Duality versus dual flatness in quantum information geometry

Abstract

We investigate questions in quantum information geometry which concern the existence and nonexistence of dual and dually flat structures on stratified sets of density operators on finite-dimensional Hilbert spaces. We show that the set of density operators of a given rank admits dually flat connections for which one connection is complete if and only if this rank is maximal. We prove, moreover, that there is never a dually flat structure on the set of pure states. Thus any general theory of quantum information geometry that involves duality concepts must inevitably be based on dual structures which are nonflat.