Weighted Two-Parameter Quantum Divergences Generated by Matrix Power-Mean Gaps

Abstract

Let ${\mathbb{P}}_n$ be the cone of $n\times n$ complex positive definite matrices. For $0<\theta <1$ and $r\ge 0$, let $M_r^{(\theta )}$ denote the weighted matrix power mean, with the logarithmic endpoint at $r=0$, defined by $$M_r^{(\theta )}(A,B)=\{\begin{array}{ll}((1-\theta )A^r+\theta B^r{)}^{1/r},& r>0,\\ \exp ((1-\theta )\log A+\theta \log B),& r=0.\end{array}$$ For $p>q\ge 0$, we introduce $${\Phi }_{p,q}^{(\theta )}(A,B)=\mathrm{Tr}[M_p^{(\theta )}(A,B)-M_q^{(\theta )}(A,B)].$$ We prove that ${\Phi }_{p,q}^{(\theta )}$ is a quantum divergence in the sense of Bhatia-Gaubert-Jain. Moreover, for $$0\le q\le 1\le p\le 2,p>q,$$ it is jointly convex and satisfies the data processing inequality for completely positive trace-preserving maps, with the output side interpreted by the regularized boundary convention whenever it is singular. The construction recovers Hellinger-type and Jensen-Shannon-type quantities as special cases or limits. We also compute the induced local quadratic form and provide scalar and matrix obstructions showing that the convexity and data-processing range cannot be extended to all parameters.