Constructing UMEB from maximally entangled basis

TL;DR

This paper introduces a new method for constructing unextendible maximally entangled bases (UMEB) from maximally entangled bases (MEB), improving existing results and providing a simple construction approach for different dimensions.

Contribution

It presents a novel construction technique for UMEB from MEB and extends the known existence results to broader dimensions, simplifying the process for mixed-dimension systems.

Findings

Established a method to generate larger UMEB in higher-dimensional spaces.

Improved previous bounds on the size of UMEB in tensor product spaces.

Provided a straightforward construction method for UMEB in systems with different local dimensions.

Abstract

A new way of constructing unextendible maximally entangled basis (UMEB) from maximally entangled basis (MEB) is proposed. Consequently, it is shown that if there is an $N$-member UMEB in $\mathbb{C}^d\otimes \mathbb{C}^d$, then there exists a $(qd)^2-q(d^2-N)$-member UMEB in $\mathbb{C}^{qd}\otimes \mathbb{C}^{qd}$ for any $q\in\mathbb{N}$. This improves the results in [Phys. Rev. A 90, 034301(2014)], which shows that there exists a $(qd)^2-(d^2-N)$-member UMEB in $\mathbb{C}^{qd}\otimes \mathbb{C}^{qd}$ provided that an $N$-member UMEB exsits in $\mathbb{C}^d\otimes \mathbb{C}^d$. In addition, a very easy way of constructing UMEB in $\mathbb{C}^d\otimes \mathbb{C}^{d'}$ with $d<d'$ is presented, which promotes and covers all the previous related work.