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arXiv:1609.04049 · Physical Review A

Constructing the unextendible maximally entangled basis from the maximally entangled basis

Yu Jing Guo

Published Nov 2, 2016• 18 citations
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Abstract

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 $N$-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 $N$-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.

DOI
10.1103/physreva.94.052302

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Connecting the UMEB in Cd⨂Cd\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}Cd⨂Cd with partial Hadamard matrices
We study the unextendible maximally entangled bases (UMEB) in Cd⨂Cd\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}Cd⨂Cd and connect it with the partial Hadamard matrix. Firstly, we show that for a given special UMEB in Cd⨂Cd\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}Cd⨂Cd, there is a partial Hadamard matrix can not extend to a complete Hadamard matrix in Cd\mathbb{C}^{d}Cd. As a corollary, any (d−1)×d(d-1)\times d(d−1)×d partial Hadamard matrix can extend to a complete Hadamard matrix. Then we obtain that for any ddd there is an UMEB except d=p or 2pd=p\ \text{or}\ 2pd=p or 2p, where p≡3mod  4p\equiv 3\mod 4p≡3mod4 and ppp is a prime. Finally, we argue that there exist different kinds of constructions of UMEB in Cnd⨂Cnd\mathbb{C}^{nd}\bigotimes\mathbb{C}^{nd}Cnd⨂Cnd for any n∈Nn\in \mathbb{N}n∈N and d=3×5×7d=3\times5 \times7d=3×5×7.