# Constructions of unextendible entangled bases

## TL;DR

This paper introduces new systematic methods for constructing special unextendible entangled bases with fixed Schmidt number in bipartite quantum systems, expanding the known classes and connecting to unextendible partial Hadamard matrices.

## Contribution

It generalizes the space decomposition method to construct SUEB$k$ in various dimensions and relates unextendible bases to partial Hadamard matrices.

## Key findings

- Constructed new SUEB$k$ in diverse dimensions.
- Provided a method to generate larger SUEB$pk$ from smaller ones.
- Linked unextendible bases with unextendible partial Hadamard matrices.

## Abstract

We provide several constructions of special unextendible entangled bases with fixed Schmidt number $k$ (SUEB$k$) in $\mathbb{C}^{d}\otimes \mathbb{C}^{d'}$ for $2\leq k\leq d\leq d'$. We generalize the space decomposition method in Guo [Phys. Rev. A 94, 052302 (2016)], by proposing a systematic way of constructing new SUEB$k$s in $\mathbb{C}^{d}\otimes \mathbb{C}^{d'}$ for $2\leq k < d \leq d'$ or $2\leq k=d< d'$. In addition, we give a construction of a $(pqdd'-p(dd'-N))$-number SUEB$pk$ in $\mathbb{C}^{pd}\otimes \mathbb{C}^{qd'}$ from an $N$-number SUEB$k$ in $\mathbb{C}^{d}\otimes \mathbb{C}^{d'}$ for $p\leq q$ by using permutation matrices. We also connect a $(d(d'-1)+m)$-number UMEB in $\mathbb{C}^{d}\otimes \mathbb{C}^{d'}$ with an unextendible partial Hadamard matrix $H_{m\times d}$ with $m<d$, which extends the result in [Quantum Inf. Process. 16(3), 84 (2017)].