Connecting UMEB in ℂ^d ⊗ ℂ^d with partial Hadamard matrices

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

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Submission date: 15. Aug. 2017
Pages: 8
published in: Quantum information processing, 16 (2017) 3, art-no. 84 
DOI number (of the published article): 10.1007/s11128-017-1537-7
Bibtex
with the following different title: Connecting unextendible maximally entangled base with partial Hadamard matrices
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Abstract:
We study the unextendible maximally entangled bases (UMEB) in ℂ^d ⊗ ℂ^d and connect the problem to the partial Hadamard matrices. We show that for a given special UMEB in ℂ^d ⊗ ℂ^d, there is a partial Hadamard matrix which can not be extended to a complete Hadamard matrix in ℂ^d. As a corollary, any (d − 1) × 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 or 2p, where p ≡ 3 mod4 and p is a prime. The existence of different kinds of constructions of UMEBs in ℂ^nd ⊗ ℂ^nd for any n ∈ ℕ and d = 3 × 5 × 7 is also discussed.