Free Probability and Operators

Date: October 14, 2022

Time: 4:00PM - 5:00PM

Location: BLOC 306

Speaker: Junchen Zhao, TAMU

Title: Optimal coupling in non-commutative probability

Abstract: In this talk, we would like to introduce optimal coupling of random variables in non-commutative probability spaces, motivated by optimal transport of measure in classical probability theory. We will focus on optimally coupling self-adjoint matrix tuples X = (X_1, ..., X_m) and Y = (Y_1, ..., Y_m) in the probability space (M_n(C), tr). It turns out (even for m > 1 and n = 2,3) they can already demonstrate the complication of non-commutativity and give us interesting examples/intuitions where explicit computations are possible. We will also point out the connections to factorizable maps in operator algebra and quantum information theory; present our new concrete example of matrix tuples in M_3(C) whose optimal coupling requires an embedding into a matrix algebra of larger dimension, namely M_3(C) ⊗ M_3(C); and sketch the proof. At last, we will talk about possible directions of future works and current obstructions. This work was advised by Todd Kemp and David Jekel.