Abstract
An orthogonal basis \({\mathcal {B}}_{9}\) for the Hilbert space C3 × C3 was presented by Bennett et al. (Phys. Rev. A 59, 1070, 1999) which was illustrated in a visual figure in their report. The character of the construction is that each base vector is a product state, thus any distinguishing operator cannot create entanglement. In this paper, we mainly focus on some new constructions of orthogonal product basis quantum states in the high-dimensional quantum systems. Especially, as for the quantum system of (2m + 1) ⊗ (2m + 1), where m ∈ Z and m ≥ 2, we have provided the direct construction in mathematical method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bennett, C.H., Divincenzo, D.P., Fuchs, C.A., Mor, T., Rains, E., Shor, P.W., Smolin, J.A., Wootters, W.K.: Phys. Rev. A 59, 1070 (1999)
Bennett, C.H., Divincenzo, D.P., Mor, T., Shor, P.W., Smolin, J.A., Terhal, B.M.: Phys. Rev. Lett. 82, 5385 (1999)
Walgate, J., Short, A.J., Hardy, L., Vedral, V.: Phys. Rev. Lett. 85, 4972 (2000)
Ghosh, S., Kar, G., Roy, A., Sen, A., Sen, U.: Phys. Rev. Lett. 87, 277902 (2001)
Walgate, J., Hardy, L.: Phys. Rev. Lett. 89, 147901 (2002)
Divincenzo, D.P., Mor, T., Shor, P.W., Smolin, J.A., Terhal, B.M.: Commun. Math. Phys. 238, 379–410 (2003)
Horodecki, M., Sen, A., Sen, U., Horodecki, K.: Phys. Rev. Lett. 90, 047902 (2003)
Chen, P., Li, C.: Phys. Rev. A 70, 022306 (2004)
Rinaldis, S.D.: Phys. Rev. A 70, 022309 (2004)
Gregoratti, M., Werner, R.F.: J. Math. Phys. 45, 2600 (2004)
Fan, H.: Phys. Rev. Lett. 92, 177905 (2004)
Ekert, A.K.: Phys. Rev. Lett. 67, 661 (1991)
Bennett, C.H., Fuchs, C.A., Smolin, J.A. In: Hirota, O., Holevo, A.S., Caves, C.M. (eds.) : Quantum communication, Computing, and Measurement, p 79. Plenum, New York (1997). e-print arXiv:quant-ph/9611006
Buhrman, H., Cleve, R., Wigderson, A.: Proceesing of the 30th Annual ACM Symposium on the Theory of Computing, Dallas, 1998, p 63. ACM, Los Alamitos (1998). e-print arXiv:quant-ph/9802040
Bennett, C.H., Brassard, G., Crepeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Phys. Rev. Lett. 70, 1985 (1993)
Shor, P.W.: SIAM J. Comput. 26, 1484 (1997) , and references therein
Grover, L.: Phys. Rev. Lett. 79, 325 (1997)
Duan, R.Y., Feng, Y., Xin, Y., Ying, M.S.: IEEE Trans. Inf. Theory 55, 1320 (2009)
Duan, R.Y., Xin, Y., Ying, M.S.: Phys. Rev. A 81, 032329 (2010)
Yu, N.K., Duan, R.Y., Ying, M.S.: Phys. Rev. A 84, 012304 (2011)
Yu, N.K., Duan, R.Y., Ying, M.S.: Phys. Rev. Lett. 109, 020506 (2012)
Bandyopadhyay, S., Ghosh, S., Kar, G.: New J. Phys. 13, 123013 (2011)
Bandyopadhyay, S.: Phys. Rev. A 85, 042319 (2012)
Cosentino, A.: Phys. Rev. A 87, 012321 (2013)
Yang, Y.H., Gao, F., Tian, G.J., Cao, T.Q., Wen, Q.Y.: Phys. Rev. A 88, 024301 (2013)
Zhang, Z.C., Wen, Q.Y., Gao, F., Tian, G.J., Cao, T.Q.: Quantum Inf. Process 13, 795 (2014)
Feng, Y., Shi, Y.Y.: IEEE Trans. Inf. Theory 55, 2799 (2009)
Childs, A.M., Leung, D., Mančinska, L., Ozols, M.: Commun. Math. Phys. 323, 1121 (2013)
Colbourn, C.J., Dinitz, J.H.: Handbook of Combinatorial Designs (2nd Edn.), Chapman and Hall/CRC, Taylor and Francis Group, Boca Raton, ISBN 1.58488.506.8
Acknowledgements
This work is supported by NSFC (Grant Nos. 61402148,61601171), Natural Science Foundation of Hebei Province (F2015205114), Doctoral Scientific Fund Project of Hebei Normal University (F2016B05).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zuo, H., Liu, S. & Yang, Y. New Constructions of Orthogonal Product Basis Quantum States. Int J Theor Phys 57, 1597–1603 (2018). https://doi.org/10.1007/s10773-018-3686-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10773-018-3686-6