Local unitary classification of sets of generalized Bell states in $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$
Cai-Hong Wang, Jiang-Tao Yuan, Mao-Sheng Li, Ying-Hui Yang, Shao-Ming Fei
TL;DR
This work addresses the LU-classification of sets of generalized Bell states (GBS) in bipartite systems $C^{d}\otimes C^{d}$ for $d\ge 3$, and determines the minimum cardinality of one-way LOCC indistinguishable GBS sets in $C^{6}\otimes C^{6}$ by establishing $f_{GBS}(6)=4$. The authors develop two Clifford-operators-based LU-classification methods that reduce LU-equivalence to $U$-equivalence of generalized Pauli sets, using essential power $P_e$ to distinguish UC-equivalence. They classify all 2-GBS and 3-GBS sets in $C^{6}\otimes C^{6}$ and provide representative elements for 4-GBS (31 classes) and 5-GBS (112 classes) sets, enabling a complete LU-analysis for these cases. A concrete 4-GBS set $\mathcal{S}_{30}=\{(0,0),(0,2),(2,0),(2,2)\}$ is shown to be one-way LOCC indistinguishable, establishing $f_{GBS}(6)=4$ and advancing the understanding of local discrimination in high-dimensional quantum systems.
Abstract
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 complete LU-classification of all sets of generalized Bell states (GBSs) in bipartite quantum systems $\mathbb{C}^{d}\otimes \mathbb{C}^{d}$ 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 propose first two classification methods based on LU-equivalence for all $l$-GBS sets for $l\geq 2$. 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 [Quant. Info. Proc. 18, 145 (2019)] or [Phys. Rev. A 91, 012329 (2015)].