Introduction to Tsirelson's Problem and Related Topics in Algebraic Quantum Theory
Organizers: Michal Banacki and David Gross. No exercises. Times and venue: Tue 12pm, Seminarraum 0.01, Theoretical Physics Building.
Announcements
- Recordings on UoC's OpenCast system are currently inoperational in the lecture theater. Sorry.
- Lecture on Tue the 21st will be by David Gross
- No lecture on the 28th.
Course Description
There are two different ways to model “locality” in quantum mechanics. In the standard approach, each subsystem is associated with a Hilbert space, and the complete system with their tensor product. In a more general model, inspired by algebraic quantum field theory, one starts with one global Hilbert space. Each subsystem then corresponds to an algebra of observables acting on the global space, and "locality" translates to the requirement that observables of different subsystems commute.
The infamous Tsirelson's problem asks whether the two models give rise to different physical predictions, in the sense that there are correlations realizable in one but not in the other. The problem was open for 20 years, during which it was shown to be equivalent to several long-standing open problems in the theory of operator algebras.
A groundbreaking result going by the name “MIP* = RE" has since resolved the problem in the affirmative (the models do make different predictions!). Remarkably, it uses computability theory. It shows that if the two sets of correlations were the same, one could obtain an algorithm for a computationally undecidable problem.
Despite this refutation, the physical and mathematical implications remain subtle. Understanding the general relationship between these paradigms continues to be an essential open problem in the foundations of quantum theory.
In this course, we provide a self-contained introduction to the theory of operator algebras, with an emphasis on C*-algebraic constructions. We will explore Tsirelson's problem, its equivalent formulations (including Connes' embedding problem and Kirchberg's conjecture), and related quantum information protocols like steering and self-testing.
Theoretical Topics Include
- Preliminaries on operator algebras: Definitions and basic properties of C*-algebras (including von Neumann algebras), positive and completely positive maps, GNS construction, and tensor/free products of C*-algebras.
- Formulation of Tsirelson's problem: Discussion on various equivalent formulations including Connes' embedding problem and Kirchberg's conjecture.
- Comparison of paradigms: Analyzing quantum commuting vs. quantum tensor product models in selected quantum information processing schemes.
Prerequisites
- Basic knowledge of quantum mechanics (Bachelor's level).
- Familiarity with functional analysis is not assumed.
Notes
Handwritten notes: