A matrix convexity approach to some celebrated quantum inequalities

2008/02/10 by Edward G. Effros · 7 citations
Mathematics · Physics and Astronomy · #Functional Equations Stability Results #Mathematical Inequalities and Applications #Statistical Mechanics and Entropy #math-ph #math.MP #msc:46L89 #msc:46N50 #msc:81Q99

paper · pdf · doi:10.1073/pnas.0807965106

8 pages

arxiv created 2008/02/10 · openalex publication_date 2009/01/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Some of the important inequalities associated with quantum entropy are immediate algebraic consequences of the Hansen-Pedersen-Jensen inequalities. A general argument is given using matrix perspectives of operator convex functions. A matrix analogue of Maréchal's extended perspectives provides additional inequalities, including a p + q < or = 1 result of Lieb.

Citations

• Multiplicativity of Completely Bounded p-Norms Implies a New Additivity Result
• Concavity of certain maps on positive definite matrices and applications to Hadamard products
• JENSEN'S OPERATOR INEQUALITY
• Matrix Analysis

Cited by

• Remembrances of Edward G. Effros
• Tomita-Takesaki Modular Theory vs. Quantum Information Theory
• User-Friendly Tail Bounds for Sums of Random Matrices
• Modulus of convexity for operator convex functions
• Convexity of quasi-entropy type functions: Lieb's and Ando's convexity theorems revisited
• A UNIFIED TREATMENT OF CONVEXITY OF RELATIVE ENTROPY AND RELATED TRACE FUNCTIONS, WITH CONDITIONS FOR EQUALITY
• Edward George Effros [wikipedia]

Related

• New Perspectives and some Celebrated Quantum Inequalities
• Choi-Davis-Jensen's inequality without convexity
• Reduced relative quantum entropy
• Some Operator Inequalities via Convexity
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