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.

Recently, Brukner and Zeilinger (Phys. Rev. Lett. 83(17) (2001) 3354) have claimed that the Shannon information is not well defined as a measure of information in quantum mechanics, adducing arguments that seek to show that it is inextricably tied to classical notions of measurement. It is shown here that these arguments do not succeed: the Shannon information does not have problematic ties to classical concepts. In a further argument, Brukner and Zeilinger compare the Shannon information unfavourably to their preferred information measure, Ið~pÞ; with regard to the definition of a notion of ‘‘total information content.’ ’ This argument is found unconvincing and the relationship between individual measures of information and notions of ‘‘total information content’ ’ investigated. We close by considering the prospects of Zeilinger’s Foundational Principle as a foundational principle for quantum mechanics.

I consider exactly what is involved in a solution to the probability problem of the Everett interpretation, in the light of recent work on applying considerations from decision theory to that problem. I suggest an overall framework for understanding probability in a physical theory, and conclude that this framework, when applied to the Everett interpretation, yields the result that that interpretation satisfactorily solves the measurement problem.

We prove generic versions of the no-cloning and no-broadcasting theorems, applicable to essentially any non-classical finite-dimensional probabilistic model that satisfies a no-signaling criterion. This includes quantum theory as well as models supporting “super-quantum ” correlations that violate the Bell inequalities to a larger extent than quantum theory. The proof of our no-broadcasting theorem is significantly more natural and more self-contained than others we have seen: we show that a set of states is broadcastable if, and only if, it is contained in a simplex whose vertices are cloneable, and therefore distinguishable by a single measurement. This necessary and sufficient condition generalizes the quantum requirement that a broadcastable set of states commute. 1

Abstract. In the present paper I show how it is possible to derive the Hilbert space formulation of Quantum Mechanics from a comprehensive definition of physical experiment and assuming experimental accessibility and simplicity as specified by five simple Postulates. This accomplishes the program presented in form of conjectures in the previous paper [1]. Pivotal roles are played by the local observability principle, which reconciles the holism of nonlocality with the reductionism of local observation, and by the postulated existence of informationally complete observables and of a symmetric faithful state. This last notion allows one to introduce an operational definition for the real version of the “adjoint"—i. e. the transposition— from which one can derive a real Hilbert-space structure via either the Mackey-Kakutani or the Gelfand-Naimark-Segal constructions. Here I analyze in detail only the Gelfand-Naimark-Segal construction, which leads to a real Hilbert space structure analogous to that of (classes of generally unbounded) selfadjoint operators in Quantum Mechanics. For finite dimensions, general dimensionality theorems that can be derived from a local observability principle, allow us to represent the elements of the real Hilbert space as operators over an underlying complex Hilbert space (see, however, a still open problem at the end of the paper). The route for the present operational axiomatization was suggested by novel ideas originated from Quantum Tomography. 1.

We present a general contextualistic statistical model for constructing quantum-like representations in physics, cognitive and social sciences, psychology, economy. In this paper we use this model to describe cognitive experiments (in particular, in psychology) to check quantum-like structures of mental processes. The crucial role is played by interference of probabilities corresponding to mental observables. Recently one of such experiments based on recognition of images was performed. This experiment confirmed my prediction on quantum-like behaviour of mind. We present the procedure of constructing the wave function of a cognitive context on the basis of statistical data for two incompatible mental observables. We discuss the structure of state spaces for cognitive systems. In fact, the general contextual probability theory predicts not only quantum-like trigonometric (cos θ) interference of probabilities, but also hyperbolic (cosh θ) interference of probabilities (as well as hyper-trigonometric). In principle, statistical data obtained in experiments with cognitive systems can produce hyperbolic (cosh θ) interference of probabilities. At the moment there are no experimental confirmations of hyperbolic interference for cognitive systems.

We prove a Gleason-type theorem for the quantum probability rule using frame functions defined on positive-operator-valued measures (POVMs), as opposed to the restricted class of orthogonal projection-valued measures used in the original theorem. The advantage of this method is that it works for two-dimensional quantum systems (qubits) and even for vector spaces over rational fields—settings where the standard theorem fails. Furthermore, unlike the method necessary for proving the original result, the present one is rather elementary. In the case of a qubit, we investigate similar results for frame functions defined upon various restricted classes of POVMs. For the so-called trine measurements, the standard quantum probability rule is again recovered. KEY WORDS: quantum measurements; quantum probability rule; frame functions; POVM.

Quantum Mechanics (QM) is a very special probabilistic theory, yet we don’t know which operational principles make it so. All axiomatization attempts suffer at least one postulate of a mathematical nature. Here I will analyze the possibility of deriving QM as the mathematical representation of a fair operational framework, i.e. a set of rules which allows the experimenter to make predictions on future events on the basis of suitable tests, e.g. without interference from uncontrollable sources. Two postulates need to be satisfied by any fair operational framework: NSF: no-signaling from the future—for the possibility of making predictions on the basis of past tests; PFAITH: existence of a preparationally faithful state—for the possibility of preparing any state and calibrating any test. I will show that all theories satisfying NSF admit a C ∗-algebra representation of events as linear transformations of effects. Based on a very general notion of dynamical independence, it is easy to see that all such probabilistic theories are non-signaling without interaction (nonsignaling for short)—another requirement for a fair operational framework. Postulate

We derive essential elements of quantum mechanics from a parametric structure extending that of traditional mathematical statistics. The basic setting is a set A of incompatible experiments, and a transformation group G on the cartesian product Π of the parameter spaces of these experiments. The set of possible parameters is constrained to lie in a subspace of Π, an orbit or a set of orbits of G. Each possible model is then connected to a parametric Hilbert space. The spaces of different experiments are linked unitarily, thus defining a common Hilbert space H. A state is equivalent to a question together with an answer: the choice of an experiment a ∈ A plus a value for the corresponding parameter. Finally, probabilities are introduced through Born’s formula, which is derived from a recent version of Gleason’s theorem. This then leads to the usual formalism of elementary quantum mechanics in important special cases. The theory is illustrated by the example of a quantum particle with spin. 1. Introduction. Both

Quantum information science is a source of task-related 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.