Given tensors T and T′ of order k and k′ respectively, the tensor product T⊗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 W-states, namely monomials of the form xd−1y, and on products of such. In particular, we prove that the partially symmetric rank of xd1−1y⊗⋯⊗xdk−1y is at most 2k−1(d1+⋯+dk).