A new bound on the rank of tensor product of W-states

TL;DR

This work bound sharpens the understanding of tensor ranks for products of generalized W-states by proving a new upper bound on the partially symmetric rank rk_{oldsymbol{d}}(W_{d_1}\otimes\cdots\otimes W_{d_k}) of $2^{k-1}(d_1+\cdots+d_k-2k+2)$. The authors develop a geometric framework based on a span of 2^{k-1} rational normal curves in the Segre-Veronese setting and employ catalecticant analysis and Sylvester's algorithm to reduce the problem to binary forms, obtaining a constructive decomposition of the desired length. The result also applies to the tensor rank and improves the prior bound by 2^k(k-1); a careful analysis shows the bound is tight in at least the k=2, d_1=d_2=3 case. Additionally, the paper proves the border rank equals 2^k and provides an explicit algorithm to compute the decomposition, highlighting practical advantages for computing partially symmetric decompositions of W-state tensors.

Abstract

A W-state is an order d symmetric tensor of the form W_d=x^{d-1}y. We prove that the partially symmetric rank of W_{d_1}\otimes \cdots \otimes W_{d_k} is at most 2^{k-1}(d_1+\cdots +d_k-2k+2). The same bound holds for the tensor rank and it is an improvement of 2^k(k-1) over the best known bound. Moreover, we provide an explicit partially symmetric decomposition achieving this bound.

Theorems & Definitions (33)

• Theorem 1.1
• Corollary 1.2
• Definition 2.1
• Definition 2.3
• Definition 2.4
• Remark 2.5
• Example 2.7
• Lemma 3.1
• proof
• Remark 3.2