A Note on Mutually Unbiased Unextendible Maximally Entangled Bases in ${\mathbb{C}}^2⨂{\mathbb{C}}^3$

We systematically study the construction of mutually unbiased bases in ${\mathbb{C}}^2⨂{\mathbb{C}}^3$, such that all the bases are unextendible maximally entangled ones. Necessary conditions of constructing a pair of mutually unbiased unextendible maximally entangled bases in ${\mathbb{C}}^2⨂{\mathbb{C}}^3$ are derived. Explicit examples are presented.