Constructions of Unextendible Maximally Entangled Bases in \(\mathbb {C}^{d}\otimes \mathbb {C}^{d^{\prime}}\)

We study unextendible maximally entangled bases (UMEBs) in $\mathbb {C}^{d}\otimes \mathbb {C}^{d^{\prime}}$ ($d<d'$). An operational method to construct UMEBs containing $d(d^{\prime}-1)$ maximally entangled vectors is established, and two UMEBs in $\mathbb {C}^{5}\otimes \mathbb {C}^{6}$ and $\mathbb {C}^{5}\otimes \mathbb {C}^{12}$ are given as examples. Furthermore, a systematic way of constructing UMEBs containing $d(d^{\prime}-r)$ maximally entangled vectors in $\mathbb {C}^{d}\otimes \mathbb {C}^{d^{\prime}}$ is presented for $r=1,2,\cdots, d-1$. Correspondingly, two UMEBs in $\mathbb {C}^{3}\otimes \mathbb {C}^{10}$ are obtained.