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Journal of Mathematical Physics

Extensions of convexity models

Enrico G. Beltrametti, Slawomir Bugajski, Veeravalli S. Varadarajan

Published May 1, 2000• 9 citations
Publisher

Abstract

The notion of convexity model is introduced to provide a general frame for statistical theories of physical interest: this frame encompasses, in particular, the classical and the quantum cases. In a convexity model the states of the physical system, and the convex structure they form, play a basic role; observables and related quantities are then naturally defined. The notion of extensions of a convexity model is studied: it appears physically relevant to cope with several needs, paradigmatically with the one of viewing the physical system as a part of a compound system. We focus attention on quantum-like extensions of both the usual classical and quantum convexity models, as well as on classical-like extensions of the quantum model. The behavior of state overlapping and state superposition under model extension is briefly examined.

DOI
10.1063/1.533256

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The notion of convexity model is introduced to provide a general frame for statistical theories of physical interest: this frame encompasses, in particular, the classical and the quantum cases. In a convexity model the states of the physical system, and the convex structure they form, play a basic role; observables and related quantities are then naturally defined. The notion of extensions of a convexity model is studied: it appears physically relevant to cope with several needs, paradigmatically with the one of viewing the physical system as a part of a compound system. We focus attention on quantum-like extensions of both the usual classical and quantum convexity models, as well as on classical-like extensions of the quantum model. The behavior of state overlapping and state superposition under model extension is briefly examined.
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.
What picture does one see, looking at a physical theory from a distance, so that the details disappear? Since quantum mechanics is a statistical theory, the most universal picture which remains after the details are forgotten is that of a convex set. Bogdan Mielnik Convex sets Our object is to understand the geometry of the set of all possible states of a quantum system that can occur in nature. This is a very general question, especially since we are not trying to define ‘state’ or ‘system’ very precisely. Indeed we will not even discuss whether the state is a property of a thing, or of the preparation of a thing, or of a belief about a thing. Nevertheless we can ask what kind of restrictions are needed on a set if it is going to serve as a space of states in the first place. There is a restriction that arises naturally both in quantum mechanics and in classical statistics: the set must be a convex set . The idea is that a convex set is a set such that one can form ‘mixtures’ of any pair of points in the set. This is, as we will see, how probability enters (although we are not trying to define ‘probability’ either). From a geometrical point of view a mixture of two states can be defined as a point on the segment of the straight line between the two points that represent the states that we want to mix. We insist that given two points belonging to the set of states, the straight line segment between them must belong to the set too. This is certainly not true of any set.
A comparison of structural features of quantum and classical physical theories, such as the information capacity of systems subject to these theories, requires a common formal framework for the presentation of corresponding concepts (such as states, observables, probability, entropy). Such a framework is provided by the notion of statistical model developed in the convexity approach to statistical physical theories. Here we use statistical models to classify and survey all possible types of embedding and extension of quantum probabilistic theories subject to certain reasonable constraints. It will be shown that the so-called canonical classical extension of quantum mechanics is essentially the only `good' representation of the quantum statistical model in a classical framework. All quantum observables are thus identified as fuzzy classical random variables.
It is well - known that the set of states of a given quantum mechanical system is to be closed from the point of view of the operational approach if we want to make mixed states or convex combinations. That is, s1s_1s1​ and s2s_2s2​ are states, then the same is to be true for λs1+(1−λ)s2,\lambda s_1 +(1-\lambda) s_2,λs1​+(1−λ)s2​, where 0<λ<1.0 < \lambda < 1.0<λ<1. We can define a convex combination of elements in a linear space, but unfortunately, in the general case the linear space is artificial for the set of states and has no physical meaning, but the procedure of forming the mixtures of states has a natural meaning. This is why we provide an abstract definition of the mixtures, which is independent of the linearity notion. We call this space a convex structure. In the work we consider state spaces, generalized state spaces, in which we select pure states, define operations and effects associated with the operations. We also consider ultraproducts of the sequences of these structures, operations and effects.
In this paper, I propose a project of enlisting quantum information science as a source of task-oriented axioms for use in the investigation of operational theories in a general framework capable of encompassing quantum mechanics, classical theory, and more. Whatever else they may be, quantum states of systems are compendia of probabilities for the outcomes of possible operations we may perform on the systems: ``operational theories.'' I discuss appropriate general frameworks for such theories, in which convexity plays a key role. 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. Understanding how 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 nature from the structure of the perspectives within it, is the key to understanding 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. The nature of information, its flow and processing, as seen from various operational persepectives, is likely to be key to understanding whether and how such coordination and unification can be achieved.