Positive definite matrices with Hermitian blocks and their partial traces

2012/08/31 by Bourin, Jean-Christophe, Lee, Eun-Young, Lin, Minghua
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1208.6494

Abstract

Let H be a positive semi-definite matrix partitioned in β× β Hermitian blocks, H=[A_s,t], 1≤ s,t,≤ β. Then, for all symmetric norms, equation* ‖ H ‖ ≤ ‖ ∑_s=1^β 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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