Constructions of unextendible entangled bases

Abstract

We provide several constructions of special unextendible entangled bases with fixed Schmidt number $k$ (SUEB$k$) in ${\mathbb{C}}^d\otimes {\mathbb{C}}^{d^{\prime }}$ for $2\le k\le d\le d^{\prime }$. 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 ${\mathbb{C}}^d\otimes {\mathbb{C}}^{d^{\prime }}$ for $2\le k<d\le d^{\prime }$ or $2\le k=d<d^{\prime }$. In addition, we give a construction of a $(pqd{d}^{\prime }-p(d{d}^{\prime }-N))$-number SUEB$pk$ in ${\mathbb{C}}^{pd}\otimes {\mathbb{C}}^{q{d}^{\prime }}$ from an $N$-number SUEB$k$ in ${\mathbb{C}}^d\otimes {\mathbb{C}}^{d^{\prime }}$ for $p\le q$ by using permutation matrices. We also connect a $(d(d^{\prime }-1)+m)$-number UMEB in ${\mathbb{C}}^d\otimes {\mathbb{C}}^{d^{\prime }}$ 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)].

arXiv comment13 pages

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