PhD Defence Sigurd Anker Laursen Storgaard

 Title: Nonlocal games: Self-testing, parallel repetition and device-independent randomness

Abstract

This thesis studies device-independent certification and the behavior of quantum advantage and self-testing under composition, using nonlocal games as a common frame work. We develop an algebraic and computational method for proving robust self
testing results. Starting from numerical semidefinite relaxations in the Navascués Pironio–Acín hierarchy, we extract exact rational sum-of-squares certificates and operator relations satisfied by optimal strategies. These relations lead to universal C∗
algebras whose representation theory can be used to establish robust self-testing. The method is applied to achieve several new robust self-testing results.

We also consider a quantum information theoretic task for which complete self-testing is neither available nor necessary: maximal device-independent randomness generation. We construct a Bell scenario capable of device-independently certifying the maximal amount of 2logd bits of private randomness using quantum systems of local dimension d.

Finally, we investigate parallel repetition and related compositions of nonlocal games. Feige’s game is shown to have a single-copy quantum advantage which disappears under every even repetition. We call this phenomenon gap-annihilation. For agenuine nonlocal-game realization of an extension of tilted CHSH, we give an exact computer assisted example in which quantum advantage is present for one copy, disappears after two copies, and re-emerges after three. We then identify a structural obstruction to such behavior. We show that vector-tight linear games satisfy perfect quantum parallel repetition and cannot exhibit gap-annihilation. Under additional self-testing assumptions, their product games self-test tensor products of single-copy ideal strategies. Together, these results demonstrate how semidefinite optimization and operator algebraic methods can be combined to study self-testing and stability of quantum advantage under composition.

Thesis will be announced later 

Advisor: Laura Mančinska, University of Copenhagen

Assessment Committee:

Chair, Associate Professor, Daniel Stilck França, Department of Mathematical Sciences, University of Copenhagen

Professor Lyudmila Turowska, Chalmers University of Technology and University of Gothenburg

Junior Professor Chair, Marc-Olivier Renou, INRIA Paris-Saclay