Connecting the UMEB in $\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}$ with partial Hadamard matrices

Abstract: We study the unextendible maximally entangled bases (UMEB) in $\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}$ and connect it with the partial Hadamard matrix. Firstly, we show that for a given special UMEB in $\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}$, there is a partial Hadamard matrix can not extend to a complete Hadamard matrix in $\mathbb{C}^{d}$. As a corollary, any $(d-1)\times d$ partial Hadamard matrix can extend to a complete Hadamard matrix. Then we obtain that for any $d$ there is an UMEB except $d=p\ \text{or}\ 2p$, where $p\equiv 3\mod 4$ and $p$ is a prime. Finally, we argue that there exist different kinds of constructions of UMEB in $\mathbb{C}^{nd}\bigotimes\mathbb{C}^{nd}$ for any $n\in \mathbb{N}$ and $d=3\times5 \times7$.