Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1802.07594 · Scientific Reports

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

Gui-Jun Zhang, Yuan-Hong Tao, Yi-Fan Han, Xin-Lei Yong, Shao-Ming Fei

Published Feb 12, 2018• 10 citations
arXivHTMLPDF

Abstract

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

DOI
10.1038/s41598-018-21561-0

Related papers

Unextendible maximally entangled bases in Cd⊗Cd

Yan-Ling Wang, Mao-Sheng Li, Shao-Ming Fei

We investigate the unextendible maximally entangled bases (UMEBs) in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ and present a 30-number UMEB construction in ${\mathbb{C}}^{6}\ensuremath{\bigotimes}{\mathbb{C}}^{6}$. For the higher-dimensional case, we show that for a given $N$-number UMEB in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$, there is an $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{N}$-number, $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{N}={(qd)}^{2}\ensuremath{-}({d}^{2}\ensuremath{-}N)$, UMEB in ${\mathbb{C}}^{qd}\ensuremath{\bigotimes}{\mathbb{C}}^{qd}$ for any $q\ensuremath{\in}\mathbb{N}$. As an example, for ${\mathbb{C}}^{12n}\ensuremath{\bigotimes}{\mathbb{C}}^{12n}$ systems, we show that there are at least two sets of UMEBs which are not equivalent.

arXiv:1409.5019•Physical Review A•Sep 15, 2014•25 citations

Unextendible maximally entangled bases

Sergey B. Bravyi, John A. Smolin

We introduce the notion of the unextendible maximally entangled basis (UMEB), a set of orthonormal maximally entangled states in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ consisting of fewer than ${d}^{2}$ vectors which have no additional maximally entangled vectors orthogonal to all of them. We prove that UMEBs do not exist for $d=2$ and give explicit constructions for a six-member UMEB with $d=3$ and a 12-member UMEB with $d=4$.

arXiv:0911.4090•Physical Review A•Oct 4, 2011•52 citations

Constructing the unextendible maximally entangled basis from the maximally entangled basis

Yu Jing Guo

A way of constructing the unextendible maximally entangled basis (UMEB) from the maximally entangled basis (MEB) is proposed. Consequently, it is shown that if there is an $N$-member UMEB in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$, then there exists a ${(qd)}^{2}\ensuremath{-}q({d}^{2}\ensuremath{-}N)$-member UMEB in ${\mathbb{C}}^{qd}\ensuremath{\bigotimes}{\mathbb{C}}^{qd}$ for any $q\ensuremath{\in}\mathbb{N}$. This improves the results in Wang et al. [Phys. Rev. A 90, 034301 (2014)], which shows that there exists a ${(qd)}^{2}\ensuremath{-}({d}^{2}\ensuremath{-}N)$-member UMEB in ${\mathbb{C}}^{qd}\ensuremath{\bigotimes}{\mathbb{C}}^{qd}$ provided that an $N$-member UMEB exsits in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$. In addition, a very easy way of constructing UMEB in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}^{\ensuremath{'}}}$ with $d<{d}^{\ensuremath{'}}$ is presented, which promotes and covers all the previous related work.

arXiv:1609.04049•Physical Review A•Nov 2, 2016•18 citations

Unextendible maximally entangled bases in Cd⊗Cd′

Mao-Sheng Li, Yan-Ling Wang, Zhu‐Jun Zheng

We solve the unextendible maximally entangled basis (UMEB) problem in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}^{\ensuremath{'}}}(d\ensuremath{\ne}{d}^{\ensuremath{'}}).$ The results show that a UMEB will always exist. In addition, there might be two sets of UMEBs with different numbers. The main difficulty is to prove the unextendibility of the set of states. We give an explicit construction of UMEBs by considering the Schmidt number of the complementary space of the states we construct.

arXiv:1407.2404•Physical Review A•Jun 11, 2014•27 citations

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

Yanling Wang, Mao-Sheng Li, Shao-Ming Fei +1 more

We study the unextendible maximally entangled bases (UMEB) in $\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}$ and connect it with the partial Hadamard matrix. Firstly, we show that for a given special UMEB in $\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}$, 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 $p\equiv 3\mod 4$ and $p$ is a prime. Finally, we argue that there exist different kinds of constructions of UMEB in $\mathbb{C}^{nd}\bigotimes\mathbb{C}^{nd}$ for any $n\in \mathbb{N}$ and $d=3\times5 \times7$.

arXiv:1604.08665•Apr 29, 2016•2 citations

Unextendible maximally entangled bases and mutually unbiased bases

Bin Chen, Shao-Ming Fei

We study an unextendible maximally entangled basis in arbitrary bipartite spaces. A systematic way of constructing a set of ${d}^{2}$ orthonormal maximally entangled states in ${\mathbb{C}}^{d}⨂{\mathbb{C}}^{{d}^{\ensuremath{'}}}(\frac{{d}^{\ensuremath{'}}}{2}<d<{d}^{\ensuremath{'}})$ is provided. The complementary space of the set of these ${d}^{2}$ orthonormal maximally entangled states contains no maximally entangled states that are orthogonal to all of them. Furthermore, we investigate mutually unbiased bases in which all the bases are unextendible maximally entangled ones. We present two unextendible maximally entangled bases in ${\mathbb{C}}^{2}⨂{\mathbb{C}}^{3}$ which are mutually unbiased.

arXiv:1309.3356•Physical Review A•Sep 4, 2013•45 citations
Constructions of Unextendible Maximally Entangled Bases in Cd⊗Cd′{{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }Cd⊗Cd′
Abstract We study unextendible maximally entangled bases (UMEBs) in Cd⊗Cd′{{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }Cd⊗Cd′ ℂ d ⊗ ℂ d ′ (d < d′). An operational method to construct UMEBs containing d(d′ − 1) maximally entangled vectors is established, and two UMEBs in C5⊗C6{{\mathbb{C}}}^{5}\otimes {{\mathbb{C}}}^{6}C5⊗C6 ℂ 5 ⊗ ℂ 6 and C5⊗C12{{\mathbb{C}}}^{5}\otimes {{\mathbb{C}}}^{12}C5⊗C12 ℂ 5 ⊗ ℂ 12 are given as examples. Furthermore, a systematic way of constructing UMEBs containing d(d′ − r) maximally entangled vectors in Cd⊗Cd′{{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d^{\prime} }Cd⊗Cd′ ℂ d ⊗ ℂ d ′ is presented for r = 1, 2, …, d − 1. Correspondingly, two UMEBs in C3⊗C10{{\mathbb{C}}}^{3}\otimes {{\mathbb{C}}}^{10}C3⊗C10 ℂ 3 ⊗ ℂ 10 are obtained.
We investigate the unextendible maximally entangled bases (UMEBs) in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ and present a 30-number UMEB construction in ${\mathbb{C}}^{6}\ensuremath{\bigotimes}{\mathbb{C}}^{6}$. For the higher-dimensional case, we show that for a given NNN-number UMEB in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$, there is an $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{N}$-number, $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{N}={(qd)}^{2}\ensuremath{-}({d}^{2}\ensuremath{-}N)$, UMEB in ${\mathbb{C}}^{qd}\ensuremath{\bigotimes}{\mathbb{C}}^{qd}$ for any $q\ensuremath{\in}\mathbb{N}$. As an example, for ${\mathbb{C}}^{12n}\ensuremath{\bigotimes}{\mathbb{C}}^{12n}$ systems, we show that there are at least two sets of UMEBs which are not equivalent.
We introduce the notion of the unextendible maximally entangled basis (UMEB), a set of orthonormal maximally entangled states in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ consisting of fewer than d2{d}^{2}d2 vectors which have no additional maximally entangled vectors orthogonal to all of them. We prove that UMEBs do not exist for d=2d=2d=2 and give explicit constructions for a six-member UMEB with d=3d=3d=3 and a 12-member UMEB with d=4d=4d=4.
A way of constructing the unextendible maximally entangled basis (UMEB) from the maximally entangled basis (MEB) is proposed. Consequently, it is shown that if there is an NNN-member UMEB in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$, then there exists a ${(qd)}^{2}\ensuremath{-}q({d}^{2}\ensuremath{-}N)$-member UMEB in ${\mathbb{C}}^{qd}\ensuremath{\bigotimes}{\mathbb{C}}^{qd}$ for any $q\ensuremath{\in}\mathbb{N}$. This improves the results in Wang et al. [Phys. Rev. A 90, 034301 (2014)], which shows that there exists a ${(qd)}^{2}\ensuremath{-}({d}^{2}\ensuremath{-}N)$-member UMEB in ${\mathbb{C}}^{qd}\ensuremath{\bigotimes}{\mathbb{C}}^{qd}$ provided that an NNN-member UMEB exsits in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$. In addition, a very easy way of constructing UMEB in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}^{\ensuremath{'}}}$ with $d<{d}^{\ensuremath{'}}$ is presented, which promotes and covers all the previous related work.
We solve the unextendible maximally entangled basis (UMEB) problem in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}^{\ensuremath{'}}}(d\ensuremath{\ne}{d}^{\ensuremath{'}}).$ The results show that a UMEB will always exist. In addition, there might be two sets of UMEBs with different numbers. The main difficulty is to prove the unextendibility of the set of states. We give an explicit construction of UMEBs by considering the Schmidt number of the complementary space of the states we construct.
Connecting the UMEB in Cd⨂Cd\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}Cd⨂Cd with partial Hadamard matrices
We study the unextendible maximally entangled bases (UMEB) in Cd⨂Cd\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}Cd⨂Cd and connect it with the partial Hadamard matrix. Firstly, we show that for a given special UMEB in Cd⨂Cd\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}Cd⨂Cd, there is a partial Hadamard matrix can not extend to a complete Hadamard matrix in Cd\mathbb{C}^{d}Cd. As a corollary, any (d−1)×d(d-1)\times d(d−1)×d partial Hadamard matrix can extend to a complete Hadamard matrix. Then we obtain that for any ddd there is an UMEB except d=p or 2pd=p\ \text{or}\ 2pd=p or 2p, where p≡3mod  4p\equiv 3\mod 4p≡3mod4 and ppp is a prime. Finally, we argue that there exist different kinds of constructions of UMEB in Cnd⨂Cnd\mathbb{C}^{nd}\bigotimes\mathbb{C}^{nd}Cnd⨂Cnd for any n∈Nn\in \mathbb{N}n∈N and d=3×5×7d=3\times5 \times7d=3×5×7.
We study an unextendible maximally entangled basis in arbitrary bipartite spaces. A systematic way of constructing a set of d2{d}^{2}d2 orthonormal maximally entangled states in ${\mathbb{C}}^{d}⨂{\mathbb{C}}^{{d}^{\ensuremath{'}}}(\frac{{d}^{\ensuremath{'}}}{2}<d<{d}^{\ensuremath{'}})$ is provided. The complementary space of the set of these d2{d}^{2}d2 orthonormal maximally entangled states contains no maximally entangled states that are orthogonal to all of them. Furthermore, we investigate mutually unbiased bases in which all the bases are unextendible maximally entangled ones. We present two unextendible maximally entangled bases in C2⨂C3{\mathbb{C}}^{2}⨂{\mathbb{C}}^{3}C2⨂C3 which are mutually unbiased.