New Weighted Spectral Geometric Mean and Quantum Divergence
Miran Jeong, Sejong Kim, Tin-Yau Tam
TL;DR
The paper introduces a new weighted spectral geometric mean $F_t(A,B)$ that interpolates between $A$ and $B$ with $F_0(A,B)=A$, $F_1(A,B)=B$, and $F_{1/2}(A,B)=A \natural B$, and situates it within the landscape of operator means. It provides operator inequalities in Löwner order, operator norm, and trace, and proves a log-majorization relationship $F_t(A,B) \prec_{\log} Q_{t,z}(A,B)$ connecting the mean to Rényi-type entropy functions. A quantum divergence $\Phi(A,B)=\mathrm{tr}[A \nabla_{t} B - F_t(A,B)]$ is defined, shown to be nonnegative and invariant under unitary conjugation and tensoring, and used to establish a strictly convex objective whose unique minimizer is the barycenter $\mathfrak{B}_t(\omega; A_1,\dots,A_n)$ with a closed Euler-type characterization. The results illuminate the structure of the new mean, link it to entropy-type measures, and enable barycentric optimization in quantum-information-geometric settings with potential practical implications for multi-variate operator means and information geometry.
Abstract
A new class of weighted spectral geometric means has recently been introduced. In this paper, we present its inequalities in terms of the Löwner order, operator norm, and trace. Moreover, we establish a log-majorization relationship between the new spectral geometric mean, and the Rényi relative operator entropy. We also give the quantum divergence of the quantity, given by the difference of trace values between the arithmetic mean and new spectral geometric mean. Finally, we study the barycenter that minimizes the weighted sum of quantum divergences for given variables.