Building on a new weighted spectral geometric mean of positive definite matrices, this work defines a quantum divergence as the trace gap between the arithmetic mean and this spectral geometric mean, proves it satisfies the divergence axioms, and constructs the barycenter that minimizes a weighted sum of such divergences. Students learn how operator inequalities (Löwner order, trace, log-majorization with Rényi relative entropy) yield a well-defined divergence and a unique barycenter characterized by an explicit fixed-point equation.
Studies a recently introduced weighted spectral geometric mean Ft(A,B) of positive definite operators and matrices. The authors prove new inequalities for this mean in the Löwner order, operator norm…