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 Guo and Wu (Phys Rev A 90:054303, 2014), \(1<k\le \min \{d_1,d_2\}\), 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 (\(m\ge 2\)). 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 \(N\ge 3\).
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The authors wish to give their thanks to the referees for their helpful comments and suggestions to improve the manuscript. This work is supported by the National Natural Science Foundation of China under Grants Nos. 11301312 and 11171249, the Natural Science Foundation of Shanxi under Grant No. 2013021001-1 and the Research start-up fund for Doctors of Shanxi Datong University under Grant No. 2011-B-01.
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Guo, Y., Jia, Y. & Li, X. Multipartite unextendible entangled basis. Quantum Inf Process 14, 3553–3568 (2015). https://doi.org/10.1007/s11128-015-1058-1
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DOI: https://doi.org/10.1007/s11128-015-1058-1