Unextendible maximally entangled bases in dxd

TL;DR

This paper explores unextendible maximally entangled bases (UMEBs) in complex d-dimensional systems, providing explicit constructions in 6x6 and methods to generate larger UMEBs in higher dimensions, revealing multiple non-equivalent sets.

Contribution

It introduces a specific 30-element UMEB in 6x6 systems and a general method to construct larger UMEBs in higher dimensions, expanding understanding of entangled bases.

Findings

Constructed a 30-element UMEB in 6x6 systems.

Established a formula to generate larger UMEBs in higher dimensions.

Identified at least two non-equivalent UMEBs in 12n x 12n systems.

Abstract

We investigate the unextendible maximally entangled bases in $\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}$ and present a $30$-number UMEB construction in $\mathbb{C}^{6}\bigotimes\mathbb{C}^{6}$. For higher dimensional case, we show that for a given $N$-number UMEB in $\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}$, there is a $\widetilde{N}$-number, $\widetilde{N}=(qd)^2-(d^2-N)$, UMEB in $\mathbb{C}^{qd}\bigotimes\mathbb{C}^{qd}$ for any $q\in\mathbb{N}$. As an example, for $\mathbb{C}^{12n}\bigotimes\mathbb{C}^{12n}$ systems, we show that there are at least two sets of UMEBs which are not equivalent.