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arXiv:1407.4362 · Physical Review A

Unextendible entangled bases with fixed Schmidt number

Yu Jing Guo, Shengjun Wu

Published Nov 19, 2014• 22 citations
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Abstract

The unextendible product basis is generalized to the unextendible entangled basis with any arbitrarily given Schmidt number $k$ (UEBk) for any bipartite system ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}^{\ensuremath{'}}}$ $(2\ensuremath{\le}k<d\ensuremath{\le}{d}^{\ensuremath{'}})$, which can also be regarded as a generalization of the unextendible maximally entangled basis. A general way of constructing such a basis with arbitrary $d$ and ${d}^{\ensuremath{'}}$ is proposed. Consequently, it is shown that there are at least $k\ensuremath{-}r$ (here $r=d\phantom{\rule{0.16em}{0ex}}\text{mod}\phantom{\rule{0.28em}{0ex}}k$ or $r={d}^{\ensuremath{'}}\phantom{\rule{0.16em}{0ex}}\text{mod}\phantom{\rule{0.28em}{0ex}}k)$ sets of UEBks when $d$ or ${d}^{\ensuremath{'}}$ is not a multiple of $k$, while there are at least $2(k\ensuremath{-}1)$ sets of UEBks when both $d$ and ${d}^{\ensuremath{'}}$ are multiples of $k$.

DOI
10.1103/physreva.90.054303

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The unextendible product basis is generalized to the unextendible entangled basis with any arbitrarily given Schmidt number kkk (UEBk) for any bipartite system Cd\ensuremath⨂Cd\ensuremath′{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}^{\ensuremath{'}}}Cd\ensuremath⨂Cd\ensuremath′ (2\ensuremath≤k<d\ensuremath≤d\ensuremath′)(2\ensuremath{\le}k<d\ensuremath{\le}{d}^{\ensuremath{'}})(2\ensuremath≤k<d\ensuremath≤d\ensuremath′), which can also be regarded as a generalization of the unextendible maximally entangled basis. A general way of constructing such a basis with arbitrary ddd and d\ensuremath′{d}^{\ensuremath{'}}d\ensuremath′ is proposed. Consequently, it is shown that there are at least k\ensuremath−rk\ensuremath{-}rk\ensuremath−r (here r=dmodkr=d\phantom{\rule{0.16em}{0ex}}\text{mod}\phantom{\rule{0.28em}{0ex}}kr=dmodk or r=d\ensuremath′modk)r={d}^{\ensuremath{'}}\phantom{\rule{0.16em}{0ex}}\text{mod}\phantom{\rule{0.28em}{0ex}}k)r=d\ensuremath′modk) sets of UEBks when ddd or d\ensuremath′{d}^{\ensuremath{'}}d\ensuremath′ is not a multiple of kkk, while there are at least 2(k\ensuremath−1)2(k\ensuremath{-}1)2(k\ensuremath−1) sets of UEBks when both ddd and d\ensuremath′{d}^{\ensuremath{'}}d\ensuremath′ are multiples of kkk.
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