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arXiv · 0910.0986

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

Nengkun Yu·Eric Chitambar·Cheng Guo·Runyao Duan
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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 $\ket{W}=\tfrac{1}{\sqrt{3}}(\ket{100}+\ket{010}+\ket{001})$. 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 $\ket{GHZ}=\tfrac{1}{\sqrt{2}}(\ket{000}+\ket{111})$.

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Nengkun Yu, Eric Chitambar, Cheng Guo, Runyao Duan. 2009-10-06. The Tensor Rank of the Tripartite State $\ket{W}^{\otimes n}$}. https://doi.org/10.1103/physreva.81.014301

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