On the partially symmetric rank of tensor products of W-states and other symmetric tensors

Given tensors $T$ and $T'$ of order $k$ and $k'$ respectively, the tensor product $T otimes T'$ is a tensor of order $k+k'$. It was recently shown that the tensor rank can be strictly submultiplicative under this operation ([Christandl-Jensen-Zuiddam]). We study this phenomenon for symmetric tensors where additional techniques from algebraic geometry are available. The tensor product of symmetric tensors results in a partially symmetric tensor and our results amount to bounds on the partially symmetric rank. Following motivations from algebraic complexity theory and quantum information theory, we focus on the so-called emph{$W$-states}, namely monomials of the form $x^{d-1}y$, and on products of such. In particular, we prove that the partially symmetric rank of $x^{d_1 -1}y ootimes x^{d_k-1} y$ is at most $2^{k-1}(d_1+ cdots +d_k)$.