When n--k systems of an n-partite permutation-invariant state are traced out, the resulting state can be approximated by a convex combination of tensor product states. This is the quantum de Finetti theorem. In this paper, we summarise the results that we have obtained in section II of : we show that an upper bound on the trace distance of this approximation is given by , where d is the dimension of the individual system, thereby improving previously known bounds. Our result follows from a more general approximation theorem for representations of the unitary group: Consider a pure state that lies in the irreducible representation of the unitary group , for highest weights μ, ν and μ+ν. Let ξμ be the state obtained by tracing out Uν. Then ξμ is close to a convex combination of the coherent states Uμ(g)|vμ〉, where and |vμ〉 is the highest weight vector in Uμ.
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