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The quantum de Finetti theorem asserts that the k-body density matrices of a N -body bosonic state approach a convex combination of Hartree states (pure tensor powers) when N is large and k fixed. In this note we review a construction due to Christandl, Mitchison, König and Renner  valid for finite dimensional Hilbert spaces, which gives a quantitative version of the theorem. We first propose a variant of their proof that leads to a slightly improved estimate. Next we provide an alternativedoi:10.1093/amrx/abu006 fatcat:tuedebblerey7cpbncdslc3ljy