Constructions of Unextendible Maximally Entangled Bases in ℂd⊗ℂd′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$$\end{document}

We study unextendible maximally entangled bases (UMEBs) in ℂd⊗ℂd′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$$\end{document} (d < d′). An operational method to construct UMEBs containing d(d′ − 1) maximally entangled vectors is established, and two UMEBs in ℂ5⊗ℂ6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{C}}}^{5}\otimes {{\mathbb{C}}}^{6}$$\end{document} and ℂ5⊗ℂ12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{C}}}^{5}\otimes {{\mathbb{C}}}^{12}$$\end{document} are given as examples. Furthermore, a systematic way of constructing UMEBs containing d(d′ − r) maximally entangled vectors in ℂd⊗ℂd′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }$$\end{document} is presented for r = 1, 2, …, d − 1. Correspondingly, two UMEBs in ℂ3⊗ℂ10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{C}}}^{3}\otimes {{\mathbb{C}}}^{10}$$\end{document} are obtained.

Quantum entanglement lies in the heart of the quantum information processing. It plays important roles in many fields such as quantum teleportation, quantum coding, quantum key distribution protocol, quantum non-locality [1][2][3][4] . Quantum teleportation, which can be used for distributed quantum learning 5 and even in organisms 6 , is a essential element in quantum information processing. Maximally entangled states attract much attention due to their importance in ensuring the highest fidelity and efficiency in quantum teleportation 7  Nonlocality is a very useful concept in quantum mechanics [9][10][11][12][13] and plays an important role in Van der Waals interaction in transformation optics 14 . It is tightly related to entanglement. While, it is proven that the unextendible product bases (UPBs) reveal some nolocality without entanglement 15,16 . The UPB is a set of incomplete orthogonal product states in bipartite quantum system   ⊗ ′ d d consisting of fewer than dd′ vectors which have no additional product states are orthogonal to each element of the set 17 .
It is obvious that Therefore, for convenience, we may just call an SV1B In deriving our main results, we need the following lemma in ref. 24 .
In this section, we will establish a flexible method to construct Proof. Without loss of generality, we can always assume the ignored d entries in V only occupy the former N columns. , and Next, we prove that all the states in (3) constitute an MEB in V.
From the definition of t mj , we also have Hence t mj ≠ t m′j for m ≠ m′. Namely, the states |φ′ j,n 〉 in (3) are all maximally entangled.
We first show that |t mj 〉 = |t mj′ 〉 if and only if j = j′.
when C(m, t m−1 , ,j ⊕ d′ 1) = 1, as proved in (i). Therefore, t m−1 , j′ = b m−1 , which contradicts to the definition of t mj . Furthermore, t mj ≠ t mj′ when t 0j ≠ t 0j′ . Therefore,    where α ω ω ω ω = 1, , ,   Remark 1. Actually both (9) and (10) are UMEBs in   ⊗ 5 6 . However, they are different although they can be unitarily transformed to each other. We will reveal the difference in the following example. One can easily get the following simple formations ′ V 1 and ′ V 2 from V 1 and V 2 by elementary transformation respectively: Then following Theorem 1 we can construct the following UMEBs φ   where α ω ω ω ω = 1, , ,

By inverse transformation
, we can obtain the following