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Wigner–Yanase–Dyson Metrics Have Monotone Curvature

A mix of human-written and AI-generated textHuman understanding: some partsmath.FA — Functional Analysismath-ph — Mathematical Physicsmath.DG — Differential Geometryquant-ph — Quantum Physics

Contributed by Suvrit Sra ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

Gibilisco and Isola conjectured that the scalar curvature of the Wigner–Yanase–Dyson metric with parameter pp on the invertible density matrices increases under mixing for all pp in some interval (1,1+ε)(1,1+\varepsilon); the case p↓1p\downarrow1 is Petz's conjecture for the Kubo–Mori metric (resolved in a companion paper). We prove that for p∈(1−ε,1+ε)p\in(1-\varepsilon,1+\varepsilon), with ε>0\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 HpH_p of three positive variables is concave, which we do by showing that −1/Hp-1/H_p is a Stieltjes transform along every line in the positive cone, uniformly in pp near 11. The value of ε\varepsilon is not explicit.

Provenance statement

The results in this paper were developed with an AI assistant (OpenAI's GPT Pro). None of the claims were obtained zero-shot: they emerged through sustained back-and-forth discussion, including failed attempts, counterexamples, audits, and repairs. The main text has not yet been vetted line by line by the author.

Tools used

OpenAI
GPT Pro

References

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Version history

  1. v1Submitted by Suvrit SraInitial depositCurrentOct 31, 2026