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

A mix of human-written and AI-generated textHuman understanding: all 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

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 hh on (0,∞)3(0,\infty)^3. We prove that hh is strictly concave, and hence Petz's conjecture. The proof writes h=−1/(8M)h=-1/(8M) with MM positive and homogeneous of degree one, and its central idea is to show that t↦1/M(X+tY)t\mapsto 1/M(X+tY) is a Stieltjes function for all X,Y>0X,Y>0; Cauchy–Schwarz and homogeneity then give concavity of hh on (0,∞)3(0,\infty)^3.

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.

Tools used

OpenAI
GPT Pro

References

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Version history

  1. v1Submitted by Suvrit SraInitial depositCurrentOct 31, 2026