Logic and the Mathematics of Quantum Physics

California Institute Of Technology, Pasadena CA

Investigators

Abstract

Quantum information theory is the mathematical theory underlying the use of the quantum mechanical states of physical systems for the purposes of information transmission and manipulation. Many questions in quantum information theory can be naturally studied within the mathematical framework of operator systems. The recent realization that operator systems can be regarded as quantum analogs of graphs has stimulated work with impact in the theories of both operator systems and quantum information. At the same time, classical notions and results from first order logic have been generalized within the framework of continuous logic to the setting where continuous structures, such as operator systems and other objects arising from quantum physics, are considered. The main goal of this research project is to build upon these developments to extend results from discrete mathematics and classical computing to the quantum setting. The investigator will pursue this project in collaboration with students at the graduate and undergraduate level. The first part of the project aims at obtaining feasible quantum algorithms to test isomorphism of operator systems. The next step in this direction is to determine the complexity of testing isomorphism of operator systems from the perspective of quantum computation. This question is of significant current interest for its implications for quantum physics and theoretical computer science. More broadly, the goal of the investigator is to extend classical descriptive complexity theory to the quantum setting, working within the framework of continuous logic. This research has the potential to address fundamental questions about quantum computing, including characterization of problems for which quantum computing provides a super-polynomial speedup over classical computing. Further aspects of the project involve the development of the theory of noncommutative Choquet simplices within the framework of operator systems, and the use of tools from model theory to build examples of C*-algebras that are not accessible via standard constructions.