Speaker
Description
In this talk, I will discuss support problems for free additive convolutions of probability measures on the real line, with emphasis on applications to quantum information theory. A central question is how the left endpoint of the support changes under free convolution powers. Using analytic subordination, global inversion, and boundary behavior of reciprocal Cauchy transforms, we obtain an explicit formula for the left endpoint of the support of a free convolution power. The formula is expressed in terms of the Nevanlinna representation and a critical integral condition associated with the representing measure. I will also explain why such support formulas are relevant to quantum information theory. In particular, they provide tools for studying positivity and $(k)$-positivity questions arising from random matrix models and positive maps. The support of the limiting spectral distribution often determines whether certain transformed random matrices are asymptotically positive, and hence whether the corresponding maps satisfy positivity-type properties. This connection gives a bridge between analytic free probability, random matrix theory, and structural questions in quantum information theory.