Unextendible maximally entangled bases

TL;DR

This paper introduces the concept of unextendible maximally entangled bases (UMEBs), proving their non-existence for two-dimensional systems and providing explicit constructions for certain higher-dimensional cases.

Contribution

It formally defines UMEBs, proves their non-existence in 2D, and constructs explicit examples in 3D and 4D systems.

Findings

UMEBs do not exist for d=2

Explicit 6-member UMEB in d=3

Explicit 12-member UMEB in d=4

Abstract

We introduce the notion of the unextendible maximally entangled basis (UMEB), a set of orthonormal maximally entangled states in d \times d consisting of fewer that d^2 vectors which have no additional maximally entangled vectors orthogonal to all of them. We prove that UMEBs don't not exist for d=2 and give an explicit constructions for a 6-member UMEB with d=3 and a 12-member UMEB with d=4.