A family of monotone quantum relative entropies

We study here the elementary properties of the relative entropyH(A,B) = Tr[φ(A)−φ(B)−φ(B)(A−B)] for φ a convex function and A,B bounded self-adjoint operators. In particular, we prove that this relative entropy is monotone if and only if φ is operator monotone. We use this to appropriately define H(A,B) in infinite dimension. In this paper we introduce a relative entropy H(A,B) of two bounded operators A and B on a Hilbert space and discuss its elementary properties. Our relative entropy H(A,B) could be useful in many practical situations including quantum information theory. In a separate work [9], we will use it to discuss the orbital stability of certain stationary states for the Hartree equation of an interacting gas containing infinitely many quantum particles. For simplicity we work on the interval I = [0, 1] but all our results can be generalized to any finite interval. We consider two N×N hermitian matrices A,B whose spectrum is a subset of I. For φ ∈ C([0, 1],R) ∩C1((0, 1),R) a convex function, we introduce the following relative entropy H(A,B) = Tr ( φ(A) − φ(B)− φ(B)(A−B) ) . (1) We remind the reader of Klein’s lemma. Lemma 1 (Klein [12, Prop. 3.16], [14, Thm 2.5.2]). If fk, gk : [a, b] → R are K measurable functions such that ∑K k=1 fk(x)gk(y) > 0 for all x, y ∈ [a, b], then we have ∑K k=1Tr (