Doctoral Defense: Ben Li

Thursday March 22, 2018 1:00 p.m. Yost, 306


Title: Convex Analysis and its application to Quantum Information Theory

Student: Ben Li

Advisors: Dr. Elisabeth Werner and Dr. Stanislaw Szarek

Abstract: My PhD thesis addresses problems on mathematical aspects of  Quantum information theory and convex geometry. This expository talk will consist of two parts. In the first part, we will discuss the non-locality problem of  the quantum world, which usually attracts most attention in this context, from the mathematical point of view it is equally striking that – at least for bipartite systems and dichotomic measurements – the discrepancy between classical and quantum correlations cannot be arbitrarily large: it can not exceed the so-called Grothendieck constant. This is a consequence of the seminal work of Tsirelson and the even more famous Grothendieck inequality from functional analysis. We calculate analytically the exact values of quantum violations for the Bell correlation inequalities that appear in the setups involving up to four measurements; they are all smaller than √ 2.

In the second part, we  discuss about extending notions from the class of convex bodies to classes of functions. Specifically, we are concerned with the notion of floating body in convex geometry. Convex floating body was first introduced independently by Baranym, Larman and Schuett, Werner. Here we introduce “floating bodies” for convex, not necessarily bounded subsets in R^n . This allows us to define floating functions for appropriate classes of functions and measures. We further establish the asymptotic behavior of the integral difference of a log-concave function and its floating function which in turn leads to a new affine invariant for functions.

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