The Tensor Rank of the Tripartite State W⊗ ⁿ

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Tensor rank refers to the number of product states needed to express a given multipartite quantum state. Its non-additivity as an entanglement measure has recently been observed. In this note, we estimate the tensor rank of multiple copies of the tripartite state W=1√3(100+010+001). Both an upper bound and a lower bound of this rank are derived. In particular, it is proven that the tensor rank of W⊗ ² is seven, thus resolving a previously open problem. Some implications of this result are discussed in terms of transformation rates between W⊗ ⁿ and multiple copies of the state GHZ=1√2(000+111).