A note on the gap between rank and border rank

Abstract

We study the tensor rank of the tensor corresponding to the algebra of n-variate complex polynomials modulo the dth power of each variable. As a result we find a sequence of tensors with a large gap between rank and border rank, and thus a counterexample to a conjecture of Rhodes. At the same time we obtain a new lower bound on the tensor rank of tensor powers of the generalised W-state tensor. In addition, we exactly determine the tensor rank of the tensor cube of the three-party W-state tensor, thus answering a question of Chen et al.

To appear in Linear Algebra and its Applications

References in corpus (2)

• Tensor Rank and Stochastic Entanglement Catalysis for Multipartite Pure States
• How hard is the tensor rank?

Cited by in corpus (8)

• Tensor rank is not multiplicative under the tensor product
• On the partially symmetric rank of tensor products of W-states and other symmetric tensors
• Tensor rank and entanglement of pure quantum states
• On algebraic branching programs of small width
• The Tensor as an Informational Resource
• Border Ranks of Positive and Invariant Tensor Decompositions: Applications to Correlations
• An Algorithmic Method of Partial Derivatives
• Entanglement in the symmetric subspace: mapping multipartite to bipartite states