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:1309.3356 · Physical Review A

Unextendible maximally entangled bases and mutually unbiased bases

Bin Chen, Shao-Ming Fei

Published Sep 4, 2013• 45 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.88.034301

Related papers

Mutually Unbiasedness between Maximally Entangled Bases and Unextendible Maximally Entangled Systems in ℂ 2 ⊗ ℂ 2 k $\mathbb {C}^{2}\otimes \mathbb {C}^{2^{k}}$

Jun Zhang, Hua Nan, Yuan-Hong Tao +1 more

The mutually unbiasedness between a maximally entangled basis (MEB) and an unextendible maximally entangled system (UMES) in the bipartite system $\mathbb {C}^{2}\otimes \mathbb {C}^{2^{k}} (k>1)$ are introduced and discussed first in this paper. Then two mutually unbiased pairs of a maximally entangled basis and an unextendible maximally entangled system are constructed; lastly, explicit constructions are obtained for mutually unbiased MEB and UMES in $\mathbb {C}^{2}\otimes \mathbb {C}^{4}$ and $\mathbb {C}^{2}\otimes \mathbb {C}^{8}$ , respectively.

International Journal of Theoretical Physics•Jul 7, 2015•6 citations

Mutually Unbiased and Unextendible Maximally Entangled Bases in C~2C~3

Qian Yang

To study unextendible maximally entangled bases and mutually unbiased bases which play important roles in quantum computation and quantum information processing,the unextendible maximally entangled bases in space C2C3are studied firstly. Then two sets of mutually unbiased bases from the unextendible maximally entangled bases in C2C3are constructed. The general form are presented,which extend the results in [9]and provide more generalized was of constructing.

Harbin Ligong Daxue xuebao•Jan 1, 2014•0 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

A note on the construction of unextendible maximally entangled bases in ℂd ⊗ ℂd′

Chenghong Wang, Kun Wang

We study the construction of unextendible maximally entangled bases in bipartite systems [Formula: see text] ([Formula: see text], [Formula: see text], [Formula: see text]). We provide a class of maximally entangled states in [Formula: see text] by constructing some unitary matrices in [Formula: see text]. In particular, we present two complete unextendible maximally entangled bases in [Formula: see text] which are mutually unbiased.

International Journal of Modern Physics B•Oct 14, 2022•0 citations

Mutually unbiased maximally entangled bases from difference matrices

Yajuan Zang, Zihong Tian, Hui-Juan Zuo +1 more

Abstract Based on maximally entangled states, we explore the constructions of mutually unbiased bases in bipartite quantum systems. We present a new way to construct mutually unbiased bases by difference matrices in the theory of combinatorial designs. In particular, we establish q mutually unbiased bases with q − 1 maximally entangled bases and one product basis in C q ⊗ C q for arbitrary prime power q . In addition, we construct maximally entangled bases for dimension of composite numbers of non-prime power, such as five maximally entangled bases in C 12 ⊗ C 12 and C 21 ⊗ C 21 , which improve the known lower bounds for d = 3 m , with (3, m ) = 1 in C d ⊗ C d . Furthermore, we construct p + 1 mutually unbiased bases with p maximally entangled bases and one product basis in C p ⊗ C p 2 for arbitrary prime number p .

arXiv:2210.01517•Journal of Physics A Mathematical and Theoretical•Sep 14, 2022•5 citations

Mutually unbiased unextendible maximally entangled bases in Cd ⊗ Cq(d+1)

Liang Bao Tang, Yi Xiang, Zhiwen Mo +1 more

Quantum measurements based on mutually unbiased base (MUB) are commonly used in quantum information processing. Firstly, we construct two completed unextendible maximally entangled bases (UMEBs) in bipartite systems [Formula: see text]. Then, the sufficient and necessary conditions of these two UMEBs develop into MUB can be obtained. Finally, we present some UMEBs that are mutually unbiased in concrete bipartite systems. These results provide a method to construct the best measurement bases.

Modern Physics Letters B•Oct 20, 2020•3 citations
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 Cd⨂Cd\ensuremath′(d\ensuremath′2<d<d\ensuremath′){\mathbb{C}}^{d}⨂{\mathbb{C}}^{{d}^{\ensuremath{'}}}(\frac{{d}^{\ensuremath{'}}}{2}<d<{d}^{\ensuremath{'}})Cd⨂Cd\ensuremath′(2d\ensuremath′​<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.
Mutually Unbiasedness between Maximally Entangled Bases and Unextendible Maximally Entangled Systems in ℂ 2 ⊗ ℂ 2 k C2⊗C2k\mathbb {C}^{2}\otimes \mathbb {C}^{2^{k}}C2⊗C2k
The mutually unbiasedness between a maximally entangled basis (MEB) and an unextendible maximally entangled system (UMES) in the bipartite system C2⊗C2k(k>1)\mathbb {C}^{2}\otimes \mathbb {C}^{2^{k}} (k>1)C2⊗C2k(k>1) are introduced and discussed first in this paper. Then two mutually unbiased pairs of a maximally entangled basis and an unextendible maximally entangled system are constructed; lastly, explicit constructions are obtained for mutually unbiased MEB and UMES in C2⊗C4\mathbb {C}^{2}\otimes \mathbb {C}^{4}C2⊗C4 and C2⊗C8\mathbb {C}^{2}\otimes \mathbb {C}^{8}C2⊗C8 , respectively.
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.
Abstract Based on maximally entangled states, we explore the constructions of mutually unbiased bases in bipartite quantum systems. We present a new way to construct mutually unbiased bases by difference matrices in the theory of combinatorial designs. In particular, we establish q mutually unbiased bases with q − 1 maximally entangled bases and one product basis in C q ⊗ C q for arbitrary prime power q . In addition, we construct maximally entangled bases for dimension of composite numbers of non-prime power, such as five maximally entangled bases in C 12 ⊗ C 12 and C 21 ⊗ C 21 , which improve the known lower bounds for d = 3 m , with (3, m ) = 1 in C d ⊗ C d . Furthermore, we construct p + 1 mutually unbiased bases with p maximally entangled bases and one product basis in C p ⊗ C p 2 for arbitrary prime number p .