{"title":"Local unitary classification of sets of generalized Bell states in \\(\\mathbb{C}^{d}\\otimes \\mathbb{C}^{d}\\)","authors":"Cai-Hong Wang, Jiang-Tao Yuan, Mao-Sheng Li, Ying-Hui Yang, Shao-Ming Fei","doi":"10.1140/epjqt/s40507-025-00393-8","DOIUrl":null,"url":null,"abstract":"<div><p>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 <span>\\(l\\ (\\geq 2)\\)</span> generalized Bell states (GBSs), high-dimensional generalizations of Bell states, in a bipartite system <span>\\(\\mathbb{C}^{d}\\otimes \\mathbb{C}^{d}\\)</span> with <span>\\(d\\geq 3\\)</span>. Based on this classification, we determine the minimal cardinality of indistinguishable GBS sets in <span>\\(\\mathbb{C}^{6}\\otimes \\mathbb{C}^{6}\\)</span> 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 <i>l</i> GBSs (<i>l</i>-GBS sets). We then establish LU-classification for all 2-GBS, 3-GBS, 4-GBS and 5-GBS sets in <span>\\(\\mathbb{C}^{6}\\otimes \\mathbb{C}^{6}\\)</span>. 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 <span>\\(d = 6\\)</span> 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).</p></div>","PeriodicalId":547,"journal":{"name":"EPJ Quantum Technology","volume":"12 1","pages":""},"PeriodicalIF":5.6000,"publicationDate":"2025-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://epjquantumtechnology.springeropen.com/counter/pdf/10.1140/epjqt/s40507-025-00393-8","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"EPJ Quantum Technology","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1140/epjqt/s40507-025-00393-8","RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"OPTICS","Score":null,"Total":0}
引用次数: 0
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 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).
期刊介绍:
Driven by advances in technology and experimental capability, the last decade has seen the emergence of quantum technology: a new praxis for controlling the quantum world. It is now possible to engineer complex, multi-component systems that merge the once distinct fields of quantum optics and condensed matter physics.
EPJ Quantum Technology covers theoretical and experimental advances in subjects including but not limited to the following:
Quantum measurement, metrology and lithography
Quantum complex systems, networks and cellular automata
Quantum electromechanical systems
Quantum optomechanical systems
Quantum machines, engineering and nanorobotics
Quantum control theory
Quantum information, communication and computation
Quantum thermodynamics
Quantum metamaterials
The effect of Casimir forces on micro- and nano-electromechanical systems
Quantum biology
Quantum sensing
Hybrid quantum systems
Quantum simulations.