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Laser Physics

Constructions of mutually unbiased maximally entangled bases in C d ⊗ C d with d an even prime power

Chenghong Wang, Kun Wang, Haitao Ma, Junjing Xing

Published Jun 12, 2026• 0 citations
Publisher

Abstract

Abstract The construction of mutually unbiased maximally entangled bases (MUMEBs) in composite dimensions is a key challenge. In this work, we addresses the problem for bipartite system C d ⊗ C d with d = 2 n . Utilizing exponential sums over finite Galois rings GR ( 4 , n ) , we explicitly construct d 2 maximally entangled bases. Crucially, we prove that a subset of d bases from this set are pairwise mutually unbiased. This finding represents a significant advancement in understanding MUMEBs for even prime power dimensions.

DOI
10.1088/1555-6611/ae68f1

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Abstract The construction of mutually unbiased maximally entangled bases (MUMEBs) in composite dimensions is a key challenge. In this work, we addresses the problem for bipartite system C d ⊗ C d with d = 2 n . Utilizing exponential sums over finite Galois rings GR ( 4 , n ) , we explicitly construct d 2 maximally entangled bases. Crucially, we prove that a subset of d bases from this set are pairwise mutually unbiased. This finding represents a significant advancement in understanding MUMEBs for even prime power dimensions.
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Mutually unbiased maximally entangled bases in Cd⊗Cd\mathbb{C}^d\otimes\mathbb{C}^dCd⊗Cd
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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 .
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.