Some basic notions of classical and quantum statistical physics

Jean Zinn‐Justin

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

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.

Shubhayan Sarkar

Abstract One of the basic observations of the classical world is that physical entities are real and can be distinguished from each other. However, within quantum theory, the idea of physical realism is not well established. A framework to analyse how observations in experiments can be described using some physical states of reality was recently developed, known as ontological models framework. Different principles when imposed on the ontological level give rise to different theories, the validity of which can be tested based on the statistics generated by these theories. Using the ontological models framework, we formulate a novel notion of classicality termed ontic-distinguishability, which is based upon the physical principles that in classical theories extremal states are physical states of reality and every sharp measurement observes the state of the system perfectly. We construct a communication task in which the success probability is bounded from above for ontological models satisfying the notion of ontic-distinguishability. Contrary to previous notions of classicality which either required systems of dimension strictly greater than two or atleast three preparations, a violation of ontic-distinguishability can be observed using just a pair of qubits and a pair of incompatible measurements. We further show that violation of previously known notions of classicality such as preparation non-contextuality and Bell’s local causality is a violation of ontic-distinguishability.

D. I. Blokhint︠s︡ev

Stephen J. Summers

A review is made of the multitude of different mathematical formalizations of the physical concept ‘two observables (or two systems) are independent’ that have been proposed in quantum theories, particularly relativistic quantum field theory. The most basic mathematical formulation of independence in any quantum theory is what one may call kinematical independence: the two observables, resp. the observables of the two quantum systems, which are represented by elements of a C*-algebra, resp. two subalgebras of a C*-algebra, are required to commute. This is related to a mathematical formulation of the notion of the coexistence (or compatibility) of two observables. Another basic notion of independence, generally called statistical independence in the literature, is, roughly speaking, two quantum systems are said to be statistically independent if each can be prepared in any state, how ever the other system has been prepared. There are numerous mathematical formulations of this notion and their interrelationships are explained. Statistical independence and kinematical independence are shown to be logically independent. Additional notions such as strict locality and their relation to statistical independence are discussed. The mathematics of a more quantitative measure of statistical independence, Bell’s inequalities, is reviewed and its relations with previously introduced notions are indicated. All of these notions are then viewed in application to relativistic quantum field theory.

Daniel R. Frankl