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

Abstract

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

Comments: 3 pages (Revtex 4). Minor corrections to Theorem 1. Presentation refined. Main results unchanged. Comments are welcome