Absolutely entangled set of pure states

TL;DR

This paper investigates the properties and constructions of absolutely entangled sets in bipartite quantum systems, providing necessary conditions and explicit methods to construct such sets in various dimensions.

Contribution

It introduces two necessary conditions for absolutely entangled sets and offers new construction methods for these sets in nonprime and specific bipartite dimensions.

Findings

Derived two necessity conditions for absolute entanglement.

Constructed absolutely entangled bases for nonprime dimensions.

Presented a new construction of absolutely entangled sets in $ ext{C}^2 imes ext{C}^n$.

Abstract

Quite recently, Cai et al. [arXiv:2006.07165v1] proposed a new concept "absolutely entangled set" for bipartite quantum systems: for any possible choice of global basis, at least one state of the set is entangled. There they presented a minimum example with a set of four states in two qubit systems and they proposed a quantitative measure for the absolute set entanglement. In this work, we derive two necessity conditions for a set of states to be an absolutely entangled set. In addition, we give a series constructions of absolutely entangled bases on $\mathbb{C}^{d_1}\otimes \mathbb{C}^{d_2}$ for any nonprime dimension $d=d_1\times d_2$. Moreover, based on the structure of the orthogonal product basis in $\mathbb{C}^2\otimes \mathbb{C}^n$, we obtain another construction of absolutely entangled set with $2n+1$ elements in $\mathbb{C}^2\otimes \mathbb{C}^n$.