Order automorphisms on positive definite operators and a few applications

We first determine the order automorphisms of the set of all positive definite operators with respect to the usual order and to the so-called chaotic order. We then apply those results to the following problems: (1) description of all bijective transformations on the space of nonsingular density operators (quantum states) which preserve the Umegaki or the Belavkin–Staszewski relative entropy; (2) characterization of the logarithmic product as the essentially unique binary operation on the set of positive definite operators that makes it an ordered commutative group with respect to the chaotic order.