# Statistical inference for Bures-Wasserstein barycenters

## TL;DR

This paper introduces the Bures-Wasserstein barycenter as a Fréchet mean of positive semi-definite operators, exploring its existence, uniqueness, and statistical properties with applications to quantum mechanics.

## Contribution

It defines the Bures-Wasserstein barycenter for operators, establishes conditions for its existence and uniqueness, and analyzes its convergence and concentration properties in statistical inference.

## Key findings

- Existence and uniqueness conditions for Bures-Wasserstein barycenters.
- Convergence and concentration results for empirical barycenters.
- Connections to optimal transportation and quantum mechanics applications.

## Abstract

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.