Positive definite matrices with Hermitian blocks and their partial traces

TL;DR

This paper proves a norm inequality for positive semi-definite matrices partitioned into Hermitian blocks, using Clifford algebra techniques, with implications for quantum states and a conjecture for complex Hilbert spaces.

Contribution

It introduces a novel decomposition approach for positive matrices and establishes a norm inequality involving partial traces and Hermitian block matrices.

Findings

The norm of the entire matrix is bounded by the sum of the norms of its diagonal blocks.

Partial trace operation increases norms of separable states on real Hilbert spaces.

The proof employs Clifford algebra generators and unitary congruences.

Abstract

Let $H$ be a positive semi-definite matrix partitioned in $\beta\times \beta$ Hermitian blocks, $H=[A_{s,t}]$, $1\le s,t,\le \beta$. Then, for all symmetric norms, {equation*} \| H \| \le \| \sum_{s=1}^{\beta} A_{s,s} \|. {equation*} The proof uses a nice decomposition for positive matrices and unitary congruences with the generators of a Clifford algebra. A few corollaries are given, in particular the partial trace operation increases norms of separable states on a real Hilbert space, leading to a conjecture for usual complex Hilbert spaces.