Some geometric properties of matrix means with respect to different metrics

The matrix Heinz mean and related divergence

“…He showed that the matrix power mean µ p (t; X, Y ) = (tX p + (1 − t)Y p ) 1 p satisfies the in-betweenness property. This property was investigated by the first author and co-authors in [4][5][6][7][8][9]. To finish this paper, we show that the matrix power mean µ p (t; X, Y ) also satisfies the in-betweenness property with respect to the new divergence in the previous section.…”

“…Finally, we also show that the matrix power mean µ p (t, A, B) satisfies the in-betweenness property (Theorem 3.2). On the in-betweenness property we refer the readers to [2,[4][5][6][7][8][9]12].…”

Relative operator entropy properties related to some weighted metrics

“…Inspired by some concepts defined in [4], the authors undertook recently a study for relative operator entropy and for Tsallis operator entropy with respect to various Hellinger metric [12,13].…”

Quantum Hellinger distances revisited

“…The Hellinger divergence is a special Csiszár-Morimoto f -divergence [10,22] generated by the convex function f (x) = x − 1 2 , and it has several possible counterparts in quantum information theory. One of them is the squared Bures distance or Wasserstein metric, see, e.g., the most recent works of Bhatia et al [8], Dinh et al [11], and Molnár [21]. Another important quantum analogue of the classical Hellinger divergence has been investigated in [7], namely the quantity…”