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Quantum Information and Computation

New bounds of mutually unbiased maximally entangled bases in ℂd ⊗ ℂkd

ChengXiaoya, ShangYun

Published Nov 1, 2018• 0 citations
Publisher

Abstract

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, w...
DOI
10.5555/3370227.3370233

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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 $k,b$ 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.

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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.

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This paper is devoted to constructing mutually unbiased maximally entangled bases (MUMEBs) in [Formula: see text], where [Formula: see text] is a prime power. We prove that [Formula: see text] when [Formula: see text] is even, and [Formula: see text] when [Formula: see text] is odd, where [Formula: see text] is the maximal size of the sets of MUMEBs in [Formula: see text]. This highly raises the lower bounds of [Formula: see text] given in D. Xu, Quant. Inf. Process. 18(7) (2019) 213; D. Xu, Quant. Inf. Process. 19(6) (2020) 175. It should de noted that the method used in the paper is completely different from that in D. Xu, Quant. Inf. Process. 18(7) (2019) 213; D. Xu, Quant. Inf. Process. 19(6) (2020) 175.

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We study mutually unbiased maximally entangled bases (MUMEB's) in bipartite system $\mathbb{C}^d\otimes\mathbb{C}^d (d \geq 3)$. We generalize the method to construct MUMEB's given in [16], by using any commutative ring $R$ with $d$ elements and generic character of $(R,+)$ instead of $\mathbb{Z}_d=\mathbb{Z}/d\mathbb{Z}$. Particularly, if $d=p_1^{a_1}p_2^{a_2}\ldots p_s^{a_s}$ where $p_1, \ldots, p_s$ are distinct primes and $3\leq p_1^{a_1}\leq\cdots\leq p_s^{a_s}$, we present $p_1^{a_1}-1$ MUMEB's in $\mathbb{C}^d\otimes\mathbb{C}^d$ by taking $R=\mathbb{F}_{p_1^{a_1}}\oplus\cdots\oplus\mathbb{F}_{p_s^{a_s}}$, direct sum of finite fields (Theorem 3.3).

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Mutually unbiased maximally entangled bases from difference matrices

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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 .

arXiv:2210.01517•Journal of Physics A Mathematical and Theoretical•Sep 14, 2022•5 citations
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.
This paper is devoted to constructing mutually unbiased maximally entangled bases (MUMEBs) in [Formula: see text], where [Formula: see text] is a prime power. We prove that [Formula: see text] when [Formula: see text] is even, and [Formula: see text] when [Formula: see text] is odd, where [Formula: see text] is the maximal size of the sets of MUMEBs in [Formula: see text]. This highly raises the lower bounds of [Formula: see text] given in D. Xu, Quant. Inf. Process. 18(7) (2019) 213; D. Xu, Quant. Inf. Process. 19(6) (2020) 175. It should de noted that the method used in the paper is completely different from that in D. Xu, Quant. Inf. Process. 18(7) (2019) 213; D. Xu, Quant. Inf. Process. 19(6) (2020) 175.
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 .