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PreprintCC BY 4.0arXiv

Alternative-mean trace divergences: geometry, data processing, and barycenters

  • Trung Dung Vuong
  • Hiroki Shudo
  • Hiroyuki Osaka
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Abstract

Let f:(0,∞)→(0,∞) be a nontrivial normalized operator monotone function and set s=f'(1). We introduce the alternative-mean trace functional altPhif(A,B) :=Tr(A∇s B) −Tr ( f(A−1♯ B) A f(A−1♯ B) ). on the positive definite cone. We prove that altPhif is a quantum divergence in the sense of Bhatia--Gaubert--Jain whose diagonal Hessian induces a positive multiple of the Bures--Wasserstein Riemannian metric. We also establish the sharp comparison s(1−s)dBW(A,B)2 ≤ altPhif(A,B) ≤ (1−s+s2)dBW(A,B)2.

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Cite

BibTeXRIS
BibTeX
@misc{vuong2026alternativemean,
  title = {{Alternative-mean trace divergences: geometry, data processing, and barycenters}},
  author = {Trung Dung Vuong and Hiroki Shudo and Hiroyuki Osaka},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00034},
  eprint = {2609.00034},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00034},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Alternative-mean trace divergences: geometry, data processing, and barycenters
AU  - Trung Dung Vuong
AU  - Hiroki Shudo
AU  - Hiroyuki Osaka
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.00034
UR  - https://arxiv.org/abs/2609.00034
AB  - Let f:(0,∞)→(0,∞) be a nontrivial normalized operator monotone function and set s=f'(1). We introduce the alternative-mean trace functional altPhi_f(A,B) :=Tr(A∇_s B) −Tr ( f(A^−1♯ B) A f(A^−1♯ B) ). on the positive definite cone. We prove that altPhi_f is a quantum divergence in the sense of Bhatia--Gaubert--Jain whose diagonal Hessian induces a positive multiple of the Bures--Wasserstein Riemannian metric. We also establish the sharp comparison s(1−s)d_BW(A,B)^2 ≤ altPhi_f(A,B) ≤ (1−s+s^2)d_BW(A,B)^2.
ER  -