Enrico G. Beltrametti

On a characterization of classical and nonclassical probabilities

In this paper a comparison of various approaches to the problem of characterization of probabilities is presented. The notion of multidimensional probability or S‐probability is introduced and it is shown that this notion can be profitably used for characterization of probability systems. The main results are given in Theorems 3 and 4. In Theorem 3 the conditions are considered under which a system of S‐probabilities admits a model on a generalized event space, while in Theorem 4 the conditions under which the system admits a classical model are considered. An interpretation for S‐probabilities is discussed and it is shown that Theorems 3 and 4 allow one to derive the logical structure of events from the observed probabilities.

Unitary measurements of discrete quantities in quantum mechanics

The pure measurements of discrete physical quantities are characterized within quantum theory of measurement and their unitary representations are given. Probabilistic aspects of measurements related to the so‐called strong correlation conditions and a probabilistic characterization of the first kind measurements are examined. The problem of the objectification of the measurement result is analyzed in terms of a classical behavior of the measuring apparatus. As a by‐product a generalization of the Wigner–Araki–Yanase theorem is given.

Quantum Observables in Classical Frameworks

A procedure of classical extension of a theory is worked out on the basis of a natural generalization of the notion of observable, the states of the extended theory being the probability measures on the pure states of the original one. Such a classical extension applies to quantum theory, and the qualifying features of quantum observables are preserved in the extended model.

On the characterization of probabilities: A generalization of Bell’s inequalities

A partial solution to the problem of generalizing Bell’s inequalities to arbitrary numbers of physical properties is proposed. It is first assumed that the considered sets of probabilities correspond to events which satisfy a postulate ensuring that they form an orthomodular partially ordered set admitting a full set of states. In this framework a theorem generalizing Bell’s inequalities to an arbitrary finite number of events is proven. An interpretation of these results in Hilbert space is indicated. Conditions characterizing the classicality of so‐called correlation probabilities are then found, and a method for verifying inequalities involving measurable correlations is discussed.

On state transformations induced by yes-no experiments, in the context of quantum logic

The standard formulation of quantum mechanics think, e.g., to the Hilbert space formulation, which has historically been the most influential and the best shaped for an analysis of its foundations is intimately related with the changes undergone by the state of the physical system as effect of the measurement of observables on the system. The probabilistic interpretation of the inner product in the Hilbert space is itself related to these state transformations: if the system is in a state represented by the unit vector 0 then the probability that the measurement of the observable represented by the self adjoint operator A gives a numerical value in the (Borel) set E C R is given by (0, PE ) where PE is the projection operator associated to the set E by the spectral decomposition of A, and PE is, up to a normalizing factor, a vector representing the state in which the system is left by the action of the measurement of A with numerical result in E. Quantum logic has taught us that a great deal of the Hilbert space structure of quantum mechanics can be traced back and, so to speak, condensed in typical properties of the sets 2 and S formed, respectively, by the nonequivalent yes-no experiments on the system, and by the states (or nonequivalent preparations) of the system. It is then natural to ask where and how the notion of state transformations caused by yes-no experiments is related and intertwined with the (2, S) structure. A review of this point is our present purpose and a great deal of what follows refers to existing literature. We shall mention results without reproducing proofs. Three aspects of the role of state transformations will be considered: first, the connection with the orthomodular lattice structure of 2; second, the connection with a structure of transition probability space which can be attached to S; third, the connection with a notion of logical implication which can be established in a language associated to 2.

On Bell-type inequalities

A Bell-type inequality is defined as an inequality of the type 0⩽L⩽1,where L is a linear combination with real coefficients of probabilities piand joint probabilities pij,pijk,...,pl,...,n corresponding to n events. A general theorem on the validity of such inequalities in correspondence to physical assumptions about commutativity or noncommutativity is given. Examples and possible physical applications are discussed.

Problem of classical and nonclassical probabilities

A characterization of classical and nonclassical probabilities expressed in terms of some inequalities between multidimensional orS-probability is given. A new criterion (not referring to correlation probabilities) for nonclassicality of the range of a completeS-probability measure on an event system is proposed.

The Bell Phenomenon in a Probabilistic Approach

The paper deals with a generalization of the standard probability theory, to be called operational probability theory: it is based on an enlargement of the usual class of random variables. Inside this framework a mathematical fact (called Bell phenomenon) is discussed which contains, as particular instances, the violations of Bell’s inequalities. A classical extension of quantum mechanics in the operational probability theory is also discussed.

On the Relationships between Classical and Quantum Mechanics

Almost a century since the birth of quantum mechanics, in presence of an impressive and still developing accumulation of empirical evidence of quantum behaviour from the atomic scale down to the subnuclear scale and up to the astronomical one, much attention is still focused on the foundations of quantum theory. In particular there are still open problems about the relationship between classical and quantum features. For instance, why putting together quantum objects (like atoms) to get a compound system one progressively looses the quantum behaviour? Why we never encounter macroscopic objects in those nonlocalised states (think, e.g., of the states staying on both arms of an interferometry device) in which its atoms, when isolated, would be able to live? The difficulty to find convincing answers is behind the puzzling aspects of the quantum theory of measurement, and behind the so-called paradoxes of quantum theory. Interesting conjectures can be found in the literature: for instance the GRW model based on the assumption of a dynamical process of spontaneous localization, or the idea that for open systems the interaction with the environment might be responsible for the transition from quantum to classical.

Operational statistical theories and effect algebras

We sketch the so called convex, or operational, framework for statistical physical theories, based on the convex set formed by the states of the physical system. A notion of observable is introduced which encompasses the ones usually adopted in classical and in quantum mechanics. In this framework a structure of effect algebra naturally arises. We also discuss a notion of extension of a descriptive physical model, pointing at the possibility of constructing classical extension of quantum mechanics.