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arXiv:1409.5019 · Physical Review A

Unextendible maximally entangled bases in Cd⊗Cd

Yan-Ling Wang, Mao-Sheng Li, Shao-Ming Fei

Published Sep 15, 2014• 25 citations
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Abstract

We investigate the unextendible maximally entangled bases (UMEBs) in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ and present a 30-number UMEB construction in ${\mathbb{C}}^{6}\ensuremath{\bigotimes}{\mathbb{C}}^{6}$. For the higher-dimensional case, we show that for a given $N$-number UMEB in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$, there is an $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{N}$-number, $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{N}={(qd)}^{2}\ensuremath{-}({d}^{2}\ensuremath{-}N)$, UMEB in ${\mathbb{C}}^{qd}\ensuremath{\bigotimes}{\mathbb{C}}^{qd}$ for any $q\ensuremath{\in}\mathbb{N}$. As an example, for ${\mathbb{C}}^{12n}\ensuremath{\bigotimes}{\mathbb{C}}^{12n}$ systems, we show that there are at least two sets of UMEBs which are not equivalent.

DOI
10.1103/physreva.90.034301

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A way of constructing the unextendible maximally entangled basis (UMEB) from the maximally entangled basis (MEB) is proposed. Consequently, it is shown that if there is an NNN-member UMEB in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$, then there exists a ${(qd)}^{2}\ensuremath{-}q({d}^{2}\ensuremath{-}N)$-member UMEB in ${\mathbb{C}}^{qd}\ensuremath{\bigotimes}{\mathbb{C}}^{qd}$ for any $q\ensuremath{\in}\mathbb{N}$. This improves the results in Wang et al. [Phys. Rev. A 90, 034301 (2014)], which shows that there exists a ${(qd)}^{2}\ensuremath{-}({d}^{2}\ensuremath{-}N)$-member UMEB in ${\mathbb{C}}^{qd}\ensuremath{\bigotimes}{\mathbb{C}}^{qd}$ provided that an NNN-member UMEB exsits in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$. In addition, a very easy way of constructing UMEB in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}^{\ensuremath{'}}}$ with $d<{d}^{\ensuremath{'}}$ is presented, which promotes and covers all the previous related work.