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

24 pages, 2 figures, final version to appear in Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl

References in corpus (5)

• Tensor Rank and Stochastic Entanglement Catalysis for Multipartite Pure States
• Tensor rank is not multiplicative under the tensor product
• Random bipartite entanglement from W and W-like states
• Border rank is not multiplicative under the tensor product
• Nondeterministic quantum communication complexity: the cyclic equality game and iterated matrix multiplication

Cited by in corpus (8)

• Partially symmetric variants of Comon's problem via simultaneous rank
• Border rank is not multiplicative under the tensor product
• Optimization at the boundary of the tensor network variety
• Rank and border rank of Kronecker powers of tensors and Strassen's laser method
• Decompositions and Terracini loci of cubic forms of low rank
• Geometric conditions for strict submultiplicativity of rank and border rank
• Border Waring Rank via Asymptotic Rank
• Linear dependent subsets of Segre varieties