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Quantum Information Processing

Mutually unbiased maximally entangled bases in $$\mathbb {C}^d\otimes \mathbb {C}^{kd}$$ C d ⊗ C k d

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

Published Apr 10, 2015• 33 citations
Publisher

Abstract

We study maximally entangled bases in bipartite systems $$\mathbb {C}^d \otimes \mathbb {C}^{kd}\ (k\in Z^{+})$$ , 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 $$\mathbb {C}^2 \otimes \mathbb {C}^{4}$$ and three maximally entangled bases in $$\mathbb {C}^2 \otimes \mathbb {C}^{6}$$ that are mutually unbiased are presented.

DOI
10.1007/s11128-015-0980-6

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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.
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Mutually unbiased maximally entangled bases in Cd⊗Cd\mathbb{C}^d\otimes\mathbb{C}^dCd⊗Cd
We study mutually unbiased maximally entangled bases (MUMEB's) in bipartite system Cd⊗Cd(d≥3)\mathbb{C}^d\otimes\mathbb{C}^d (d \geq 3)Cd⊗Cd(d≥3). We generalize the method to construct MUMEB's given in [16], by using any commutative ring RRR with ddd elements and generic character of (R,+)(R,+)(R,+) instead of Zd=Z/dZ\mathbb{Z}_d=\mathbb{Z}/d\mathbb{Z}Zd​=Z/dZ. Particularly, if d=p1a1p2a2…psasd=p_1^{a_1}p_2^{a_2}\ldots p_s^{a_s}d=p1a1​​p2a2​​…psas​​ where p1,…,psp_1, \ldots, p_sp1​,…,ps​ are distinct primes and 3≤p1a1≤⋯≤psas3\leq p_1^{a_1}\leq\cdots\leq p_s^{a_s}3≤p1a1​​≤⋯≤psas​​, we present p1a1−1p_1^{a_1}-1p1a1​​−1 MUMEB's in Cd⊗Cd\mathbb{C}^d\otimes\mathbb{C}^dCd⊗Cd by taking R=Fp1a1⊕⋯⊕FpsasR=\mathbb{F}_{p_1^{a_1}}\oplus\cdots\oplus\mathbb{F}_{p_s^{a_s}}R=Fp1a1​​​⊕⋯⊕Fpsas​​​, direct sum of finite fields (Theorem 3.3).
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 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.
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.