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Zhongguo kexue. Wulixue Lixue Tianwenxue

The concrete construction of mutually unbiased maximally entangled bases in C~dC~(kd)

Tao Yuan

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

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.

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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.
In this paper, we study the concrete construction of mutually unbiased maximally entangled bases in bipartite systems Cd ⊗Ckd (k ∈ Z+). We first analyze and simplify the sufficient and necessary conditions of two maximally entangled bases to be mutually unbiased in Cd ⊗Ckd, then we use matrix forms to illustrate these conditions in C2 ⊗C2k and C3 ⊗C3k and generalize them in Cd ⊗Ckd. Thus we find that the sufficient and necessary conditions of two maximally entangled bases to be mutually unbiased in Cd ⊗Ckd are translating to the conditions of transit matrices Tkd between two orthonormal bases in Ckd satisfy, that is, Tkd can be divided into k2 submatrices of d ×d satisfying same equations. So the problem of constructing mutually unbiased maximally entangle bases in Cd ⊗Ckd (k ∈ Z+) are changing to the choice of transit matrices in Ckd. According the above equations of transit matrices in Ckd, we first construct two transit matrices in C2, C3 and C4, then we find that using any unitary matrix with nonzero entities in Ck to tensor product the above chosen transit matrices in Cd(d= 2;3;4) from left, we can easily get the transit matrices in Ckd. Hence the choice of transit matrices in Ckd is changing to the choice of unitary matrices with nonzero entities in Ck. Until now, the problem of constructing mutually unbiased maximally entangled bases in Cd ⊗Ckd is really simplified to the choice of unitary matrices in Ck. Since the unitary matrices with nonzero entities in Ck are always exist, we can confirm that mutually unbiased maximally entangled bases in Cd ⊗Ckd(d = 2;3; 4;d ∈ Z+) always exist. Based on the two transit matrices we construct in each space in Ckd(d =2;3; 4;k =1;2;3), we want to describe the concrete construction of two mutually unbiased maximally entangled bases in Cd ⊗Ckd(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.
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 .
Mutually unbiased bases which is also maximally entangled bases is called mutually unbiased maximally entangled bases (MUMEBs). We study the construction of MUMEBs in bipartite system. In detail, we construct 2(p^a-1) MUMEBs in C^d\otimes C^d by properties of Guss sums for arbitrary odd d. It improves the known lower bound p^a-1 for odd d. Certainly, it also generalizes the lower bound 2(p^a-1) for d being a single prime power. Furthermore, we construct MUMEBs in C^d\otimes C^{kd} for general k>= 2 and odd d. We get the similar lower bounds as k,bk,bk,b are both single prime powers. Particularly, when k is a square number, by using mutually orthogonal Latin squares, we can construct more MUMEBs in C^d\otimes C^{kd}, and obtain greater lower bounds than reducing the problem into prime power dimension in some cases.
Mutually unbiased bases which is also maximally entangled bases is called mutually unbiased maximally entangled bases (MUMEBs). We study the construction of MUMEBs in bipartite system. In detail, we construct 2(p^a-1) MUMEBs in \cd by properties of Guss sums for arbitrary odd d. It improves the known lower bound p^a-1 for odd d. Certainly, it also generalizes the lower bound 2(p^a-1) for d being a single prime power. Furthermore, we construct MUMEBs in \ckd for general k\geq 2 and odd d. We get the similar lower bounds as k,b are both single prime powers. Particularly, when k is a square number, by using mutually orthogonal Latin squares, we can construct more MUMEBs in \ckd, and obtain greater lower bounds than reducing the problem into prime power dimension in some cases.