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

Abstract

We study the unextendible maximally entangled bases (UMEB) in ${\mathbb{C}}^d⨂{\mathbb{C}}^d$ and connect it with the partial Hadamard matrix. Firstly, we show that for a given special UMEB in ${\mathbb{C}}^d⨂{\mathbb{C}}^d$, 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 $p\equiv 3\mathrm{mod}4$ and $p$ is a prime. Finally, we argue that there exist different kinds of constructions of UMEB in ${\mathbb{C}}^{nd}⨂{\mathbb{C}}^{nd}$ for any $n\in \mathbb{N}$ and $d=3\times 5\times 7$.

6 pages, no figures