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Published in:
01-10-2019
Constructions of unextendible entangled bases
Published in: Quantum Information Processing | Issue 10/2019
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Abstract
We provide several constructions of special unextendible entangled bases with fixed Schmidt number k (\(\hbox {SUEB}k\)) in \({\mathbb {C}}^{d}\otimes {\mathbb {C}}^{d'}\) for \(2\le k\le d\le d'\). We generalize the space decomposition method in Guo (Phys Rev A 94:052302, 2016), by proposing a systematic way of constructing new \(\hbox {SUEB}k\hbox {s}\) in \({\mathbb {C}}^{d}\otimes {\mathbb {C}}^{d'}\) for \(2\le k < d \le d'\) or \(2\le k=d< d'\). In addition, we give a construction of a \((pqdd'-p(dd'-N))\)-number SUEBpk in \({\mathbb {C}}^{pd}\otimes {\mathbb {C}}^{qd'}\) from an N-number \(\hbox {SUEB}k\) in \({\mathbb {C}}^{d}\otimes {\mathbb {C}}^{d'}\) for \(p\le q\) 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 Wang et al. (Quantum Inf Process 16:84, 2017).