Unextendible entangled bases with fixed Schmidt number

TL;DR

This paper introduces a generalized concept of unextendible entangled bases with fixed Schmidt number in bipartite systems, providing a construction method and establishing the existence of multiple such bases under various dimensional conditions.

Contribution

It generalizes unextendible product bases to unextendible entangled bases with arbitrary Schmidt number and offers a systematic construction method for these bases.

Findings

Existence of at least k-r sets of UEBk when dimensions are not multiples of k

At least 2(k-1) sets of UEBk when both dimensions are multiples of k

A general construction method for UEBk in bipartite systems

Abstract

The unextendible product basis (UPB) is generalized to the unextendible entangled basis with any arbitrarily given Schmidt number $k$ (UEBk) for any bipartite system $\mathbb{C}^d\otimes\mathbb{C}^{d'}$ ($2\leq k<d\leq d'$), which can also be regarded as a generalization of the unextendible maximally entangled basis (UMEB). A general way of constructing such a basis with arbitrary $d$ and $d'$ is proposed. Consequently, it is shown that there are at least $k-r$ (here $r=d$ mod $k$, or $r=d'$ mod $k$) sets of UEBk when $d$ or $d'$ is not the multiple of $k$, while there are at least $2(k-1)$ sets of UEBk when both $d$ and $d'$ are the multiples of $k$.