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Construction of mutually unbiased bases in Cd⊗C2^l d

Jun Zhang, Shao-Ming Fei, Hua Nan, Yuan-Hong Tao

Published Jan 1, 2015• 2 citations
OpenAlex

Abstract

We study mutually unbiased bases in C ⊗ C2d . A systematic way of constructing mutually unbiased maximally entangled bases (MUMEBs) in C ⊗ C2d(l ∈ Z) from MUMEBs in Cd⊗Cd(d′ = kd, k ∈ Z), and a general approach to construct mutually unbiased unextendible maximally entangled bases (MUUMEBs) in Cd⊗C2d(l ∈ Z) from MUUMEBs in Cd⊗Cd(d′ = kd+ r, 0 < r < d) have been presented. Detailed examples are given.

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We study mutually unbiased bases in C ⊗ C2d . A systematic way of constructing mutually unbiased maximally entangled bases (MUMEBs) in C ⊗ C2d(l ∈ Z) from MUMEBs in Cd⊗Cd(d′ = kd, k ∈ Z), and a general approach to construct mutually unbiased unextendible maximally entangled bases (MUUMEBs) in Cd⊗C2d(l ∈ Z) from MUUMEBs in Cd⊗Cd(d′ = kd+ r, 0 < r < d) have been presented. Detailed examples are given.
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 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.
We construct two mutually unbiased bases by maximally entangled states (MUMEBsss) in C2⊗C3\mathbb{C}^{2}\otimes \mathbb{C}^{3}C2⊗C3. This is the first example of MUMEBsss in Cd⊗Cd′\mathbb{C}^{d}\otimes \mathbb{C}^{d'}Cd⊗Cd′ when d∤d′d\nmid d'd∤d′, namely d′d'd′ is not divisible by ddd. We show that they cannot be extended to four MUBs in C6\mathbb{C}^6C6. We propose a recursive construction of mutually unbiased bases formed by special entangled states with a fixed Schmidt number kkk (MUSEBkkks). It shows that min⁡{t1,t2}\min \{t_{1},t_{2}\}min{t1​,t2​} MUSEBk1k2k_{1}k_{2}k1​k2​s in Cpd⊗Cqd′\mathbb{C}^{pd}\otimes \mathbb{C}^{qd'}Cpd⊗Cqd′ can be constructed from t1t_{1}t1​ MUSEBk1k_{1}k1​s in Cd⊗Cd′\mathbb{C}^{d}\otimes \mathbb{C}^{d'}Cd⊗Cd′ and t2t_{2}t2​ MUSEBk2k_{2}k2​s in Cp⊗Cq\mathbb{C}^{p}\otimes \mathbb{C}^{q}Cp⊗Cq for any d,d′,p,qd,d',p,qd,d′,p,q. Further, we show that three MUMEBsss exist in Cd⊗Cd′\mathbb{C}^{d}\otimes \mathbb{C}^{d'}Cd⊗Cd′ for any d,d′d,d'd,d′ with d∣d′d\mid d'd∣d′, and two MUMEBsss exist in Cd⊗Cd′\mathbb{C}^{d}\otimes \mathbb{C}^{d'}Cd⊗Cd′ for infinitely many d,d′d,d'd,d′ with d∤d′d\nmid d'd∤d′.