Abstract
A theoretical scheme is proposed to implement bidirectional quantum controlled teleportation (BQCT) by using a nine-qubit entangled state as a quantum channel, where Alice may transmit an arbitrary two-qubit state called qubits \(A_1\) and \(A_2\) to Bob; and at the same time, Bob may also transmit an arbitrary two-qubit state called qubits \(B_1\) and \(B_2\) to Alice via the control of the supervisor Charlie. Based on our channel, we explicitly show how the bidirectional quantum controlled teleportation protocol works. And we show this bidirectional quantum controlled teleportation scheme may be determinate and secure. Taking the amplitude-damping noise and the phase-damping noise as typical noisy channels, we analytically derive the fidelities of the BQCT process and show that the fidelities in these two cases only depend on the amplitude parameter of the initial state and the decoherence noisy rate.
Similar content being viewed by others
References
Bennett, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys. Rev. Lett. 70(13), 1895–1899 (1993)
Zhang, Z.J.: Controlled teleportation of an arbitrary n-qubit quantum information using quantum secret sharing of classical message. Phys. Lett. A 352(1), 55–58 (2006)
Zhang, Z.J., Man, Z.X.: Multiparty quantum secret sharing of classical messages based on entanglement swapping. Phys. Rev. A 72(2), 022303 (2005)
Shukla, C., Pathak, A.: Hierarchical quantum communication. Phys. Lett. A 377(19), 1337–1344 (2013)
Luo, M.-X., Yun, Deng, Chen, X.-B., Yang, Y.-X.: The faithful remote preparation of general quantum states. Quantum Inf. Process. 12(1), 279–294 (2013)
Sharma, V., Shukla, C., Banerjee, S., Pathak, A.: Controlled Bidirectional Remote State Preparation in Noisy Environment: A Generalized View. arXiv:1409.0833 (2014)
Liu, L.L., Hwang, T.: Controlled remote state preparation protocols via AKLT states. Quantum Inf. Process. 13(7), 1639–1650 (2014)
Wang, D., Ye, L.: Multiparty-controlled joint remote state preparation. Quantum Inf. Process. 12(10), 3223–3237 (2013)
Choudhury, B.S., Dhara, A.: Joint remote state preparation for two-qubit equatorial states. Quantum Inf. Process. 14(1), 373–379 (2015)
Zhang, Z.-H., Shu, L., Mo, Z.-W., Zheng, J., Ma, S.-Y., Luo, M.-X.: Joint remote state preparation between multi-sender and multi-receiver. Quantum Inf. Process. 13(9), 1979–2005 (2014)
Wang, S.F., Liu, Y.M., Chen, J.L., Liu, X.S., Zhang, Z.J.: Deterministic single-qubit operation sharing with five-qubit cluster state. Quantum Inf. Process. 12(7), 2497–2507 (2013)
Ji, Q.B., Liu, Y.M., Yin, X.F., Liu, X.S., Zhang, Z.J.: Quantum operation sharing with symmetric and asymmetric W states. Quantum Inf. Process. 12(7), 2453–2464 (2013)
Hassanpour, S., Houshmand, M.: Efficient controlled quantum secure direct communication based on GHZ-like states. Quantum Inf. Process. 14(2), 739–753 (2015)
Zha, X.W., Zou, Z.C., Qi, J.X., Song, H.Y.: Bidirectional quantum controlled teleportation via five-qubit cluster state. Int. J. Theor. Phys. 52(6), 1740–1744 (2013)
Shukla, C., Banerjee, A., Pathak, A.: Bidirectional controlled teleportation by using 5-qubit states: a generalized view. Int. J. Theor. Phys. 52(10), 3790–3796 (2013)
Thapliyal, K., Pathak, A.: Applications of quantum cryptographic switch: various tasks related to controlled quantum communication can be performed using Bell states and permutation of particles. Quantum Inf. Process. 14(7), 2599–2616 (2015)
Thapliya, K., Verma, A., Pathak, A.: A general method for selecting quantum channel for bidirectional controlled state teleportation and other schemes of controlled quantum communication. Quantum Inf. Process. (2015). doi:10.1007/s11128-015-1124-8
Chen, Y.: Bidirectional quantum controlled teleportation by using a genuine six-qubit entangled state. Int. J. Theor. Phys. 54(1), 269–272 (2015)
Yan, A.: Bidirectional controlled teleportation via six-qubit cluster state. Int. J. Theor. Phys. 52(11), 3870–3873 (2013)
Duan, Y.-J., Zha, X.-W., Sun, X.-M., Xia, J.-F.: Bidirectional quantum controlled teleportation via a maximally seven-qubit entangled state. Int. J. Theor. Phys. 53(8), 2697–2707 (2014)
Fu, H.-Z., Tian, X.-L., Hu, Y.: A general method of selecting quantum channel for bidirectional quantum teleportation. Int. J. Theor. Phys. 53(6), 1840–1847 (2014)
Li, Y.H., Li, X.L., Sang, M.H., Nie, Y.Y., Wang, Z.S.: Bidirectional controlled quantum teleportation and secure direct communication using five-qubit entangled state. Quantum Inf. Process. 12(12), 3835–3844 (2013)
Srinatha, N., Omkar, S., Srikanth, R., Banerjee, S., Pathak, A.: The quantum cryptographic switch. Quantum Inf. Process. 13(1), 59–70 (2014)
O’Brien, J.L., Pryde, G.J., White, A.G., Ralph, T.C., Branning, D.: Demonstration of an all-optical quantum controlled-NOT gate. Nature 426, 264–267 (2003)
Xian-Ting, L.: Classical information capacities of some single qubit quantum noisy channels. Commun. Theor. Phys. 39, 537–542 (2003)
Zheng, S.B.: Scheme for approximate conditional teleportation of an unknown atomic state without the Bell-state measurement. Phys. Rev. A 69(6), 064302 (2004)
Riebe, M., et al.: Deterministic quantum teleportation with atoms. Nature 429, 734–737 (2004)
Bouwmeester, D., Pan, J.W., Mattle, K., et al.: Experimental quantum teleportation. Nature 390, 575–579 (1997)
Cerè, A., Lucamarini, M., Giuseppe, G.D., Tombesi, P.: Experimental test of two-way quantum key distribution in the presence of controlled noise. Phys. Rev. Lett. 96(20), 200501 (2006)
Watrous, J.: PSPACE has constant-round quantum interactive proof systems. Theor. Comput. Sci. 292(3), 575–588 (2003)
Yuan, H., Liu, Y.M., Zhang, W., Zhang, Z.J.: Optimizing resource consumption, operation complexity and efficiency in quantum state sharing. J. Phys. B 41, 145506 (2008)
Acknowledgments
This work is supported by the Natural Science Foundation of Jiangxi Province, China (Grant No. 20142BAB202005), the Research Foundation of state key laboratory of advanced optical communication systems and networks, Shanghai Jiao Tong University, China.
Author information
Authors and Affiliations
Corresponding author
Appendix
Appendix
Alice’s and Bob’s possible measurement result, Charlie’s possible measurement result, and the corresponding locally unitary transformations performed by Alice and Bob on qubits 3, 5, 7 and 9, respectively.
Alice’s and Bob’s result | Charlie’s result | Unitary transformation |
---|---|---|
\(|{\Phi ^{\pm }}\rangle _{A_1 2}|{\Phi ^{\pm }}\rangle _{A_2 4}|{\Phi ^{\pm }}\rangle _{B_1 6}|{\Phi ^{\pm }} \rangle _{B_2 8} \) | \(|0 \rangle _1 \) | \(\left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _3 \otimes \left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _5 \otimes \left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _7 \otimes \left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _9 \) |
\(|{\Phi ^{\pm }}\rangle _{A_1 2}|{\Phi ^{\pm }}\rangle _{A_2 4}|{\Phi ^{\pm }}\rangle _{B_1 6}|{\Psi ^{\pm }} \rangle _{B_2 8} \) | \(|0 \rangle _1 \) | \(\left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _3 \otimes \left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _5 \otimes \left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _7 \otimes \left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _9 \) |
\(|{\Phi ^{\pm }}\rangle _{A_1 2}|{\Phi ^{\pm }}\rangle _{A_2 4}|{\Psi ^{\pm }}\rangle _{B_1 6}|{\Phi ^{\pm }} \rangle _{B_2 8} \) | \(|0 \rangle _1 \) | \(\left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _3 \otimes \left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _5 \otimes \left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _7 \otimes \left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _9 \) |
\(|{\Phi ^{\pm }}\rangle _{A_1 2}|{\Phi ^{\pm }}\rangle _{A_2 4}|{\Psi ^{\pm }}\rangle _{B_1 6}|{\Psi ^{\pm }} \rangle _{B_2 8} \) | \(|0 \rangle _1 \) | \(\left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _3 \otimes \left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _5 \otimes \left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _7 \otimes \left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _9 \) |
\(|{\Phi ^{\pm }}\rangle _{A_1 2}|{\Psi ^{\pm }}\rangle _{A_2 4}|{\Phi ^{\pm }}\rangle _{B_1 6}|{\Phi ^{\pm }} \rangle _{B_2 8} \) | \(|0 \rangle _1 \) | \(\left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _3 \otimes \left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _5 \otimes \left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _7 \otimes \left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _9 \) |
\(|{\Phi ^{\pm }}\rangle _{A_1 2}|{\Psi ^{\pm }}\rangle _{A_2 4}|{\Phi ^{\pm }}\rangle _{B_1 6}|{\Psi ^{\pm }} \rangle _{B_2 8} \) | \(|0 \rangle _1 \) | \(\left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _3 \otimes \left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _5 \otimes \left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _7 \otimes \left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _9 \) |
\(|{\Phi ^{\pm }}\rangle _{A_1 2}|{\Psi ^{\pm }}\rangle _{A_2 4}|{\Psi ^{\pm }}\rangle _{B_1 6}|{\Phi ^{\pm }} \rangle _{B_2 8} \) | \(|0 \rangle _1 \) | \(\left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _3 \otimes \left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _5 \otimes \left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _7 \otimes \left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _9 \) |
\(|{\Phi ^{\pm }}\rangle _{A_1 2}|{\Psi ^{\pm }}\rangle _{A_2 4}|{\Psi ^{\pm }}\rangle _{B_1 6}|{\Psi ^{\pm }} \rangle _{B_2 8} \) | \(|0 \rangle _1 \) | \(\left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _3 \otimes \left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _5 \otimes \left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _7 \otimes \left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _9 \) |
\(|{\Psi ^{\pm }}\rangle _{A_1 2}|{\Phi ^{\pm }}\rangle _{A_2 4}|{\Phi ^{\pm }}\rangle _{B_1 6}|{\Phi ^{\pm }} \rangle _{B_2 8} \) | \(|0 \rangle _1 \) | \(\left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _3 \otimes \left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _5 \otimes \left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _7 \otimes \left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _9 \) |
\(|{\Psi ^{\pm }}\rangle _{A_1 2}|{\Phi ^{\pm }}\rangle _{A_2 4}|{\Phi ^{\pm }}\rangle _{B_1 6}|{\Psi ^{\pm }} \rangle _{B_2 8} \) | \(|0 \rangle _1 \) | \(\left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 1|}\right) _3 \otimes \left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _5 \otimes \left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _7 \otimes \left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _9 \) |
\(|{\Psi ^{\pm }}\rangle _{A_1 2}|{\Psi ^{\pm }}\rangle _{A_2 4}|{\Phi ^{\pm }}\rangle _{B_1 6}|{\Phi ^{\pm }} \rangle _{B_2 8} \) | \(|0 \rangle _1 \) | \(\left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _3 \otimes \left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _5 \otimes \left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _7 \otimes \left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _9 \) |
\(|{\Psi ^{\pm }}\rangle _{A_1 2}|{\Psi ^{\pm }}\rangle _{A_2 4}|{\Phi ^{\pm }}\rangle _{B_1 6}|{\Psi ^{\pm }} \rangle _{B_2 8} \) | \(|0 \rangle _1 \) | \(\left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _3 \otimes \left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _5 \otimes \left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _7 \otimes \left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _9 \) |
\(|{\Psi ^{\pm }}\rangle _{A_1 2}|{\Phi ^{\pm }}\rangle _{A_2 4}|{\Psi ^{\pm }}\rangle _{B_1 6}|{\Phi ^{\pm }} \rangle _{B_2 8} \) | \(|0 \rangle _1 \) | \(\left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _3 \otimes \left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _5 \otimes \left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _7 \otimes \left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _9 \) |
\(|{\Psi ^{\pm }}\rangle _{A_1 2}|{\Phi ^{\pm }}\rangle _{A_2 4}|{\Psi ^{\pm }}\rangle _{B_1 6}|{\Psi ^{\pm }} \rangle _{B_2 8} \) | \(|0 \rangle _1 \) | \(\left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _3 \otimes \left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _5 \otimes \left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _7 \otimes \left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _9 \) |
\(|{\Psi ^{\pm }}\rangle _{A_1 2}|{\Psi ^{\pm }}\rangle _{A_2 4}|{\Psi ^{\pm }}\rangle _{B_1 6}|{\Phi ^{\pm }} \rangle _{B_2 8} \) | \(|0 \rangle _1 \) | \(\left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _3 \otimes \left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _5 \otimes \left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _7 \otimes \left( {|0 \rangle \langle 0|\pm |1 \rangle \langle 1|}\right) _9 \) |
\(|{\Psi ^{\pm }}\rangle _{A_1 2}|{\Psi ^{\pm }}\rangle _{A_2 4}|{\Psi ^{\pm }}\rangle _{B_1 6}|{\Psi ^{\pm }} \rangle _{B_2 8} \) | \(|0 \rangle _1 \) | \(\left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _3 \otimes \left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _5 \otimes \left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _7 \otimes \left( {|0 \rangle \langle 1|\pm |1 \rangle \langle 0|}\right) _9 \) |
\(|{\Phi ^{\pm }}\rangle _{A_1 2}|{\Phi ^{\pm }}\rangle _{A_2 4}|{\Phi ^{\pm }}\rangle _{B_1 6}|{\Phi ^{\pm }} \rangle _{B_2 8} \) | \(|1 \rangle _1 \) | \(\left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _3 \otimes \left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _5 \otimes \left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _7 \otimes \left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _9 \) |
\(|{\Phi ^{\pm }}\rangle _{A_1 2}|{\Phi ^{\pm }}\rangle _{A_2 4}|{\Phi ^{\pm }}\rangle _{B_1 6}|{\Psi ^{\pm }} \rangle _{B_2 8} \) | \(|1 \rangle _1 \) | \(\left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _3 \otimes \left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _5 \otimes \left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _7 \otimes \left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _9 \) |
\(|{\Phi ^{\pm }}\rangle _{A_1 2}|{\Phi ^{\pm }}\rangle _{A_2 4}|{\Psi ^{\pm }}\rangle _{B_1 6}|{\Phi ^{\pm }} \rangle _{B_2 8} \) | \(|1 \rangle _1 \) | \(\left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _3 \otimes \left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _5 \otimes \left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _7 \otimes \left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _9 \) |
\(|{\Phi ^{\pm }}\rangle _{A_1 2}|{\Phi ^{\pm }}\rangle _{A_2 4}|{\Psi ^{\pm }}\rangle _{B_1 6}|{\Psi ^{\pm }} \rangle _{B_2 8} \) | \(|1 \rangle _1 \) | \(\left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _3 \otimes \left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _5 \otimes \left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _7 \otimes \left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _9 \) |
\(|{\Phi ^{\pm }}\rangle _{A_1 2}|{\Psi ^{\pm }}\rangle _{A_2 4}|{\Phi ^{\pm }}\rangle _{B_1 6}|{\Phi ^{\pm }} \rangle _{B_2 8} \) | \(|1 \rangle _1 \) | \(\left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _3 \otimes \left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _5 \otimes \left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _7 \otimes \left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _9 \) |
\(|{\Phi ^{\pm }}\rangle _{A_1 2}|{\Psi ^{\pm }}\rangle _{A_2 4}|{\Phi ^{\pm }}\rangle _{B_1 6}|{\Psi ^{\pm }} \rangle _{B_2 8} \) | \(|1 \rangle _1 \) | \(\left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _3 \otimes \left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _5 \otimes \left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _7 \otimes \left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _9 \) |
\(|{\Phi ^{\pm }}\rangle _{A_1 2}|{\Psi ^{\pm }}\rangle _{A_2 4}|{\Psi ^{\pm }}\rangle _{B_1 6}|{\Phi ^{\pm }} \rangle _{B_2 8} \) | \(|1 \rangle _1 \) | \(\left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _3 \otimes \left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _5 \otimes \left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _7 \otimes \left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _9 \) |
\(|{\Phi ^{\pm }}\rangle _{A_1 2}|{\Psi ^{\pm }}\rangle _{A_2 4}|{\Psi ^{\pm }}\rangle _{B_1 6}|{\Psi ^{\pm }} \rangle _{B_2 8} \) | \(|1 \rangle _1 \) | \(\left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _3 \otimes \left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _5 \otimes \left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _7 \otimes \left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _9 \) |
\(|{\Psi ^{\pm }}\rangle _{A_1 2}|{\Phi ^{\pm }}\rangle _{A_2 4}|{\Phi ^{\pm }}\rangle _{B_1 6}|{\Phi ^{\pm }} \rangle _{B_2 8} \) | \(|1 \rangle _1 \) | \(\left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _3 \otimes \left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _5 \otimes \left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _7 \otimes \left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _9 \) |
\(|{\Psi ^{\pm }}\rangle _{A_1 2}|{\Phi ^{\pm }}\rangle _{A_2 4}|{\Phi ^{\pm }}\rangle _{B_1 6}|{\Psi ^{\pm }} \rangle _{B_2 8} \) | \(|1 \rangle _1 \) | \(\left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _3 \otimes \left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _5 \otimes \left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _7 \otimes \left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _9 \) |
\(|{\Psi ^{\pm }}\rangle _{A_1 2}|{\Psi ^{\pm }}\rangle _{A_2 4}|{\Phi ^{\pm }}\rangle _{B_1 6}|{\Phi ^{\pm }} \rangle _{B_2 8} \) | \(|1 \rangle _1 \) | \(\left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _3 \otimes \left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _5 \otimes \left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _7 \otimes \left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _9 \) |
\(|{\Psi ^{\pm }}\rangle _{A_1 2}|{\Psi ^{\pm }}\rangle _{A_2 4}|{\Phi ^{\pm }}\rangle _{B_1 6}|{\Psi ^{\pm }} \rangle _{B_2 8} \) | \(|1 \rangle _1 \) | \(\left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _3 \otimes \left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _5 \otimes \left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _7 \otimes \left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _9 \) |
\(|{\Psi ^{\pm }}\rangle _{A_1 2}|{\Phi ^{\pm }}\rangle _{A_2 4}|{\Psi ^{\pm }}\rangle _{B_1 6}|{\Phi ^{\pm }} \rangle _{B_2 8} \) | \(|1 \rangle _1 \) | \(\left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _3 \otimes \left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _5 \otimes \left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _7 \otimes \left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _9 \) |
\(|{\Psi ^{\pm }}\rangle _{A_1 2}|{\Phi ^{\pm }}\rangle _{A_2 4}|{\Psi ^{\pm }}\rangle _{B_1 6}|{\Psi ^{\pm }} \rangle _{B_2 8} \) | \(|1 \rangle _1 \) | \(\left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _3 \otimes \left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _5 \otimes \left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _7 \otimes \left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _9 \) |
\(|{\Psi ^{\pm }}\rangle _{A_1 2}|{\Psi ^{\pm }}\rangle _{A_2 4}|{\Psi ^{\pm }}\rangle _{B_1 6}|{\Phi ^{\pm }} \rangle _{B_2 8} \) | \(|1 \rangle _1 \) | \(\left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _3 \otimes \left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _5 \otimes \left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _7 \otimes \left( {|0 \rangle \langle 1|\mp |1 \rangle \langle 0|}\right) _9 \) |
\(|{\Psi ^{\pm }}\rangle _{A_1 2}|{\Psi ^{\pm }}\rangle _{A_2 4}|{\Psi ^{\pm }}\rangle _{B_1 6}|{\Psi ^{\pm }} \rangle _{B_2 8} \) | \(|1 \rangle _1 \) | \(\left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _3 \otimes \left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _5 \otimes \left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _7 \otimes \left( {-|0 \rangle \langle 0|\mp |1 \rangle \langle 1|}\right) _9 \) |
Rights and permissions
About this article
Cite this article
Li, Yh., Jin, Xm. Bidirectional controlled teleportation by using nine-qubit entangled state in noisy environments. Quantum Inf Process 15, 929–945 (2016). https://doi.org/10.1007/s11128-015-1194-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11128-015-1194-7