PhD Defense Alexander Frei

This thesis consists of two main parts:

Part I on the classification of relative Cuntz-Pimsner algebras.

Part II on operator algebras and nonlocal games in quantum information.

Part I centers around the classification of relative Cuntz-Pimsner algebras from preprint A:  More precisely, the attached preprint gives a systematic classification of gauge-equivariant representations (and whence in turn also of gauge-invariant invariant ideals) and further unravels Katsura's construction as a canonical dilation given by the maximal covariance.

We give a short summary on its main results in the first section. As a first application we then classify pullbacks for relative Cuntz-Pimsner algebras and use these to provide examples for the failure of pullbacks for absolute Cuntz-Pimsner algebras. This section constitutes a summary of upcoming work with Piotr M. Hajac and Mariusz Tobolski.

The second application concerns Morita equivalence for relative Cuntz-Pimsner algebras: Using our classification we swiftly recover a classical result by Fowler-Muhly-Raeburn and reveal the failure for short exact sequences by relative Cuntz-Pimsner algebras. We then outline Morita equivalences arising from higher tensor powers of correspondences and those Morita equivalences for relative Cuntz-Pimsner algebras not arising by any finite tensor power at all. This section serves as an outlook on upcoming work by the author.

Part II centers around operator algebras and nonlocal games: The first section constitutes a summary of Connes implies Tsirelson from preprint B. The following sections concentrate on computing quantum values using operator algebraic techniques and the classification of optimal states and their correlations. We begin this with a summary on uniqueness of optimal states from preprint C. We then elaborate on genuine self-testing for their correlations based on order-two moments and follow with an outlook on robust self-testing for optimal states in the quantum commuting model. Both of these constitute ongoing joint work with Azin Shahiri.

We finish the second part with an operator algebraic classification of optimal states and their quantum value for the tilted CHSH games. This is based on an upcoming preprint with Azin Shahiri. This further serves as a primer on ongoing work around the I3322 inequality (from a new perspective) as well as on nonlocal games exhibiting a separation between finite dimensional and quantum spatial correlations using operator algebraic techniques.

Thesis for download

Supervisor: Professor Søren Eilers, MATH, University of Copenhagen 

Assessment committee:
Associate professor, Laura Mancinska (chair), MATH, University of Copenhagen 
Professor emeritus Vern I. Paulsen, University of Waterloo
Senior lecturer, Evgenios Kakariadis, University of Newcastle