A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy

We consider the following trace function on n-tuples of positive operators: \Phi_p(A_1,A_2,...,A_n) = Trace (\sum_{j=1}^n A_j^p)^{1/p} and prove that it is jointly concave for 0<p\le 1 and convex for p=2. We then derive from this a Minkowski type inequality for operators on a tensor product of three Hilbert spaces, and show how this implies the strong subadditivity of quantum mechanical entropy. For p>2, \Phi_p is neither convex nor concave. We conjecture that \Phi_p is convex for 1<p<2, but our methods do not show this.