bibkey: hiai2025spectralnonconcavity
authors: Fumio Hiai
year: 2025
title: Log-majorizations between quasi-geometric type means for matrices
doi: null
url: https://arxiv.org/abs/2510.04691v3
claim: Equations (2.3)--(2.4) define the original spectral mean and its powered quasi-extension; equations (2.5)--(2.6) define the separate TILDE family. Proposition 5.19(1) and Problem 5.20 concern trace joint concavity.
strata_touched:
  - D5/S3/Quantum/Hiai
license: citation-only
triage: anchor

The UNTILDE trace functional

Equation (2.3) defines the original spectral geometric mean using L=A^(-1) #(1/2) B and L^alpha A L^alpha. Its powered quasi-extension uses A^p and B^p followed by the outer spectral 1/p power. Thus its trace is S(alpha,p,A,B)=Re Tr[(L^alpha A^p L^alpha)^(1/p)], with L=A^(-p) #(1/2) B^p. All matrix powers are spectral powers. Equation (2.4) is this original family’s powered quasi-extension. Equations (2.5)–(2.6) define the separate weighted spectral family, denoted with a tilde in the source. The two definitions are not interchanged.

Proposition 5.19(1) gives necessary restrictions for concavity. Problem 5.20 reports that sufficient conditions are unknown apart from the midpoint alpha=1/2. These are source statements, rather than sources for the off-midpoint proof.

The repository’s off-midpoint obstruction uses real symmetric positive definite density matrices A_t=diag(1-t,t) and B_s=(1/2)[[1,1-2s],[1-2s,1]]. Its mathematical content is the unequal-scale endpoint t=s^2, the positive-power interpolation limit, and an affine Jensen violation for every p>0 and every 0<alpha<1 other than 1/2. The derivation is repository content; it is not attributed as a theorem of Hiai’s paper.

Vuong, Shudo and Osaka, arXiv:2609.00034v1, Theorem 4.1 covers the p=1 slice through F_f(A,B)=Tr[A f(A^(-1)#B)^2]. Its inspected statement has neither powered inputs nor the outer 1/p exponent. The all-p obstruction requires its own endpoint proof. This bounded source comparison does not certify global priority. No positive midpoint classification is asserted here.

Verified locator

https://arxiv.org/abs/2510.04691v3, equations (2.3)–(2.6), Proposition 5.19(1), and Problem 5.20.

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