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

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, ..., A_m) = (Tr[(∑_j=1^m A_j^p) ^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.