Connecting unextendible maximally entangled base with partial Hadamard matrices

Abstract

We study the unextendible maximally entangled bases (UMEB) in $\mathbb {C}^{d}\bigotimes \mathbb {C}^{d}$ 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.