Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Log-majorizations between quasi-geometric type means for matrices

Fumio Hiai

TL;DR

The paper investigates log-majorization relations between families of quasi-geometric matrix means $\\mathcal{M}_{\\alpha,p}$, spanning $R_{\\alpha,p}$, $G_{\\alpha,p}$, $SG_{\\alpha,p}$, $\\widetilde{S}G_{\\alpha,p}$, and $LE_{\\alpha}$, for $\\alpha>0$, $\\alpha\neq1$, and $p>0$. It develops a comprehensive framework linking log-majorization to norm inequalities, joint concavity/convexity of trace functionals, and quantum Renyi-type divergences, employing determinant invariance and anti-symmetric tensor powers to derive sufficiency/necessity results, with explicit characterizations for several pairs (notably between $SG_{\\alpha,p}$ and $R_{\\alpha,q}$) and partial results in others. The work further connects these matrix inequalities to monotonicity properties of quantum divergences under CPTP maps, yielding sandwiched inequalities that constrain joint concavity/convexity of the trace means. While complete characterizations remain open for certain pairs (e.g., $SG$ and $\\tilde{SG}$), the paper provides a cohesive set of results, identifies sharp necessary conditions, and outlines significant open problems, contributing to both matrix analysis and quantum information theory. The findings have potential implications for operator means, matrix norm inequalities, and quantum information divergences, informing both theory and applications in these domains.

Abstract

In this paper, for $α\in(0,\infty)\setminus\{1\}$, $p>0$ and positive semidefinite matrices $A$ and $B$, we consider the quasi-extension $\mathcal{M}_{α,p}(A,B):=\mathcal{M}_α(A^p,B^p)^{1/p}$ of several $α$-weighted geometric type matrix means $\mathcal{M}_α(A,B)$ such as the $α$-weighted geometric mean in Kubo--Ando's sense, the Rényi mean, etc. The log-majorization $\mathcal{M}_{α,p}(A,B)\prec_{\log}\mathcal{N}_{α,q}(A,B)$ is examined for pairs $(\mathcal{M},\mathcal{N})$ of those $α$-weighted geometric type means. The joint concavity/convexity of the trace functions $\mathrm{Tr}\,\mathcal{M}_{α,p}$ is also discussed based on theory of quantum divergences.

Log-majorizations between quasi-geometric type means for matrices

TL;DR

The paper investigates log-majorization relations between families of quasi-geometric matrix means , spanning , , , , and , for , , and . It develops a comprehensive framework linking log-majorization to norm inequalities, joint concavity/convexity of trace functionals, and quantum Renyi-type divergences, employing determinant invariance and anti-symmetric tensor powers to derive sufficiency/necessity results, with explicit characterizations for several pairs (notably between and ) and partial results in others. The work further connects these matrix inequalities to monotonicity properties of quantum divergences under CPTP maps, yielding sandwiched inequalities that constrain joint concavity/convexity of the trace means. While complete characterizations remain open for certain pairs (e.g., and ), the paper provides a cohesive set of results, identifies sharp necessary conditions, and outlines significant open problems, contributing to both matrix analysis and quantum information theory. The findings have potential implications for operator means, matrix norm inequalities, and quantum information divergences, informing both theory and applications in these domains.

Abstract

In this paper, for , and positive semidefinite matrices and , we consider the quasi-extension of several -weighted geometric type matrix means such as the -weighted geometric mean in Kubo--Ando's sense, the Rényi mean, etc. The log-majorization is examined for pairs of those -weighted geometric type means. The joint concavity/convexity of the trace functions is also discussed based on theory of quantum divergences.

Paper Structure

This paper contains 16 sections, 35 theorems, 167 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Log-majorizations
  4. $SG_{\alpha,p}\prec_{\log}R_{\alpha,q}$ and $R_{\alpha,q}\prec_{\log}SG_{\alpha,p}$
  5. $\widetilde{S}G_{\alpha,p}\prec_{\log}R_{\alpha,q}$ and $R_{\alpha,q}\prec_{\log}\widetilde{S}G_{\alpha,p}$
  6. Regarding $G_{\alpha,p}\prec_{\log}G_{\alpha,q}$, $SG_{\alpha,p}\prec_{\log}SG_{\alpha,q}$ and $\widetilde{S}G_{\alpha,p}\prec_{\log}\widetilde{S}G_{\alpha,q}$
  7. More log-majorizations
  8. Equality cases in norm inequalities derived from log-majorizations
  9. Joint concavity/convexity
  10. Joint concavity/convexity and monotonicity
  11. Quantum Rényi type divergences
  12. Joint concavity/convexity of trace functions
  13. Concluding remarks
  14. Equality case in norm inequality for $G_{\alpha,p}$
  15. Proof of Theorem \ref{['T-5.3']}
  16. ...and 1 more sections

Key Result

Proposition 2.1

Let $\alpha\in(0,\infty)\setminus\{1\}$ and $p>0$. Let $\mathcal{M}_{\alpha,p}$ be any of $G_{\alpha,p}$, $R_{\alpha,p}$, $SG_{\alpha,p}$, $\widetilde{S}G_{\alpha,p}$ and $LE_\alpha$.

Theorems & Definitions (72)

  • Proposition 2.1
  • proof
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • proof : Proof of Theorem \ref{['T-2.2']}
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Theorem 3.1
  • ...and 62 more