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 language | English (US) |
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Pages (from-to) | 107-126 |
Number of pages | 20 |
Journal | Letters in Mathematical Physics |
Volume | 83 |
Issue number | 2 |
DOIs | |
State | Published - Feb 2008 |
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics
Keywords
- Concavity
- Convexity
- Entropy
- Operator norms
- Trace inequality