Counterexamples for the Convexity of Certain Matricial Inequalities

In [CL99] Carlen and Lieb considered Minkowski type inequalities in the context of operators on a Hilbert space. More precisely, they considered the homogenous expression fpq(x1, ..., xn) = ( tr ( ( n ∑ k=1 x q k) p/q ))1/p defined for positive matrices. The concavity for q = 1 and p < 1 yields strong subadditivity for quantum entropy. We discuss the convexity of fpq and show that, contrary to the commutative case, there exists a q0 > 1 such that f1q is not convex for all 1 < q < q0. This is achieved by constructing a family of interesting channels on 2× 2 matrices.