  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, K\"onig 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 alternative proof of an explicit formula due to
Chiribella, which gives the density matrices of the constructed state as a
function of those of the original state.
