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New Journal of Physics

Construction of local unitary equivalent quantum states via local unitary equivalent orthogonal arrays

Zhang Xiao, Shanqi Pang, Mengqian Chen, Shao-Ming Fei

Published Aug 1, 2026• 0 citations
Publisher

Abstract

Abstract Two orthogonal arrays (OAs) are local unitary (LU) equivalent if they lead to LU-equivalent quantum states. Finding LU-equivalent OAs is an open problem posed by Goyeneche et al ( Phys. Rev. A 90 022316). In this paper, we develop methods of constructing infinite families of LU-equivalent OAs, both fixed-level and mixed-level, by investigating the constructions, structures and properties of OAs. We consequently provide a solution to the open problem. The results significantly advance our understanding of LU-equivalent quantum states and their potential applications in quantum information processing.

DOI
10.1088/1367-2630/ae99fa

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Abstract Two orthogonal arrays (OAs) are local unitary (LU) equivalent if they lead to LU-equivalent quantum states. Finding LU-equivalent OAs is an open problem posed by Goyeneche et al ( Phys. Rev. A 90 022316). In this paper, we develop methods of constructing infinite families of LU-equivalent OAs, both fixed-level and mixed-level, by investigating the constructions, structures and properties of OAs. We consequently provide a solution to the open problem. The results significantly advance our understanding of LU-equivalent quantum states and their potential applications in quantum information processing.
Orthogonal arrays (OAs) are combinatorial configurations with applications in experimental design, error-correcting codes, and quantum information. Local unitary (LU) equivalence provides a natural framework for classifying multipartite entangled states. Using the correspondence between OAs and quantum states, Goyeneche and Życzkowski [Phys. Rev. A, 2014, 90: 022316] posed the problem of determining when OAs are LULULU equivalent. In this paper, we construct OAs from generator matrices and establish conditions for Fourier-based LULULU equivalence of OAs and irredundant orthogonal arrays (IrOAs). For prime alphabets, we identify the Fourier partners of the linear OAs considered here with the arrays of their corresponding dual codes. This gives an explicit connection between LULULU equivalence and coding theory. We also identify families of LULULU equivalent OAs arising from specific classes of linear codes.
The necessary and sufficient conditions for the equivalence of arbitrary nnn-qubit pure quantum states under local unitary (LU) operations derived in [B. Kraus, Phys. Rev. Lett. 104, 020504 (2010)] are used to determine the different LU-equivalence classes of up to five-qubit states. Due to this classification new parameters characterizing multipartite entanglement are found and their physical interpretation is given. Moreover, the method is used to derive examples of two nnn-qubit states (with n>2n>2n>2 arbitrary) which have the properties that all the entropies of any subsystem coincide; however, the states are neither LU equivalent nor can be mapped into each other by general local operations and classical communication.
The problem of determining whether two states are equivalent by local unitary (LU) operations is important for quantum information processing. In this paper, we propose an alternative perspective to study this problem by comparing the decidabilities of LU equivalence (also known as LU decidabilities for short) between entanglement witnesses and states. We introduce a relation between sets of Hermitian operators in terms of the LU decidability. Then, we compare the LU decidability for the set of entanglement witnesses to the LU decidabilities for several sets of states. By comparison, we establish a hierarchy of these sets in terms of LU decidabilities. Moreover, we realize that the simultaneous LU (SLU) equivalence between tuples of mutually orthogonal projectors is crucial to LU equivalent operators. We reveal by examples that for two tuples of projectors, the partial SLU equivalence cannot ensure the overall SLU equivalence. Generally, we present a necessary and sufficient condition such that two tuples of states are SLU equivalent.
Since two quantum states that are local unitary (LU) equivalent have the same amount of entanglement, it is meaningful to find a practical method to determine the LU equivalence of given quantum states. In this paper, we present a valid process to find the unitary tensor product decomposition for an arbitrary unitary matrix. Then, by using this process, the conditions for determining the local unitary equivalence of quantum states are obtained. A numerical verification is carried out, which shows the practicability of our protocol. We also present a property of LU invariants by using the universality of quantum gates which can be used to construct the complete set of LU invariants.
Local unitary classification of sets of generalized Bell states in Cd⊗Cd\mathbb{C}^{d}\otimes \mathbb{C}^{d}Cd⊗Cd
Two sets of quantum entangled states that are equivalent under local unitary transformations may exhibit identical effectiveness and versatility in various quantum information processing tasks. Consequently, classification under local unitary transformations has become a fundamental issue in the theory of quantum entanglement. The primary objective of this work is to establish a practical LU-classification for all sets of l (≥2)l\ (\geq 2)l (≥2) generalized Bell states (GBSs), high-dimensional generalizations of Bell states, in a bipartite system Cd⊗Cd\mathbb{C}^{d}\otimes \mathbb{C}^{d}Cd⊗Cd with d≥3d\geq 3d≥3 . Based on this classification, we determine the minimal cardinality of indistinguishable GBS sets in C6⊗C6\mathbb{C}^{6}\otimes \mathbb{C}^{6}C6⊗C6 under one-way local operations and classical communication (one-way LOCC). We first propose two classification methods based on LU-equivalence for all sets of l GBSs (l-GBS sets). We then establish LU-classification for all 2-GBS, 3-GBS, 4-GBS and 5-GBS sets in C6⊗C6\mathbb{C}^{6}\otimes \mathbb{C}^{6}C6⊗C6 . Since LU-equivalent sets share identical local distinguishability, it suffices to examine representative GBS sets from equivalent classes. Notably, we identify a one-way LOCC indistinguishable 4-GBS set among these representatives, thereby resolving the case of d=6d = 6d=6 for the problem of determining the minimum cardinality of one-way LOCC indistinguishable GBS sets in (Yuan et al. in Quantum Inf Process. 18:145, 2019) or (Zhang et al. in Phys Rev A 91:012329, 2015).