Inequalities for Trace Norms of 2 ? 2 Block Matrices

Abstract

This paper derives an inequality relating the p-norm of a positive 2×2 block matrix to the p-norm of the 2×2 matrix obtained by replacing each block by its p-norm. The inequality had been known for integer values of p, so the main contribution here is the extension to all values p≥1. In a special case the result reproduces Hanner’s inequality. A weaker inequality which applies also to non-positive matrices is presented. As an application in quantum information theory, the inequality is used to obtain some results concerning maximal p-norms of product channels.