A variation on von Neumann Entropy and a result of Varadarajan

Abstract

Thirty years ago, I had just completed my Ph.D. under Varadarajan when, as part of a subsequent reading course, he and I considered a generalization of von Neumann entropy, here called a matrix entropy, computed by using the classical entropy function on the diagonals of density matrices. I had asked whether the value of von Neumann entropy was the maximum of the matrix entropy on a given unitary equivalence class. Varadarajan soon sketched a proof of this, which is presented here. It serves as a nice way to see classical entropy sitting in the von Neumann entropy context, and a reminder for this short note’s author of a pleasant time spent working with a remarkable scholar and teacher.