Abstract
So far, very little is known about local indistinguishability of multipartite orthogonal product bases except some special cases. We first give a method to construct an orthogonal product basis with n parties each holding a \(\frac{1}{2}(n+1)\)-dimensional system, where \(n\ge 5\) and n is odd. The proof of the local indistinguishability of the basis exhibits that it is a sufficient condition for the local indistinguishability of an orthogonal multipartite product basis that all the positive operator-valued measure elements of each party can only be proportional to the identity operator to make further discrimination feasible. Then, we construct a set of n-partite product states, which contains only 2n members and cannot be perfectly distinguished by local operations and classic communication. All the results lead to a better understanding of the phenomenon of quantum nonlocality without entanglement in multipartite and high-dimensional quantum systems.
Similar content being viewed by others
References
Liu, B., Gao, F., Huang, W., et al.: QKD-based quantum private query without a failure probability. Sci. China Phys. Mech. Astron. 58, 100301 (2015)
Luo, M.X., Wang, X.J.: Universal quantum computation with qudits. Sci. China Phys. Mech. Astron. 57(9), 1712–1717 (2014)
Dong, H.H., Guo, B.Y., Yin, B.S.: Generalized fractional supertrace identity for hamiltonian structure of Nls–Mkdv hierarchy with self-consistent sources. Anal. Math. Phys. 6(2), 199–209 (2016)
Bennett, C.H., et al.: Quantum nonlocality without entanglement. Phys. Rev. A 59, 1070 (1999)
Bennett, C.H., DiVincenzo, D.P., Mor, T., Shor, P.W., Smolin, J.A., Terhal, B.M.: Unextendible product bases and bound entanglement. Phys. Rev. Lett. 82, 5385 (1999)
DiVincenzo, D.P., Mor, T., Shor, P.W., Smolin, J.A., Terhal, M.: Unextendible product bases, uncompleteable product bases and bound entanglement. Commun. Math. Phys. 238, 379 (2003)
Zhao, Q.L., Li, X.Y.: A bargmann system and the involutive solutions associated with a new 4-order lattice hierarchy. Anal. Math. Phys. 6(3), 237–254 (2016)
Walgate, J., Short, A.J., Hardy, L., Vedral, V.: Local distinguishability of multipartite orthogonal quantum states. Phys. Rev. Lett. 85, 4972 (2000)
Walgate, J., Hardy, L.: Nonlocality, asymmetry, and distinguishing bipartite states. Phys. Rev. Lett. 89, 147901 (2002)
Feng, Y., Shi, Y.Y.: Characterizing locally indistinguishable orthogonal product states. IEEE Trans. Inf. Theory 55, 2799 (2009)
Niset, J., Cerf, N.J.: Multipartite nonlocality without entanglement in many dimensions. Phys. Rev. A 74, 052103 (2006)
Rinaldis, S.D.: Distinguishability of complete and unextendible product bases. Phys. Rev. A 70, 022309 (2004)
Cao, T., Gao, F., Tian, G., et al.: Local discrimination scheme for some unitary operations. Sci. China Phys. Mech. Astron. 59, 690311 (2016). doi:10.1007/s11433-016-0121-8
Zhang, T.Q., Ma, W.B., Meng, X.Z., Zhang, T.H.: Periodic solution of a prey–predator model with nonlinear state feedback control. Appl. Math. Comput. 266, 95–107 (2015)
Jiang, W., Ren, X.J., Wu, Y.C., Zhou, Z.W., Guo, G.C., Fan, H.: A sufficient and necessary condition for \(2n-1\) orthogonal states to be locally distinguishable in a \(C^{2}\otimes C^{n}\) system. J. Phys. A Math. Theor. 43, 325303 (2010)
Yang, Y.H., Gao, F., Tian, G.J., Cao, T.Q., Wen, Q.Y.: Local distinguishability of orthogonal quantum states in a \(2\otimes 2\otimes 2\) system. Phys. Rev. A 88, 024301 (2013)
Childs, A.M., et al.: A framework for bounding nonlocality of state discrimination. Commun. Math. Phys. 323, 1121–1153 (2013)
Ma, T., Zhao, M.J., Wang, Y.K., Fei, S.M.: Noncommutativity and local indistinguishability of quantum states. Sci. Rep. 4, 6336 (2014)
Zhang, Z.C., Gao, F., Tian, G.J., Cao, T.Q., Wen, Q.Y.: Nonlocality of orthogonal product basis quantum states. Phys. Rev. A 90, 022313 (2014)
Wang, Y.L., Li, M.S., Zheng, Z.J., Fei, S.M.: Nonlocality of orthogonal product-basis quantum states. Phys. Rev. A 92, 032313 (2015)
Zhang, Z.C., Gao, F., Qin, S.J., Yang, Y.H., Wen, Q.Y.: Nonlocality of orthogonal product states. Phys. Rev. A 92, 012332 (2015)
Yu, S.X., Oh, C.H.: Detecting the local indistinguishability of maximally entangled states. arXiv:1502.01274v1 [quant-ph] (2015)
Zhang, Z.C., Gao, F., Cao, Y., Qin, S.J., Wen, Q.Y.: Local indistinguishability of orthogonal product states. Phys. Rev. A 93, 012314 (2016)
Johnston, N.: The structure of qubit unextendible product bases. J. Phys. A Math. Theor. 47, 424034 (2014)
Chen, J.X., Johnston, N.: The minimum size of unextendible product bases in the bipartite case (and some multipartite cases). Commun. Math. Phys. 333, 351–365 (2015)
Johnston, N.: In: Severini, S., Brandao, F. (eds.) Proceedings of the Eighth Conference on the Theory of Quantum Computation, Communication and Cryptography (Schloss DagstuhlCLeibniz-Zentrum für Informatik, Dagstuhl, 2013), vol. 22, pp. 93–105 (2013)
Lebl, J., Shakeel, A., Wallach, N.: Local distinguishability of generic unentangled orthonormal bases. Phys. Rev. A 93, 012330 (2016)
Wang, Y.L., Li, M.S., Zheng, Z.J., Fei, S.M.: The local indistinguishability of multipartite product states. arXiv:1603.01731v1 (2016)
Halder, S.: On a class of small nonlocal set of n-party orthogonal product states. arXiv:1603.08438v1 (2016)
Xu, G.B., Wen, Q.Y., Qin, S.J., Yang, Y.H., Gao, F.: Quantum nonlocality of multipartite orthogonal product states. Phys. Rev. 93, 032341 (2016)
Acknowledgements
This work is supported by NSFC (Grant Nos. 61572081, 61672110, 61402148).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Xu, GB., Wen, QY., Gao, F. et al. Local indistinguishability of multipartite orthogonal product bases. Quantum Inf Process 16, 276 (2017). https://doi.org/10.1007/s11128-017-1725-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11128-017-1725-5