Extensions and Restrictions of Generalized Probabilistic Theories

Generalized probabilistic theories (GPTs) provide a mathematical framework for quantum theory in an operational language, but they do not directly incorporate the notion of subsystems. Sections are a generalization of subsystems that describe situations where not all possible observables can be implemented. Jonathan Steinberg discusses the mathematical foundations of GPTs using Archimedean order unit spaces and investigates the algebraic nature of sections, including their category theoretic structure and transformation properties. He shows how the tensor product can be interpreted as a special type of section and applies this concept to quantum theory, comparing it with the algebraic approach. Steinberg also gives a complete characterization of low-dimensional sections of arbitrary quantum systems using the theory of matrix pencils.

Format: Paperback / softback
Length: 79 pages
Publication date: 17 May 2022
Publisher: Springer Fachmedien Wiesbaden

Generalized probabilistic theories (GPTs) are a powerful tool that allows us to write quantum theory in a purely operational language, enabling us to formulate a wide range of different theories. However, one challenge that arises is the integration of the notion of subsystems within the framework of convex operational theories. While sections can be seen as a generalization of subsystems, they describe situations where not all possible observables can be implemented.
Jonathan Steinberg delves into the mathematical foundations of GPTs by utilizing the language of Archimedean order unit spaces. He investigates the algebraic nature of sections, including an analysis of the category theoretic structure and the transformation properties of the state space. Steinberg demonstrates how the Hilbert space formulation of quantum mechanics uses tensor products to describe subsystems, and he shows how one can interpret the tensor product as a special type of a section.
Furthermore, Steinberg applies this concept to quantum theory and compares it with the formulation in the algebraic approach. He then provides a complete characterization of low-dimensional sections of arbitrary quantum systems using the theory of matrix pencils.
In conclusion, GPTs offer a powerful framework for writing quantum theory in a purely operational language, and by investigating the algebraic nature of sections, we can gain a deeper understanding of the mathematical foundations of these theories. Steinberg's work provides valuable insights into the integration of subsystems within convex operational theories and the characterization of low-dimensional sections of arbitrary quantum systems.

Weight: 131g
Dimension: 210 x 148 (mm)
ISBN-13: 9783658375805
Edition number: 1st ed. 2022