A Generalized Matrix Power Mean and a New Quantum Hellinger Divergence

In this paper, we study a matrix equation involving the matrix geometric mean \(A\natural_{t}B=A^{1/2}(A^{-1/2}BA^{-1/2})^{t}A^{1/2},\) \(t\in(1,2)\) . We study several properties and inequalities for the unique solution of such an equation. We also use the weighted geometric mean \(A\natural_{t}B\) to define a new quantum Hellinger divergence and show that the new quantum divergence satisfies the Data Processing Inequality.