A new bound on the rank of tensor product of W-states

Abstract

A W-state is an order d symmetric tensor of the form W_d=x^{d-1}y. We prove that the partially symmetric rank of W_{d_1}\otimes \cdots \otimes W_{d_k} is at most 2^{k-1}(d_1+\cdots +d_k-2k+2). The same bound holds for the tensor rank and it is an improvement of 2^k(k-1) over the best known bound. Moreover, we provide an explicit partially symmetric decomposition achieving this bound.