The Tensor Rank of the Tripartite State ${\mathrm{\mid }W⟩}^{\otimes n}$}

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 $\mathrm{\mid }W⟩=\frac{1}{\sqrt{3}}(\mathrm{\mid }100⟩+\mathrm{\mid }010⟩+\mathrm{\mid }001⟩)$. Both an upper bound and a lower bound of this rank are derived. In particular, it is proven that the tensor rank of ${\mathrm{\mid }W⟩}^{\otimes 2}$ is seven, thus resolving a previously open problem. Some implications of this result are discussed in terms of transformation rates between ${\mathrm{\mid }W⟩}^{\otimes n}$ and multiple copies of the state $\mathrm{\mid }GHZ⟩=\frac{1}{\sqrt{2}}(\mathrm{\mid }000⟩+\mathrm{\mid }111⟩)$.