# Symmetry, Self-Duality and the Jordan Structure of Quantum Theory

### APA

Wilce, A. (2011). Symmetry, Self-Duality and the Jordan Structure of Quantum Theory . Perimeter Institute. https://pirsa.org/11050037

### MLA

Wilce, Alexander. Symmetry, Self-Duality and the Jordan Structure of Quantum Theory . Perimeter Institute, May. 10, 2011, https://pirsa.org/11050037

### BibTex

@misc{ pirsa_PIRSA:11050037, doi = {10.48660/11050037}, url = {https://pirsa.org/11050037}, author = {Wilce, Alexander}, keywords = {Quantum Foundations}, language = {en}, title = {Symmetry, Self-Duality and the Jordan Structure of Quantum Theory }, publisher = {Perimeter Institute}, year = {2011}, month = {may}, note = {PIRSA:11050037 see, \url{https://pirsa.org}} }

Susquehanna University

Talk Type

**Subject**

Abstract

This talk reviews recent and on-going work, much of it joint with Howard Barnum, on the origins of the Jordan-algebraic structure of finite-dimensional quantum theory. I begin by describing a simple recipe for constructing highly symmetrical probabilistic models, and discuss the ordered linear spaces generated by such models. I then consider the situation of a probabilistic theory consisting of a symmetric monoidal *-category of finite-dimensional such models: in this context, the state and effect cones are self-dual. Subject to a further ``steering" axiom, they are also homogenous, and hence, by the Koecher-Vinberg Theorem, representable as the cones of formally real Jordan algebras. Finally, if the theory contains a single system with the structure of a qubit, then (by a result of H. Hanche-Olsen), each model in the category is the self-adjoint part of a C*-algebra.