The $p$-Bures-Wasserstein distance via $3\times 3$ block matrices (with application to $p$-Bures-Wasserstein barycenters)

We study the $p$-Bures-Wasserstein distance $D_p(A,B)=(\mathrm{t}\mathrm{r}{A}^p+\mathrm{t}\mathrm{r}{B}^p-2\mathrm{\parallel }{A}^{1\mathrm{/}2}{B}^{1\mathrm{/}2}{\mathrm{\parallel }}_p^p{)}^{1\mathrm{/}2}$ on positive semidefinite matrices; it equals the Bures-Wasserstein distance for $p=1$, and the Hilbert-Schmidt distance for $p=2$. We prove that $D_p$ is a metric for every $1\le p\le 2$, by proving a stronger 2014 conjecture of Lin and van den Driessche on $3\times 3$ block matrices. Our main tool is a partition inequality for Schatten norms, proved by complex interpolation. We also characterize Fréchet means for $D_p$ by a fixed-point equation and prove geometric convergence of the associated iteration.