A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy

and prove that it is jointly concave for 0 < p ≤ 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, Φp is neither convex nor concave. We conjecture that Φp is convex for 1 < p < 2, but our methods do not show this. ∗ Work supported by U.S. National Science Foundation grant no. DMS 923097 ∗∗ Work supported by U.S. National Science Foundation grant no. PHY95–19433–A01. Copyright c ©1997 in image and content by the authors. Reproduction of this article in its entirety by any means is permitted.