Matrix trace inequalities related to the Tsallis relative entropies of real order, II

Abstract

In this paper, as a continuation of two previous papers, we show matrix trace inequalities related to two Tsallis relative entropies of all real order: For density matrices ρ and σ, and each α∈R﹨{0}, the Tsallis relative entropy Dα(ρ|σ) is defined by Dα(ρ|σ)=−Tr(ρ1−ασα−ρα) and NTα(ρ|σ) is defined by NTα(ρ|σ)=−Tr[ρ♮ασ−ρα], where ♮α is defined by ρ♮ασ=ρ1/2(ρ−1/2σρ−1/2)αρ1/2. Then we show the order relation between two Tsallis relative entropies Dα and NTα of all real order α∈R and a 1-parameter extension of the path connecting them of all real order. Moreover, we show estimates of the difference between two Tsallis relative entropies of all real order.