Wigner–Yanase–Dyson Metrics Have Monotone Curvature
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…
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