Generalized probabilistic theories (GPTs) allow us to write quantum
theory in a purely operational language and enable us to formulate
other, vastly different theories. As it turns out, there is no
canonical way to integrate the notion of subsystems within the
framework of convex operational theories. Sections can be seen as
generalization of subsystems and describe situations where not all
possible observables can be implemented. Jonathan Steinberg
discusses the mathematical foundations of GPTs using the language
of Archimedean order unit spaces and investigates the algebraic
nature of sections. This includes an analysis of the category
theoretic structure and the transformation properties of the state
space. Since the Hilbert space formulation of quantum mechanics
uses tensor products to describe subsystems, he shows how one can
interpret the tensor product as a special type of a section. In
addition he applies this concept to quantum theory and compares it
with the formulation in the algebraic approach. Afterwards he gives
a complete characterization of low dimensional sections of
arbitrary quantum systems using the theory of matrix pencils.
General
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