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International Journal of Theoretical Physics

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, Shao-Ming Fei

Published Jul 7, 2015• 6 citations
Publisher

Abstract

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.

DOI
10.1007/s10773-015-2731-y

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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
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Mutually unbiased maximally entangled bases in Cd⊗Ckd\mathbb {C}^d\otimes \mathbb {C}^{kd}Cd⊗Ckd C d ⊗ C k d
We study maximally entangled bases in bipartite systems Cd⊗Ckd (k∈Z+)\mathbb {C}^d \otimes \mathbb {C}^{kd}\ (k\in Z^{+})Cd⊗Ckd (k∈Z+) , which are mutually unbiased. By systematically constructing maximally entangled bases, we present an approach in constructing mutually unbiased maximally entangled bases. In particular, five maximally entangled bases in C2⊗C4\mathbb {C}^2 \otimes \mathbb {C}^{4}C2⊗C4 and three maximally entangled bases in C2⊗C6\mathbb {C}^2 \otimes \mathbb {C}^{6}C2⊗C6 that are mutually unbiased are presented.
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.