Tensor Rank and Stochastic Entanglement Catalysis for Multipartite Pure States

TL;DR

This paper investigates the tensor rank of symmetric multipartite states like the W state, providing improved estimates and revealing that multiple copies or catalysis can enable transformations impossible for single copies.

Contribution

It introduces new bounds on tensor ranks of multipartite states and demonstrates how multiple copies or catalysis enable entanglement transformations not possible otherwise.

Findings

Three copies of W_3 state have rank 15 or 16

Two copies of W_N state have rank 3N-2

Multiple copies enable transformations impossible for single copies

Abstract

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.