Connecting the UMEB in Cᵈᵈ with partial Hadamard matrices

الأصل · EN

We study the unextendible maximally entangled bases (UMEB) in Cᵈᵈ and connect it with the partial Hadamard matrix. Firstly, we show that for a given special UMEB in Cᵈᵈ, there is a partial Hadamard matrix can not extend to a complete Hadamard matrix in Cᵈ. As a corollary, any (d-1)× d partial Hadamard matrix can extend to a complete Hadamard matrix. Then we obtain that for any d there is an UMEB except d=por2p, where p≡ 3 4 and p is a prime. Finally, we argue that there exist different kinds of constructions of UMEB in Cnd for any n∈ N and d=3×5 ×7.