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Harbin Ligong Daxue xuebao

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

Qian Yang

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

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.

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International Journal of Theoretical Physics•Jul 7, 2015•6 citations
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.
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.
In this paper, we study the concrete construction of mutually unbiased maximally entangled bases in bipartite systems C d C kd(k ∈ Z+). We first analyze and simplify the sufficient and necessary conditions of two maximally entangled bases to be mutually unbiased in C d C kd, then we use matrix forms to illustrate these conditions in C2 C2 k and C3 C3 k and generalize them in C d C kd. Thus we find that the sufficient and necessary conditions of two maximally entangled bases to be mutually unbiased in C d C kd are translating to the conditions of transit matrices T kd between two orthonormal bases in C kd satisfy, that is, T kd can be divided into k2 submatrices of d × d satisfying same equations. So the problem of constructing mutually unbiased maximally entangle bases in C d C kd(k ∈ Z+) are changing to the choice of transit matrices in C kd. According the above equations of transit matrices in C kd, we first construct two transit matrices in C2, C3 and C4, then we find that using any unitary matrix with nonzero entities in C k to tensor product the above chosen transit matrices in C d(d = 2, 3, 4) from left, we can easily get the transit matrices in C kd. Hence the choice of transit matrices in C kd is changing to the choice of unitary matrices with nonzero entities in C k. Until now, the problem of constructing mutually unbiased maximally entangled bases in C d C kd is really simplified to the choice of unitary matrices in C k.Since the unitary matrices with nonzero entities in C k are always exist, we can confirm that mutually unbiased maximally entangled bases in C d C kd(d = 2, 3, 4; k ∈ Z+) always exist. Based on the two transit matrices we construct in each space in C kd(d = 2, 3, 4; k = 1, 2, 3), we want to describe the concrete construction of two mutually unbiased maximally entangled bases in C d C kd(d = 2, 3, 4; k = 1, 2, 3). As examples, we present a pair of mutually unbiased maximally entangled bases in C2 C6 and C3 C6, whose dimensions are not prime powers.
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.