Connecting unextendible maximally entangled base with partial Hadamard matrices

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


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 cannot be extended to a Hadamard matrix in \(\mathbb {C}^{d}\). As a corollary, any \((d-1)\times d\) partial Hadamard matrix can be extended to a 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 \(d=3\times 5 \times 7\) is also discussed.


Unextendible maximally entangled bases Partial Hadamard matrices Maximally entangled state 



This work is supported by the NSFC 11475178, NSFC 11571119 and NSFC 11275131.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Yan-Ling Wang
    • 1
  • Mao-Sheng Li
    • 2
  • Shao-Ming Fei
    • 3
    • 4
  • Zhu-Jun Zheng
    • 1
  1. 1.Department of MathematicsSouth China University of TechnologyGuangzhouChina
  2. 2.Department of Mathematical SciencesTsinghua UniversityBeijingChina
  3. 3.School of Mathematical SciencesCapital Normal UniversityBeijingChina
  4. 4.Max-Planck-Institute for Mathematics in the SciencesLeipzigGermany

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