A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy II: Convexity and Concavity

2007/10/31 by Eric A. Carlen, Elliott H. Lieb · 8 citations
Computer Science · Mathematics · Physics and Astronomy · #Mathematical Inequalities and Applications #Optimization and Variational Analysis #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.OA #msc:15A90 #msc:47A63

paper · pdf · doi:10.1007/s11005-008-0223-1

published as Lett. Math. Phys., Vol. 83, No. 2, pp. 107-126 (2008) · Proof of a conjecture in math/0701352. Revised version replaces earlier draft. 18 pages, latex

arxiv created 2007/11/30 · openalex publication_date 2008/02/01 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We revisit and prove some convexity inequalities for trace functions conjectured in the earlier part I. The main functional considered is Φ_p,q(A₁,A₂,...,A_m) = (trace((∑_j=1^m A_j^p)^q/p))^1/q for m positive definite operators A_j. In part I we only considered the case q=1 and proved the concavity of Φ_p,1 for 0 < p ≤ 1 and the convexity for p=2. We conjectured the convexity of Φ_p,1 for 1< p < 2. Here we not only settle the unresolved case of joint convexity for 1 ≤ p ≤ 2, we are also able to include the parameter q≥ 1 and still retain the convexity. Among other things this leads to a definition of an L^q(L^p) norm for operators when 1 ≤ p ≤ 2 and a Minkowski inequality for operators on a tensor product of three Hilbert spaces -- which leads to another proof of strong subadditivity of entropy. We also prove convexity/concavity properties of some other, related functionals.

Citations

• Concavity of certain maps on positive definite matrices and applications to Hadamard products
• The Wigner-Yanase Entropy is not Subadditive
• Structure of States Which Satisfy Strong Subadditivity of Quantum Entropy with Equality

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