Local unitary classification of sets of generalized Bell states in \(\mathbb{C}^{d}\otimes \mathbb{C}^{d}\)

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 \(l\ (\geq 2)\) generalized Bell states (GBSs), high-dimensional generalizations of Bell states, in a bipartite system \(\mathbb{C}^{d}\otimes \mathbb{C}^{d}\) with \(d\geq 3\). Based on this classification, we determine the minimal cardinality of indistinguishable GBS sets in \(\mathbb{C}^{6}\otimes \mathbb{C}^{6}\) 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 \(\mathbb{C}^{6}\otimes \mathbb{C}^{6}\). 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 \(d = 6\) 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).