Extensions and Restrictions of Generalized Probabilistic Theories

by Jonathan Steinberg

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 inthe algebraic approach. Afterwards he gives a complete characterization of low dimensional sections of arbitrary quantum systems using the theory of matrix pencils.

120 pages, Kindle Edition

Published May 16, 2022

About the author

Jonathan Steinberg is the Walter H. Annenberg Professor of European History and former Chair of the Department of History at the University of Pennsylvania. He received his A. B. from Harvard University and his Ph.D. from Cambridge University.