Highly entangled tensors
Provides tighter lower bounds on maximal entanglement for general tensors, relevant to quantum information theory and tensor network theory.
The paper introduces a geometric measure for entanglement of tensors and proves existence of tensors with entanglement exceeding k log2(n) - log2(k) - o(log2(k)), improving on previous bounds for symmetric tensors.
A geometric measure for the entanglement of a unit length tensor $T \in (\mathbb{C}^n)^{\otimes k}$ is given by $- 2 \log_2 ||T||_σ$, where $||.||_σ$ 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.