QLunch: Haonan Zhang

Speaker: Haonan Zhang, IST Austria

Title: From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture and beyond

Abstract: In a celebrated paper in 1973, Lieb proved what we now call Lieb's Concavity Theorem and resolved a conjecture of Wigner-Yanase-Dyson. This result has found many applications in mathematical physics and quantum information. In recent years, Audenaert and Datta conjectured that certain trace functionals are jointly convex, which is equivalent to the data processing inequalities for alpha-z Rényi relative entropies. Later on, a stronger conjecture was made by Carlen-Frank-Lieb when reviewing the related problems. In this talk, I will present a very simple variational method to study the concavity/convexity of trace functionals. This allows us to reduce many concavity/convexity problems to certain fundamental results: mainly this 1973 concavity theorem of Lieb and its complementary convexity result of Ando in 1979. Along the way, we settle the conjecture of Carlen-Frank-Lieb concerning "double convexity".  We may also prove "triple convexity" theorems, which extend some results of Hiai-Petz and Carlen-Frank-Lieb. Such "triple convexity" results are related to a conjecture by Al-Rashed and Zegarliński, which we can prove using complex interpolation. Time permitting, more applications will be discussed. The talk is based on arXiv:1811.01205, arXiv:2007.06644 and arXiv:2108.05785.