Tensor Rank and Stochastic Entanglement Catalysis for Multipartite Pure States

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 vertical bar W-3 > = 1/root 3(vertical bar 100 > + vertical bar 010 > + vertical bar 001 >) and its N-partite generalization vertical bar W-N >. Previous tensor rank estimates are dramatically improved and we show that (i) three copies of vertical bar W-3 > have a rank of either 15 or 16, (ii) two copies of vertical bar W-N > have a rank of 3N - 2, and (iii) n copies of vertical bar W-N > have a rank of 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.