Research Preprints · Matrix analysis

The $p$-Bures–Wasserstein distance via $3\times3$ block matrices (with application to $p$-Bures–Wasserstein barycenters)

Suvrit Sra · TU Munich, Department of Mathematics, School of CIT

Version v1 · released 11 October 2026 · 20 pages · The $p$-Bures–Wasserstein distance and the Lin–van den Driessche conjecture

PDFarXiv:2610.15889v1GitHubTimestamp

Abstract

We study the $p$-Bures–Wasserstein distance $D_p(A,B)=(\mathrm{tr}\,A^p+\mathrm{tr}\,B^p-2\|A^{1/2}B^{1/2}\|_p^p)^{1/2}$ 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 $1\le p\le2$, by proving a stronger 2014 conjecture of Lin and van den Driessche on $3\times3$ 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.

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Cite

@misc{sra2026pbures-v1,
  author        = {Suvrit Sra},
  title         = {{The $p$-Bures--Wasserstein distance via $3\times3$ block matrices (with application to $p$-Bures--Wasserstein barycenters)}},
  year          = {2026},
  month         = oct,
  howpublished  = {\url{https://arxiv.org/abs/2610.15889v1}},
  eprint        = {2610.15889v1},
  archiveprefix = {arXiv},
  note          = {Also released as pbures v1 (11 October 2026) at \url{https://github.com/suvrit/preprints/tree/main/preprints/pbures/v1}},
}

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