Kubo-Mori Metric

The Kubo-Mori metric is the Riemannian metric on the space of invertible density matrices given by

where is the Fréchet derivative of the matrix logarithm at applied to , and denotes the matrix trace (Petz 1994). The tangent space consists of trace-zero Hermitian matrices. In a basis diagonalizing , the derivative multiplies the -entry of by the divided difference of the logarithm, with value when the eigenvalues coincide.

Petz's conjecture asks whether the scalar curvature is nondecreasing under mixing (Petz 1994). If the eigenvalue list of majorizes that of , the conjecture predicts that the scalar curvature at is at least that at . Dittmann (2000) reduced the conjecture to concavity of an auxiliary function of three positive variables. Sra (2026) reports strict concavity of that function and hence the conjectured curvature monotonicity. Sra credits GPT Pro with assistance in developing and repairing the proof.