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

Given tensors $T$ and $T^{\mathrm{\prime }}$ of order $k$ and $k^{\mathrm{\prime }}$ respectively, the tensor product $T\otimes T^{\mathrm{\prime }}$ is a tensor of order $k+k^{\mathrm{\prime }}$. 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 become 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 “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 $x^{d_1-1}y\otimes \mathrm{.}\mathrm{.}\mathrm{.}\otimes x^{d_k-1}y$ is at most $2^{k-1}(d_1+ … +d_k)$.