Extension of the unextendible product bases of 5-qubit under coarsening the system. (English) Zbl 1509.81227
Summary: It has been shown that there exist unextendible product bases (UPBs) with sizes of 6, 8, 9, and 10 in the 5-qubit system. These product vectors are not UPB any more when they are regarded as product vectors in the coarsened systems (i.e., some subsystems are merged, e.g., if two subsystems are merged, the system becomes \(2\otimes 2\otimes 2\otimes 4\)). In this paper, we show that these product states can be expanded to UPBs in the coarsened system by adding some product vectors. Comparing with the existing UPBs, we get a series of novel UPBs which have different sizes or have the same size as that of the existing ones but are not equivalent to them.
MSC:
| 81P55 | Special bases (entangled, mutual unbiased, etc.) |
Software:
QETLABReferences:
| [1] | Bennett, CH; DiVincenzo, DP; Mor, T., Unextendible product bases and bound entanglement, Phys. Rev. Lett., 82, 5385 (1999) · doi:10.1103/PhysRevLett.82.5385 |
| [2] | DiVincenzo, DP; Mor, T.; Shor, PW, Unextendible product bases, uncompletable product bases and bound entanglement, Commun. Math. Phys., 238, 379-410 (2003) · Zbl 1027.81004 · doi:10.1007/s00220-003-0877-6 |
| [3] | Pittenger, AO, Unextendible product bases and the construction of inseparable states, Lin. Alg. Appl., 359, 235-248 (2003) · Zbl 1017.15011 · doi:10.1016/S0024-3795(02)00423-8 |
| [4] | Sollid, PY; Leinaas, JM; Myrheim, J., Unextendible product bases and extremal density matrices with positive partial transpose, Phys. Rev. A, 84 (2011) · doi:10.1103/PhysRevA.84.042325 |
| [5] | Terhal, BM, A family of indecomposable positive linear maps based on entangled quantum states, Lin. Alg. Appl., 323, 61-73 (2001) · Zbl 0977.15014 · doi:10.1016/S0024-3795(00)00251-2 |
| [6] | Demianowicz, M.; Augusiak, R., From unextendible product bases to genuinely entangled subspaces, Phys. Rev. A, 98 (2018) · doi:10.1103/PhysRevA.98.012313 |
| [7] | Wang, K.; Chen, L.; Zhao, L., \(4\times 4\) unextendible product basis and genuinely entangled space, Quant. Inf. Process., 18, 1-28 (2019) · Zbl 1508.81212 · doi:10.1007/s11128-019-2324-4 |
| [8] | Shen, Y.; Chen, L., Construction of genuine multipartite entangled states, J. Phys. A Math. Theor., 53 (2020) · Zbl 1514.81059 · doi:10.1088/1751-8121/ab7521 |
| [9] | Augusiak, R.; Stasińska, J.; Hadley, C., Bell inequalities with no quantum violation and unextendable product bases, Phys. Rev. Lett., 107 (2011) · doi:10.1103/PhysRevLett.107.070401 |
| [10] | Chen, J.; Chen, L.; Zeng, B., Unextendible product basis for fermionic systems, J. Math. Phys, 55 (2014) · Zbl 1295.81027 · doi:10.1063/1.4893358 |
| [11] | Chen, J.; Johnston, N., The minimum size of unextendible product bases in the bipartite case (and some multipartite cases), Commun. Math. Phys., 333, 351-365 (2015) · Zbl 1306.81010 · doi:10.1007/s00220-014-2186-7 |
| [12] | Shi, F.; Li, MS; Chen, L., Strong quantum nonlocality for unextendible product bases in heterogeneous systems, J. Phys. A Math. Theor., 55 (2021) · Zbl 1499.81019 · doi:10.1088/1751-8121/ac3bea |
| [13] | Cohen, SM, Understanding entanglement as resource: Locally distinguishing unextendible product bases, Phys. Rev. A, 77 (2008) · doi:10.1103/PhysRevA.77.012304 |
| [14] | Zhang, ZC; Wu, X.; Zhang, X., Locally distinguishing unextendible product bases by using entanglement efficiently, Phys. Rev. A, 101 (2020) · doi:10.1103/PhysRevA.101.022306 |
| [15] | Duan, R.; Xin, Y.; Ying, M., Locally indistinguishable subspaces spanned by three-qubit unextendible product bases, Phys. Rev. A, 81 (2010) · doi:10.1103/PhysRevA.81.032329 |
| [16] | Shi, F.; Li, MS; Zhang, X., Unextendible and uncompletable product bases in every bipartition, New J. Phys., 24 (2022) · doi:10.1088/1367-2630/ac9e14 |
| [17] | Johnston, N., The structure of qubit unextendible product bases, J. Phys. A Math. Theor., 47 (2014) · Zbl 1304.81038 · doi:10.1088/1751-8113/47/42/424034 |
| [18] | Chen, L.; Dokovic, DZ, Nonexistence of n-qubit unextendible product bases of size \(2^n-5\), Quant. Inf. Process., 17, 1-10 (2018) · Zbl 1402.81046 · doi:10.1007/s11128-017-1791-8 |
| [19] | Chen, L.; Dokovic, DZ, Multiqubit UPB: the method of formally orthogonal matrices, J. Phys. A Math. Theor., 51 (2018) · Zbl 1396.81029 · doi:10.1088/1751-8121/aac53b |
| [20] | Wang, K.; Chen, L.; Shen, Y., Constructing \(2\times 2\times 4\) and \(4\times 4\) unextendible product bases and positive-partial-transpose entangled states, Lin. Alg. Appl., 69, 131-146 (2021) · Zbl 1458.81009 |
| [21] | Zhang, T., Chen, L.: Constructing unextendible product bases from multiqubit ones. arXiv:2203.08397 |
| [22] | Chen, L.; Dokovic, DZ, Orthogonal product bases of four qubits, J. Phys. A Math. Theor., 50 (2017) · Zbl 1375.81050 · doi:10.1088/1751-8121/aa8546 |
| [23] | Chen, L.; Dokovic, DZ, Multiqubit UPB: the method of formally orthogonal matrices, J. Phys. A Math. Theor., 51 (2018) · Zbl 1396.81029 · doi:10.1088/1751-8121/aac53b |
| [24] | De Baerdemacker, S.; De Vos, A.; Chen, L., The Birkhoff theorem for unitary matrices of arbitrary dimensions, Lin. Alg. Appl., 514, 151-164 (2017) · Zbl 1349.15034 · doi:10.1016/j.laa.2016.10.028 |
| [25] | Feng, K., Unextendible product bases and 1-factorization of complete graphs, Disc. Appl. Math., 154, 942-949 (2006) · Zbl 1094.05043 · doi:10.1016/j.dam.2005.10.011 |
| [26] | Wang, Y.L., Li, M.S., Fei, S.M., et al.: Constructing unextendible product bases from the old ones. arXiv:1703.06542 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
