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By exploiting the permutation symmetry of Dick states, we derive closed analytical expressions of Schmidt decompositions for {\it all} possible bipartitions of a system described by this kind of state.
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We could have defined the more general state | ϕ n ( k ) ⟩ = ( 1 − a 2 ) 1 2 | D n ( k ) ⟩ + ( a / ℓ ) ∑ q = 1 ℓ P ^ q ( n ) | 0 … 0 ⏟ n − k 1 … 1 ⏟ k ⟩ . |\phi_{n}^{(k)}\rangle=(1-a^{2})^{\frac{1}{2}}|D_{n}^{(k)}\rangle+(a/\sqrt{\ell})\sum_{q=1}^{\ell}\hat{P}_{q}^{(n)}|\underbrace{0...0}_{n-k}\underbrace{1...1}_{k}\rangle. However, for an increasing number of terms the level of asymmetry diminishes as, for ℓ = ( n k ) \ell=\left.n\choose k\right. we would get | ϕ n ( k ) ⟩ = | D n ( k ) ⟩ |\phi_{n}^{(k)}\rangle=|D_{n}^{(k)}\rangle . So the state used in the body of the paper, with ℓ = 1 \ell=1 is the most demanding for the witness
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