Highly entangled tensors

Abstract

A geometric measure for the entanglement of a unit length tensor $T\in ({\mathbb{C}}^n{)}^{\otimes k}$ is given by $-2{\log }_2||T|{|}_{\sigma }$, where $||.|{|}_{\sigma }$ denotes the spectral norm. A simple induction gives an upper bound of $(k-1){\log }_2(n)$ for the entanglement. We show the existence of tensors with entanglement larger than $k{\log }_2(n)-{\log }_2(k)-o({\log }_2(k))$. Friedland and Kemp have similar results in the case of symmetric tensors. Our techniques give improvements in this case.

arXiv comment13 pages, improved results and a section added on symmetric tensors