Constructions of unextendible entangled bases

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)].