A trace inequality arising from quantum information theory
The author introduces another informational quantity for real-valued (continuous) functions \(f\) and \(g\): \[ S_{f,g}(A,x)=T_r f(A)Xg(A)X-T_r f(A)g(A)X^2 \] for selfadjoint matrices \(A\) and \(X\) which is an extension of \textit{E. H. Lieb}'s skew information [Adv. Math. 11, 267--288 (1973; Zbl 0267.46055)]. Conditions for \(f\) and \(g\) such that \(S_{f,g}(A,x)\) should be negative or positive, are given. As applications, the problem posed by \textit{K. Yanagi, S. Furuichi} and \textit{K. Kuriyama} [Linear Algebra Appl. 395, 351--359 (2005; Zbl 1067.94006)] from quantum information theory is settled.
scientific article; zbMATH DE number 2173021