A Minkowski type trace inequality and strong subadditivity of quantum entropy II: Convexity and concavity

Eric A. Carlen, Elliott H. Lieb

Research output: Contribution to journalArticle

52 Scopus citations

Abstract

We revisit and prove some convexity inequalities for trace functions conjectured in this paper's antecedent. The main functional considered is Φ{p,q} (A1, A2, ..., Am) = (Tr[(∑j=1m Ajp) q/p])1/q for m positive definite operators A j . In our earlier paper, 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.

Original languageEnglish (US)
Pages (from-to)107-126
Number of pages20
JournalLetters in Mathematical Physics
Volume83
Issue number2
DOIs
StatePublished - Feb 1 2008

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Keywords

  • Concavity
  • Convexity
  • Entropy
  • Operator norms
  • Trace inequality

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