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

Abstract

In this paper, we show matrix trace inequalities related to the Tsallis relative entropy of real order: For positive definite matrices ρ and σ, and each 0<α≤1Dα(ρ|σ)≤−Tr[ρ1−qqTαq(ρq|σq)] for all q≥α>0, where the Tsallis relative entropy Dα(ρ|σ) is defined by Dα(ρ|σ)=−Tr(ρ1−ασα−ρα) and the Tsallis relative operator entropy Tα(ρ|σ) is defined by Tα(ρ|σ)=ρ♯ασ−ρα, where ♯α is the matrix α-geometric mean. Moreover, we show estimates of the difference between two Tsallis relative entropies of real order.