  In this work we introduce the concept of Bures-Wasserstein barycenter $Q_*$,
that is essentially a Fr\'echet mean of some distribution $\mathbb{P}$
supported on a subspace of positive semi-definite Hermitian operators
$\mathbb{H}_{+}(d)$. We allow a barycenter to be restricted to some affine
subspace of $\mathbb{H}_{+}(d)$ and provide conditions ensuring its existence
and uniqueness. We also investigate convergence and concentration properties of
an empirical counterpart of $Q_*$ in both Frobenius norm and Bures-Wasserstein
distance, and explain, how obtained results are connected to optimal
transportation theory and can be applied to statistical inference in quantum
mechanics.
