bibkey: hiai2015meanconcavity
authors: Fumio Hiai
year: 2015
title: Concavity of certain matrix trace and norm functions. II
doi: null
url: https://arxiv.org/abs/1507.00853v3
claim: Lemma 3.3 supplies the classical joint trace concavity of positive-map operator means with the outer reciprocal power.
strata_touched:
  - D5/S3/Quantum/Hiai
license: citation-only
triage: anchor

Mean-trace concavity

Lemma 3.3 assumes strictly positive linear maps, an operator mean, exponents in [0,1] with at least one nonzero, and an operator-monotone scalar function. At equal exponents p>0, the identity scalar function and the same invertible congruence map X -> Z adjoint X Z, it gives joint concavity of Re Tr((Z adjoint (B^p #theta A^p) Z)^(1/p)). The genuine mean has its base in the first argument: X #theta Y = X^(1/2)(X^(-1/2) Y X^(-1/2))^theta X^(1/2). The local operatorPowerMean has its base in its second argument instead.

The connected Hilbert-Schmidt derivation uses the classical arbitrary-vector Lieb lift, genuine-mean joint concavity and both-input order monotonicity, the common-right-factor identity, and forward attained trace Young. The triple trace pairing has arbitrary Z. Its strictly positive optimizing matrix and the final trace-power result require invertible Z. Noncommuting inputs are retained: only left/right actions or powers of one input commute.

The source result is classical; it is not the open Problem 5.18 in arXiv:2510.04691v3. That original problem also requires inverse-geometric identification, the original parameter assembly, and a common-epsilon-I positive semidefinite limit preserving the support constraint s(A)>=s(B). The paper is cited; no paper proof text is transcribed.

Verified locator

https://arxiv.org/abs/1507.00853v3, Lemma 3.3.