Connecting unextendible maximally entangled base with partial Hadamard matrices

Abstract

We study the unextendible maximally entangled bases (UMEB) in $$\mathbb {C}^{d}\bigotimes \mathbb {C}^{d}$$Cd?Cd and connect the problem to the partial Hadamard matrices. We show that for a given special UMEB in $$\mathbb {C}^{d}\bigotimes \mathbb {C}^{d}$$Cd?Cd, there is a partial Hadamard matrix which cannot be extended to a Hadamard matrix in $$\mathbb {C}^{d}$$Cd. As a corollary, any $$(d-1)\times d$$(d-1)×d partial Hadamard matrix can be extended to a Hadamard matrix, which answers a conjecture about $$d=5$$d=5. We obtain that for any d there is a UMEB except for $$d=p\ \text {or}\ 2p$$d=por2p, where $$p\equiv 3\mod 4$$p?3mod4 and p is a prime. The existence of different kinds of constructions of UMEBs in $$\mathbb {C}^{nd}\bigotimes \mathbb {C}^{nd}$$Cnd?Cnd for any $$n\in \mathbb {N}$$n?N and $$d=3\times 5 \times 7$$d=3×5×7 is also discussed.