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

TL;DR

This paper investigates the tensor rank of multiple copies of the tripartite W state, establishing bounds and proving that two copies have a tensor rank of seven, which advances understanding of multipartite entanglement.

Contribution

It provides the first exact tensor rank for two copies of the W state and discusses implications for entanglement transformation rates.

Findings

Tensor rank of $ ext{W}^{ ext{tensor} 2}$ is exactly 7.

Derived bounds for tensor rank of multiple W states.

Discussed implications for transformations between W and GHZ states.

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 $\ket{GHZ}=\tfrac{1}{\sqrt{2}}(\ket{000}+\ket{111})$.