Connecting UMEB in ℂd ⊗ ℂd 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
published in: Quantum information processing, 16 (2017) 3, art-no. 84
DOI number (of the published article): 10.1007/s11128-017-1537-7
with the following different title: Connecting unextendible maximally entangled base with partial Hadamard matrices
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We study the unextendible maximally entangled bases (UMEB) in ℂd ⊗ ℂd and connect the problem to the partial Hadamard matrices. We show that for a given special UMEB in ℂd ⊗ ℂd, there is a partial Hadamard matrix which can not be extended to a complete Hadamard matrix in ℂd. 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 ℂnd ⊗ ℂnd for any n ∈ ℕ and d = 3 × 5 × 7 is also discussed.