Metric adjusted skew information: Convexity and restricted forms of superadditivity

We give a truly elementary proof of the convexity of metric adjusted skew information following an idea of Effros. We extend earlier results of weak forms of superadditivity to general metric adjusted skew informations. Recently, Luo and Zhang introduced the notion of semi-quantum states on a bipartite system and proved superadditivity of the Wigner-Yanase-Dyson skew informations for such states. We extend this result to general metric adjusted skew informations. We finally show that a recently introduced extension to parameter values $ 1<p\le 2 $ of the WYD-information is a special case of (unbounded) metric adjusted skew information.