Highly entangled tensors

TL;DR

This paper introduces a geometric measure for tensor entanglement, establishes bounds, and demonstrates the existence of highly entangled tensors, advancing understanding in quantum information and tensor analysis.

Contribution

It defines a spectral norm-based entanglement measure for tensors and proves the existence of tensors with entanglement exceeding previous bounds, including symmetric cases.

Findings

Existence of tensors with entanglement larger than $k \, \log_2(n)$ minus small corrections.

An upper bound of $(k-1) \log_2(n)$ for tensor entanglement.

Improved bounds for symmetric tensors compared to prior work.

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.