Convexity of quasi-entropy type functions: Lieb's and Ando's convexity theorems revisited

Given a positive function $f$ on $(0,\mathrm{\infty })$ and a non-zero real parameter $\theta$, we consider a function $I_f^\theta(A,B,X)=Tr X^*(f(L_AR_B^{-1})R_B)^\theta(X)$inthreematrices$A,B>0$and$X$. In the literature $\theta =\pm 1$ has been typical. The concept unifies various quantum information quantities such as quasi-entropy, monotone metrics, etc. We characterize joint convexity/concavity and monotonicity properties of the function $I_f^{\theta }$, thus unifying some known results for various quantum quantities.