Physics – Quantum Physics
Scientific paper
2010-03-16
Phys. Rev. Lett.105, 200501 (2010)
Physics
Quantum Physics
PhysRevLett.105.200501
Scientific paper
10.1103/PhysRevLett.105.200501
The tensor rank (also known as generalized Schmidt rank) of multipartite pure states plays an important role in the study of entanglement classifications and transformations. We employ powerful tools from the theory of homogeneous polynomials to investigate the tensor rank of symmetric states such as the tripartite state $\ket{W_3}=\tfrac{1}{\sqrt{3}}(\ket{100}+\ket{010}+\ket{001})$ and its $N$-partite generalization $\ket{W_N}$. Previous tensor rank estimates are dramatically improved and we show that (i) three copies of $\ket{W_3}$ has rank either 15 or 16, (ii) two copies of $\ket{W_N}$ has rank $3N-2$, and (iii) $n$ copies of $\ket{W_N}$ has rank O(N). A remarkable consequence of these results is that certain multipartite transformations, impossible even probabilistically, can become possible when performed in multiple copy bunches or when assisted by some catalyzing state. This effect is impossible for bipartite pure states.
Chen Lin
Chitambar Eric
Duan Runyao
Ji Zhengfeng
Winter Andreas
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