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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
53/2017

Connecting UMEB in $\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}$ with partial Hadamard matrices

Yan-Ling Wang, Mao-Sheng Li, Shao-Ming Fei and Zhu-Jun Zheng

Abstract

We study the unextendible maximally entangled bases (UMEB) in $\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}$ and connect the problem to the partial Hadamard matrices. We show that for a given special UMEB in $\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}$, there is a partial Hadamard matrix which can not be extended to a complete Hadamard matrix in $\mathbb{C}^{d}$. As a corollary, any $(d-1)\times d$ partial Hadamard matrix can be extended to a complete Hadamard matrix, which answers a conjecture about $d=5$. We obtain that for any $d$ there is a UMEB except for $d=p\ \text{or}\ 2p$, where $p\equiv 3\mod 4$ and $p$ is a prime. The existence of different kinds of constructions of UMEBs in $\mathbb{C}^{nd}\bigotimes\mathbb{C}^{nd}$ for any $n\in \mathbb{N}$ and $d=3\times5 \times7$ is also discussed.

Received:
Aug 15, 2017
Published:
Aug 16, 2017

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inJournal
2017 Repository Open Access
Yan-Ling Wang, Mao-Sheng Li, Shao-Ming Fei and Zhu-Jun Zheng

Connecting unextendible maximally entangled base with partial Hadamard matrices

In: Quantum information processing, 16 (2017) 3, p. 84