Sandwiched Rényi relative entropy on density operators

Abstract

Relative entropies play important roles in classical and quantum information theory. In this paper, we discuss the sandwiched Rényi relative entropy for [Formula: see text] on [Formula: see text] (the cone of positive trace-class operators acting on an infinite-dimensional complex Hilbert space [Formula: see text]) and characterize all surjective maps preserving the sandwiched Rényi relative entropy on [Formula: see text]. Such transformations have the form [Formula: see text] for each [Formula: see text], where [Formula: see text] and [Formula: see text] is either a unitary or an anti-unitary operator on [Formula: see text]. Particularly, all surjective maps preserving sandwiched Rényi relative entropy on [Formula: see text] (the set of all quantum states on [Formula: see text]) are necessarily implemented by either a unitary or an anti-unitary operator.