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math.FA — Functional Analysis

Wigner–Yanase–Dyson Metrics Have Monotone Curvature

Contributed by Suvrit Sra

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.

A mix of human-written and AI-generated textHuman understanding: some partsKubo-Mori metricStieltjes transformWigner-Yanase-Dyson metricdivided differencesmajorizationscalar curvature

Nov 02, 2026PDF ↗View work →

hexagon:2610.00717v1

math.FA — Functional Analysis

A Proof of Petz's Conjecture on the Scalar Curvature of the Kubo–Mori Metric

Contributed by Suvrit Sra

Petz (1994) conjectured that the scalar curvature of the Kubo–Mori metric on invertible density matrices increases as the state becomes more mixed. Dittmann (2000) reduced the conjecture to the concavity of an explicit symmetric function $h$ on $(0,\infty)^3$ . We prove that $h$ is strictly concave, and hence Petz's conjecture. The proof writes $h=-1/(8M)$ with $M$ positive and homogeneous of degree one, and its central idea is to show that $t\mapsto 1/M(X+tY)$ is a Stieltjes function for all $X,Y>0$ ; Cauchy–Schwarz and homogeneity then give concavity of $h$ on $(0,\infty)^3$ .

A mix of human-written and AI-generated textHuman understanding: all partsKubo-Mori metricMorozova-Chentsov kernelPetz conjectureStieltjes functionlogarithmic meanscalar curvaturestrict concavity

Nov 02, 2026PDF ↗View work →

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Wigner–Yanase–Dyson Metrics Have Monotone Curvature

A Proof of Petz's Conjecture on the Scalar Curvature of the Kubo–Mori Metric

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