In this work we introduce the concept of Bures-Wasserstein barycenter \(Q_*\), that is essentially a Fr\'echet mean of some distribution \(\P\) supported on a subspace of positive semi-definite Hermitian operators \(\H_{+}(d)\). We allow a barycenter to be restricted to some affine subspace of \(\H_{+}(d)\) and provide conditions ensuring its existence and uniqueness. We also investigate convergence and concentration properties of an empirical counterpart of \(Q_*\) in both Frobenious 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.
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