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arXiv · 1208.6494

Positive definite matrices with Hermitian blocks and their partial traces

Jean-Christophe Bourin·Eun-Young Lee·Minghua Lin
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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.

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Jean-Christophe Bourin, Eun-Young Lee, Minghua Lin. 2012-08-31. Positive definite matrices with Hermitian blocks and their partial traces. https://arxiv.org/abs/1208.6494

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