DECOMPOSITION AND PARTIAL TRACE OF POSITIVE MATRICES WITH HERMITIAN BLOCKS
Jean-Christophe Bourin, Eunyoung Lee · International Journal of Mathematics · 2013
Let H = [As, t] be a positive definite matrix written in β × β Hermitian blocks and let Δ = A1, 1 + ⋯ + Aβ, β be its partial trace. Assume that β = 2p for some p ∈ ℕ. Then, up to a direct sum operation, H is the average of β matrices isometrically congruent to Δ. A few corollaries are given, related to important inequalities in quantum information theory such as the Nielsen–Kempe separability criterion.