Remarks on the quantum de Finetti theorem for bosonic systems

The quantum de Finetti theorem asserts that the k-body density matrices of an 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 et al. [8] 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 [7], which gives the density matrices of the constructed state as a function of those of the original state.