Frank Hansen

We study operator means more general than the Kubo-Ando means. They are given as minima of geodesically convex functions and allow uniquely defined extensions to any number of variables. Multivariate hyper-means is a class of means of this type bounded from below by the arithmetic mean. We extend the definition of a hyper-man in the bivariate case and discover bivariate means that are not restrictions of the multivariate means studied earlier. We introduce the notion of reduced relative quantum entropy and prove that it is convex. The result is used to give a simplified proof of a theorem of Lieb and Seiringer.

We investigate the geometric properties for a class of trace functions expressed in terms of the deformed logarithmic and exponential functions. We extend earlier results of Epstein, Hiai, Carlen and Lieb.

We investigate geometric properties of a class of trace functions expressed in terms of the deformed logarithmic and exponential functions. These trace functions and their properties may be of independent interest. We use them in particular to extend earlier results of Epstein, Hiai, Carlen and Lieb.

By using a variational principle we find a necessary and sufficient condition for an operator to majorise the parallel sum of two positive definite operators. This result is then used as a vehicle to create new operator inequalities involving the parallel sum.

Local quantum uncertainty and interferometric power were introduced by Girolami et al. as geometric quantifiers of quantum correlations. The aim of the present paper is to discuss their properties in a unified manner by means of the metric adjusted skew information defined by Hansen.