Matrix monotonicity and self-concordance: how to handle quantum entropy in optimization problems
The authors propose a self-concordant barrier for the so-called quantum or von Neumann entropy which is the function \(\text{Tr}(X\, \text{Ln}(X))\) where \(X\) is a positive semi-definite complex Hermitian matrix. This result opens the way for including many optimization problems (arising, e.g., in quantum information theory, quantum statistical physics, information geometry) into a general interior point polynomial framework. The authors consider a broad class of functions, namely, primitives of matrix monotone functions on positive semi-axis. It turns out that the standard \(\ln \det\) barrier is compatible (in the sense of the theory of self-concordant functions) with functions from this class. The authors also establish the self-concordance of associated barriers (along with barrier parameter estimate) on the epigraphs of functions of this class.
scientific article; zbMATH DE number 6819050