  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 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 $x^{d_1 -1}y \otimes ... \otimes x^{d_k-1} y$ is at most $2^{k-1}(d_1+ ...
+d_k)$.
