Multipartite unextendible entangled basis

TL;DR

This paper extends the concept of unextendible entangled bases with arbitrary Schmidt number to multipartite systems, providing a general construction method and proving the existence of infinitely many such bases in systems with three or more parties.

Contribution

It introduces a method to construct multipartite unextendible entangled bases from bipartite ones, broadening the scope of entanglement structures in quantum systems.

Findings

Infinite UEBk sets exist in multipartite systems with any dimensions.

A recursive construction method for multipartite UEBk is proposed.

The work generalizes bipartite UEBk to multipartite cases.

Abstract

The unextendible entangled basis with any arbitrarily given Schmidt number $k$ (UEBk) in $\mathbb{C}^{d_1}\otimes\mathbb{C}^{d_2}$ is proposed in [Phys. Rev. A 90 (2014) 054303], $1<k\leq \min\{d_1,d_2\}$, which is a set of orthonormal entangled states with Schmidt number $k$ in a $d_1\otimes d_2$ system consisting of fewer than $d_1d_2$ vectors which have no additional entangled vectors with Schmidt number $k$ in the complementary space. In this paper, we extend it to multipartite case and a general way of constructing $(m+1)$-partite UEBk from $m$-partite UEBk is proposed ($m\geq 2$). Consequently, we show that there are infinitely many UEBks in $\mathbb{C}^{d_1}\otimes\mathbb{C}^{d_2}\otimes\cdots\otimes\mathbb{C}^{d_N}$ with any dimensions and any $N\geq3$.