Constructing 2 × 2 × 4 and 4 × 4 unextendible product bases and positive-partial-transpose entangled states. Issue 1 (2nd January 2021)

• Record Type: Journal Article
• Authors: Wang, Kai; Chen, Lin; Shen, Yi; Sun, Yize; Zhao, Li-Jun
• Is Part Of: Linear & multilinear algebra. Volume 69:Issue 1(2021)
• Page Start: 131
• Page End: 146
• Publication Date: 2021-01-02
• Subjects: Unextendible product basis (UPB) -- positive partial transpose (PPT) -- entanglement -- four qubit system; 81P45; 03.65.Ud -- 03.67.Mn; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2019.1588849
• Languages: English
• ISSNs: 0308-1087

Abstract

Abstract : The 4-qubit unextendible product bases (UPBs) has been recently studied by [Johnston, J. Phys. A: Math. Theor. 47 (2014) 424034]. Based on this study we show that there is only one UPB of size 6 and six UPBs of size 9 in H = C 2 ⊗ C 2 ⊗ C 4, three UPBs of size 9 in K = C 4 ⊗ C 4, and no UPB of size 7 in H and K . Furthermore we construct a new family of 4-qubit positive-partial-transpose (PPT) entangled states of rank seven, and show that they are also PPT entangled states in H and K, respectively. We analytically derive a geometric measure of a special PPT entangled states.