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 Phi(p,q)(A(1), A(2),..., A(m)) = (Tr[(Sigma(m)(j=1)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 Phi(p,1) for 0 < p <= 1 and the convexity for p = 2. We conjectured the convexity of Phi(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.