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

Abstract

Given tensors $T$ and $T^{\prime }$ of order $k$ and $k^{\prime }$ respectively, the tensor product $T\otimes T^{\prime }$ is a tensor of order $k+k^{\prime }$. 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 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 $x^{d-1}y$, and on products of such. In particular, we prove that the partially symmetric rank of $x^{d_1-1}y\otimes \cdots \otimes x^{d_k-1}y$ is at most $2^{k-1}(d_1+\cdots +d_k)$.