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 $2^{k-1}(d_1+ ... +d_k)$.
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