Inequalities for Trace Norms of 2 ? 2 Block Matrices

Junjian Yang, Huan Xu

Koenraad M. R. Audenaert

We convert a conjectured inequality from quantum information theory, due to He and Vidal, into a block matrix inequality and prove a very special case. Given matrices , , of the same size, let and be the block matrices and , respectively. Then, the conjectured inequality iswhere and denote the trace norm and the Hilbert–Schmidt norm, respectively. We prove this inequality for the already challenging case with .

Tsun-Ming Cheung, Hamed Hatami, Kaave Hosseini +3 more

We present a simple method based on a variant of Hölder’s inequality to lower-bound the trace norm of Boolean matrices. As the main result, we obtain an exponential separation between the randomized decision tree depth and the spectral norm (i.e. the Fourier $$L_1$$ -norm) of a Boolean function. This answers an open question of Cheung, Hatami, Hosseini, and Shirley (CCC 2023). As immediate consequences, we obtain the following results.

David Sutter

Alexey E. Rastegin

Christopher King

This paper derives an inequality relating the p-norm of a positive 2 x 2 block matrix to the p-norm of the 2 x 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. As an application in quantum information theory, the inequality is used to obtain some results concerning maximal p-norms of product channels.