Yvette, France; e-mail: [email protected]. (shrink)

This paper presents a minimal formulation of nonrelativistic quantum mechanics, by which is meant a formulation which describes the theory in a succinct, self-contained, clear, unambiguous and of course correct manner. The bulk of the presentation is the so-called “microscopic theory”, applicable to any closed system S of arbitrary size N, using concepts referring to S alone, without resort to external apparatus or external agents. An example of a similar minimal microscopic theory is the standard formulation of classical mechanics, which (...) serves as the template for a minimal quantum theory. The only substantive assumption required is the replacement of the classical Euclidean phase space by Hilbert space in the quantum case, with the attendant all-important phenomenon of quantum incompatibility. Two fundamental theorems of Hilbert space, the Kochen–Specker–Bell theorem and Gleason’s theorem, then lead inevitably to the well-known Born probability rule. For both classical and quantum mechanics, questions of physical implementation and experimental verification of the predictions of the theories are the domain of the macroscopic theory, which is argued to be a special case or application of the more general microscopic theory. (shrink)

The law of gravitation, an example of physical law The relation of mathematics to physics The great conservation principles Symmetry in physical law The distinction of past and future Probability and uncertainty: the quantum mechanical view of nature Seeking new laws.

I consider the fact that there are a number of interesting ways to ‘reconstruct’ quantum theory, and suggest that, very broadly speaking, a form of ‘instrumentalism’ makes good sense of the situation. This view runs against some common wisdom, which dismisses instrumentalism as ‘cheap’. In contrast, I consider how an instrumentalist might think about the reconstruction theorems, and, having made a distinction between ‘reconstructing’ quantum theory and ‘reinventing’ quantum theory, I suggest that there is an adequate instrumentalist approach to the (...) theory that invokes both. (shrink)

Professor Merrie Bergmann presents an accessible introduction to the subject of many-valued and fuzzy logic designed for use on undergraduate and graduate courses in non-classical logic. Bergmann discusses the philosophical issues that give rise to fuzzy logic - problems arising from vague language - and returns to those issues as logical systems are presented. For historical and pedagogical reasons, three-valued logical systems are presented as useful intermediate systems for studying the principles and theory behind fuzzy logic. The major fuzzy logical (...) systems - Lukasiewicz, Gödel, and product logics - are then presented as generalisations of three-valued systems that successfully address the problems of vagueness. A clear presentation of technical concepts, this book includes exercises throughout the text that pose straightforward problems, that ask students to continue proofs begun in the text, and that engage students in the comparison of logical systems. (shrink)