Research Preprints · Mathematical physics
Wigner–Yanase–Dyson Metrics Have Monotone Curvature
Abstract
Gibilisco and Isola conjectured that the scalar curvature of the Wigner–Yanase–Dyson metric with parameter $p$ on the invertible density matrices increases under mixing for all $p$ in some interval $(1,1+\varepsilon)$; the case $p\downarrow1$ is Petz's conjecture for the Kubo–Mori metric (resolved in a companion paper). We prove that for $p\in(1-\varepsilon,1+\varepsilon)$, with $\varepsilon>0$ independent of the dimension, the scalar curvature is a concave function of the eigenvalues and hence increases under mixing. By Dittmann's formula it suffices to prove that one symmetric function $H_p$ of three positive variables is concave, which we do by showing that $-1/H_p$ is a Stieltjes transform along every line in the positive cone, uniformly in $p$ near $1$. The value of $\varepsilon$ is not explicit.
Versions
Version numbers are this site's own; arXiv numbering, where present, is separate. Every released file carries an OpenTimestamps proof made on its release day.
| Version | Released | Length | File | Proof | arXiv |
|---|---|---|---|---|---|
| v1 | 9 October 2026 | 20 pp | .ots | — |
Cite
@misc{sra2026wyd-v1,
author = {Suvrit Sra},
title = {{Wigner--Yanase--Dyson Metrics Have Monotone Curvature}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/wyd/v1}},
note = {Version v1, released 9 October 2026},
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pip install opentimestamps-client curl -O https://suvrit.github.io/papers/wyd/wyd-v1.pdf -O https://suvrit.github.io/papers/wyd/wyd-v1.pdf.ots ots verify wyd-v1.pdf.ots