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

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Yu, Nengkun
Chitambar, Eric
Guo, Cheng
Duan, Runyao
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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 $\ket{W}=\tfrac{1}{\sqrt{3}}(\ket{100}+\ket{010}+\ket{001})$. Both an upper bound and a lower bound of this rank are derived. In particular, it is proven that the tensor rank of $\ket{W}^{\otimes 2}$ is seven, thus resolving a previously open problem. Some implications of this result are discussed in terms of transformation rates between $\ket{W}^{\otimes n}$ and multiple copies of the state $\ket{GHZ}=\tfrac{1}{\sqrt{2}}(\ket{000}+\ket{111})$.
Comment: Comments: 3 pages (Revtex 4). Minor corrections to Theorem 1. Presentation refined. Main results unchanged. Comments are welcome
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Quantum Physics
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