Operator algebras and nonlocal games: The quantum commuting model: Optimal states and self-testing
Research output: Book/Report › Ph.D. thesis › Research
This thesis consists of two main parts: Part I on relative Cuntz–Pimsner algebras,and 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-equivariantrepresentations (and whence in turn also of gauge-invariant invariant ideals) and further unravelsKatsura’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 theseto provide examples for the failure of pullbacks for absolute Cuntz–Pimsner algebras. This sectionconstitutes 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 revealthe failure for short exact sequences by relative Cuntz–Pimsner algebras. We then outlineMorita equivalences arising from higher tensor powers of correspondences and those Moritaequivalences 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 on “Connes implies Tsirelson” from preprint B.The following sections concentrate on computing quantum values using operator algebraictechniques and the classification of optimal states and their correlations. We begin for this with asummary on uniqueness of optimal states from preprint C.We then elaborate on genuine self-testing for their correlations based on order-two moments andfollow 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 a representation–theoretic classification of optimal states and their quantum value for the tilted CHSH games based on an upcoming preprint with Azin Shahiri(currently under preparation).This further serves as a primer on ongoing work around the I3322 inequality (from a newperspective) as well as on nonlocal games exhibiting a separation between finite dimensional andquantum spatial correlations using operator algebraic techniques.
|Department of Mathematical Sciences, Faculty of Science, University of Copenhagen
|Published - 2023