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arXiv:1407.2404 · Physical Review A

Unextendible maximally entangled bases in Cd⊗Cd′

Mao-Sheng Li, Yan-Ling Wang, Zhu‐Jun Zheng

Published Jun 11, 2014• 27 citations
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Abstract

We solve the unextendible maximally entangled basis (UMEB) problem in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}^{\ensuremath{'}}}(d\ensuremath{\ne}{d}^{\ensuremath{'}}).$ The results show that a UMEB will always exist. In addition, there might be two sets of UMEBs with different numbers. The main difficulty is to prove the unextendibility of the set of states. We give an explicit construction of UMEBs by considering the Schmidt number of the complementary space of the states we construct.

DOI
10.1103/physreva.89.062313

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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.
Constructions of Unextendible Maximally Entangled Bases in Cd⊗Cd′{{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }Cd⊗Cd′
Abstract We study unextendible maximally entangled bases (UMEBs) in Cd⊗Cd′{{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }Cd⊗Cd′ ℂ d ⊗ ℂ d ′ (d < d′). An operational method to construct UMEBs containing d(d′ − 1) maximally entangled vectors is established, and two UMEBs in C5⊗C6{{\mathbb{C}}}^{5}\otimes {{\mathbb{C}}}^{6}C5⊗C6 ℂ 5 ⊗ ℂ 6 and C5⊗C12{{\mathbb{C}}}^{5}\otimes {{\mathbb{C}}}^{12}C5⊗C12 ℂ 5 ⊗ ℂ 12 are given as examples. Furthermore, a systematic way of constructing UMEBs containing d(d′ − r) maximally entangled vectors in Cd⊗Cd′{{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }Cd⊗Cd′ ℂ d ⊗ ℂ d ′ is presented for r = 1, 2, …, d − 1. Correspondingly, two UMEBs in C3⊗C10{{\mathbb{C}}}^{3}\otimes {{\mathbb{C}}}^{10}C3⊗C10 ℂ 3 ⊗ ℂ 10 are obtained.
The unextendible product basis is generalized to the unextendible entangled basis with any arbitrarily given Schmidt number kkk (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 ddd and ${d}^{\ensuremath{'}}$ is proposed. Consequently, it is shown that there are at least $k\ensuremath{-}r$ (here r=dmodkr=d\phantom{\rule{0.16em}{0ex}}\text{mod}\phantom{\rule{0.28em}{0ex}}kr=dmodk or $r={d}^{\ensuremath{'}}\phantom{\rule{0.16em}{0ex}}\text{mod}\phantom{\rule{0.28em}{0ex}}k)$ sets of UEBks when ddd or ${d}^{\ensuremath{'}}$ is not a multiple of kkk, while there are at least $2(k\ensuremath{-}1)$ sets of UEBks when both ddd and ${d}^{\ensuremath{'}}$ are multiples of kkk.