A Note on Mutually Unbiased Unextendible Maximally Entangled Bases in ℂ2⊗ℂ3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {C}^{2}\bigotimes \mathbb {C}^{3}$\end{document}

We systematically study the construction of mutually unbiased bases in ℂ2⊗ℂ3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {C}^{2}\bigotimes \mathbb {C}^{3}$\end{document}, such that all the bases are unextendible maximally entangled ones. Necessary conditions of constructing a pair of mutually unbiased unextendible maximally entangled bases in ℂ2⊗ℂ3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {C}^{2}\bigotimes \mathbb {C}^{3}$\end{document} are derived. Explicit examples are presented.