Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Effect of quantum noise on deterministic remote state preparation of an arbitrary two-particle state via various quantum entangled channels

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

As one of important research branches of quantum communication, deterministic remote state preparation (DRSP) plays a significant role in quantum network. Quantum noises are prevalent in quantum communication, and it can seriously affect the safety and reliability of quantum communication system. In this paper, we study the effect of quantum noise on deterministic remote state preparation of an arbitrary two-particle state via different quantum channels including the \(\chi \) state, Brown state and GHZ state. Firstly, the output states and fidelities of three DRSP algorithms via different quantum entangled channels in four noisy environments, including amplitude-damping, phase-damping, bit-flip and depolarizing noise, are presented, respectively. And then, the effects of noises on three kinds of preparation algorithms in the same noisy environment are discussed. In final, the theoretical analysis proves that the effect of noise in the process of quantum state preparation is only related to the noise type and the size of noise factor and independent of the different entangled quantum channels. Furthermore, another important conclusion is given that the effect of noise is also independent of how to distribute intermediate particles for implementing DRSP through quantum measurement during the concrete preparation process. These conclusions will be very helpful for improving the efficiency and safety of quantum communication in a noisy environment.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Zhou, Z.L., Yang, C.N., Chen, B.J., Sun, X.M., Liu, Q., Jonathan, Wu Q.M.: Effective and efficient image copy detection with resistance to arbitrary rotation. IEICE Trans. Inf. Syst. E99–D(6), 1531–1540 (2016)

    Article  ADS  Google Scholar 

  2. Chen, X.Y., Chen, S., Wu, Y.L.: Coverless information hiding method based on the Chinese character encoding. J. Internet Technol. 18(2), 313–320 (2017)

    Google Scholar 

  3. Xia, Z.H., Wang, X.H., Zhang, L.G., Qin, Z., Sun, X.M., Ren, K.: A privacy-preserving and copy-deterrence content-based image retrieval scheme in cloud computing. IEEE Trans. Inf. Forensics Secur. 11(11), 2594–2608 (2016)

    Article  Google Scholar 

  4. Fu, Z.J., Ren, K., Shu, J.G., Sun, X.M., Huang, F.X.: Enabling personalized search over encrypted outsourced data with efficiency improvement. IEEE Trans. Parallel Distrib. Syst. 27(9), 25462559 (2016)

    Article  Google Scholar 

  5. Bennett, C.H., Brassard, G.: Quantum cryptography: public key distribution and coin tossing. Proc. IEEE Int. Conf. Comput. Syst. Signal Process. 175, 175–179 (1984)

    MATH  Google Scholar 

  6. Hillery, M., Buzek, V., Berthiaume, A.: Quantum secret sharing. Phys. Rev. A 59(3), 1829–1834 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Curty, M., Santos, D.J.: Quantum authentication of classical messages. Phys. Rev. A 64(6), 062309 (2001)

    Article  ADS  Google Scholar 

  8. Bennett, C.H., Brassard, G., Crepeau, C.: Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys. Rev. Lett. 70(13), 1895–1899 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Lo, H.K.: Classical-communication cost in distributed quantum-information processing: a generalization of quantum-communication complexity. Phys. Rev. A 62(1), 012313 (2000)

    Article  ADS  Google Scholar 

  10. Bennett, C.H., DiVincenzo, D.P., Shor, P.W.: Remote state preparation. Phys. Rev. Lett. 87(7), 077902 (2001)

    Article  ADS  Google Scholar 

  11. Leung, D.W., Shor, P.W.: Oblivious remote state preparation. Phys. Rev. Lett. 90(12), 127905 (2003)

    Article  ADS  Google Scholar 

  12. Berry, D.W., Sanders, B.C.: Optimal remote state preparation. Phys. Rev. Lett. 90(5), 057901 (2003)

    Article  ADS  Google Scholar 

  13. Devetak, I., Berger, T.: Low-entanglement remote state preparation. Phys. Rev. Lett. 87(19), 197901 (2001)

    Article  ADS  Google Scholar 

  14. Chen, Q.Q., Xia, Y., An, N.B.: Joint remote preparation of an arbitrary three-qubit state via EPR-type pairs. Opt. Commun. 284(10–11), 2617–2621 (2011)

    Article  ADS  Google Scholar 

  15. Hou, K., Wang, J., Lu, Y.L.: Joint remote preparation of a multipartite GHZ-class state. Int. J. Theor. Phys. 48(7), 2005–2015 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Luo, M.X., Chen, X.B., Ma, S.Y.: Joint remote preparation of an arbitrary three-qubit state. Opt. Commun. 283(23), 4796–4801 (2010)

    Article  ADS  Google Scholar 

  17. Yang, K.Y., Xia, Y.: Joint remote preparation of a general three-qubit state via non-maximally GHZ states. Int. J. Theor. Phys. 51(5), 1647–1654 (2012)

    Article  MATH  Google Scholar 

  18. Chen, X.B., Ma, S.Y., Su, Y.: Controlled remote state preparation of arbitrary two and three qubit states via the Brown state. Quantum Inf. Process. 11(6), 1653–1667 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Gao, C., Ma, S.Y., Chen, W.L.: Controlled remote preparation via the Brown state with no restriction. Int. J. Theor. Phys. 55(5), 2643–2652 (2016)

    Article  MATH  Google Scholar 

  20. Zhang, P., Li, X., Ma, S.Y., Qu, Z.G.: Deterministic remote state preparation via the \(\chi \) state. Commun. Theor. Phys. 00, 0000 (2017)

    Google Scholar 

  21. Ma, S.Y., Gao, C., Zhang, P.: Deterministic remote preparation via the Brown state. Quantum Inf. Process. 16(4), 93 (2017). UNSP

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Wang, M.M., Qu, Z.G., Wang, W.: Effect of noise on deterministic joint remote preparation of an arbitrary two-qubit state. Quantum Inf. Process. 16(5), 140 (2017). UNSP

    Article  ADS  MATH  Google Scholar 

  23. Wang, D., Hoehn, R.D., Ye, L., Sabre, K.: Generalized remote preparation of arbitrary m-qubit entangled states via genuine entanglements. Entropy 17, 1755–1774 (2015)

    Article  ADS  Google Scholar 

  24. Wang, D., Hu, Y.D., Wang, Z.Q., Ye, L.: Efficient and faithful remote preparation of arbitrary three- and four-particle W-class entangled states. Quantum Inf. Process. 14, 2135–2151 (2015)

    Article  ADS  MATH  Google Scholar 

  25. Wang, D., Huang, A.J., Sun, W.Y., Shi, J.D., Ye, L.: Practical single-photon-assisted remote state preparation with non-maximally entanglement. Quantum Inf. Process. 15, 3367–3381 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Brown, I.D.K., Stepney, S., Sudbery, A.: Searching for highly entangled multi-qubit states. J. Phys. A Math. Gen. 38(5), 1119–1131 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Wang, M.M., Qu, Z.G.: Effect of quantum noise on deterministic joint remote state preparation of a qubit state via a GHZ channel. Quantum Inf. Process. 15(11), 4805–4818 (2016)

    Article  ADS  MATH  Google Scholar 

  28. Guan, X.W., Chen, X.B., Wang, L.C.: Joint remote preparation of an arbitrary two-qubit state in noisy environments. Int. J. Theor. Phys. 53(7), 2236–2245 (2014)

    Article  MATH  Google Scholar 

  29. Wang, M.M., Qu, Z.G., Wang, W., Chen, J.G.: Effect of noise on joint remote preparation of multi-qubit state. Int. J. Theor. Phys. 15(2), 1750012 (2017)

    MathSciNet  MATH  Google Scholar 

  30. Chen, Z.F., Liu, J.M., Ma, L.: Deterministic joint remote preparation of an arbitrary two-qubit state in the presence of noise. Chin. Phys. B 23(2), 020312 (2014)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 61373131, 61303039, 61232016, 61501247, 61702274), the Natural Science Foundation of Jiangsu Province (No. BK20170958), Sichuan Youth Science and Technique Foundation (No. 2017JQ0048), NUIST Research Foundation for Talented Scholars (2015r014), PAPD and CICAEET funds.

Author information

Authors and Affiliations

  1. Jiangsu Engineering Center of Network Monitoring, Nanjing University of Information Science and Technology, Nanjing, China

    Zhiguo Qu & Le Sun

  2. School of Computer and Software, Nanjing University of Information Science and Technology, Nanjing, China

    Shengyao Wu

  3. School of Computer Science, Xi’an Polytechnic University, Xi’an, China

    Mingming Wang

  4. School of Electronic Engineering, Dublin City University, Dublin, Ireland

    Xiaojun Wang

Authors
  1. Zhiguo Qu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shengyao Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mingming Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Le Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Xiaojun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhiguo Qu.

Appendix

Appendix

$$\begin{aligned} \rho _{\mathrm{out}}^{XD} =\,&\frac{1}{2}\left\{ {{{\left( {1 - \lambda } \right) }^2}\left[ {\left( {{t_0} + {t_3}} \right) \left| {00} \right\rangle +\, \left( {{t_2} - {t_1}} \right) \left| {01} \right\rangle + \left( {{t_1} + {t_2}} \right) \left| {10} \right\rangle } \right. } \right. \\&\left. { +\, \left( {{t_3} - {t_0}} \right) \left| {11} \right\rangle } \right] \times \left[ {{{\left( {{t_0} + {t_3}} \right) }^ * }\left\langle {00} \right| + {{\left( {{t_2} - {t_1}} \right) }^ * }\left\langle {01} \right| + {{\left( {{t_1} + {t_2}} \right) }^ * }\left\langle {10} \right| } \right. \\&\left. { +\, {{\left( {{t_3} - {t_0}} \right) }^ * }\left\langle {11} \right| } \right] + \frac{{\lambda \left( {1 - \lambda } \right) }}{3}\left[ {\left( {{t_2} - {t_1}} \right) \left| {00} \right\rangle +\, \left( {{t_0} + {t_3}} \right) \left| {01} \right\rangle } \right. \\&\left. { +\, \left( {{t_3} - {t_0}} \right) \left| {10} \right\rangle + \left( {{t_1} + {t_2}} \right) \left| {11} \right\rangle } \right] \times \left[ {{{\left( {{t_2} - {t_1}} \right) }^ * }\left\langle {00} \right| + {{\left( {{t_0} + {t_3}} \right) }^ * }\left\langle {01} \right| } \right. \\&\left. { +\, {{\left( {{t_3} - {t_0}} \right) }^ * }\left\langle {10} \right| + {{\left( {{t_1} + {t_2}} \right) }^ * }\left\langle {11} \right| } \right] + \frac{{\lambda \left( {1 - \lambda } \right) }}{3}\left[ {\left( {{t_0} + {t_3}} \right) \left| {00} \right\rangle } \right. \\&\left. { +\, \left( {{t_1} - {t_2}} \right) \left| {01} \right\rangle + \left( {{t_1} + {t_2}} \right) \left| {10} \right\rangle + \left( {{t_0} - {t_3}} \right) \left| {11} \right\rangle } \right] \times \left[ {{{\left( {{t_0} + {t_3}} \right) }^ * }\left\langle {00} \right| } \right. \\&\left. { +\, {{\left( {{t_1} - {t_2}} \right) }^ * }\left\langle {01} \right| + {{\left( {{t_1} + {t_2}} \right) }^ * }\left\langle {10} \right| + {{\left( {{t_0} - {t_3}} \right) }^ * }\left\langle {11} \right| } \right] \\&+\, \frac{{\lambda \left( {1 - \lambda } \right) }}{3}\left[ {\left( {{t_1} - {t_2}} \right) \left| {00} \right\rangle + \left( {{t_0} + {t_3}} \right) \left| {01} \right\rangle + \left( {{t_0} - {t_3}} \right) \left| {10} \right\rangle } \right. \\&\left. { +\, \left( {{t_1} + {t_2}} \right) \left| {11} \right\rangle } \right] \times \left[ {{{\left( {{t_1} - {t_2}} \right) }^ * }\left\langle {00} \right| + {{\left( {{t_0} + {t_3}} \right) }^ * }\left\langle {01} \right| + {{\left( {{t_0} - {t_3}} \right) }^ * }\left\langle {10} \right| } \right. \\&\left. { +\, {{\left( {{t_1} + {t_2}} \right) }^ * }\left\langle {11} \right| } \right] + \frac{{\lambda \left( {1 - \lambda } \right) }}{3}\left[ {\left( {{t_1} + {t_2}} \right) \left| {00} \right\rangle + \left( {{t_3} - {t_0}} \right) \left| {01} \right\rangle } \right. \\&\left. { +\, \left( {{t_0} + {t_3}} \right) \left| {10} \right\rangle + \left( {{t_2} - {t_1}} \right) \left| {11} \right\rangle } \right] \times \left[ {{{\left( {{t_1} + {t_2}} \right) }^ * }\left\langle {00} \right| + {{\left( {{t_3} - {t_0}} \right) }^ * }\left\langle {01} \right| } \right. \\&\left. { +\, {{\left( {{t_0} + {t_3}} \right) }^ * }\left\langle {10} \right| + {{\left( {{t_2} - {t_1}} \right) }^ * }\left\langle {11} \right| } \right] + \frac{{{\lambda ^2}}}{9}\left[ {\left( {{t_3} - {t_0}} \right) \left| {00} \right\rangle + \left( {{t_1} + {t_2}} \right) \left| {01} \right\rangle } \right. \\&\left. { +\, \left( {{t_2} - {t_1}} \right) \left| {10} \right\rangle + \left( {{t_0} + {t_3}} \right) \left| {11} \right\rangle } \right] \times \left[ {{{\left( {{t_3} - {t_0}} \right) }^ * }\left\langle {00} \right| + {{\left( {{t_1} + {t_2}} \right) }^ * }\left\langle {01} \right| } \right. \\&\left. { +\, {{\left( {{t_2} - {t_1}} \right) }^ * }\left\langle {10} \right| + {{\left( {{t_0} + {t_3}} \right) }^ * }\left\langle {11} \right| } \right] + \frac{{{\lambda ^2}}}{9}\left[ {\left( {{t_1} + {t_2}} \right) \left| {00} \right\rangle + \left( {{t_0} - {t_3}} \right) \left| {01} \right\rangle } \right. \\&\left. { +\, \left( {{t_0} + {t_3}} \right) \left| {10} \right\rangle + \left( {{t_1} - {t_2}} \right) \left| {11} \right\rangle } \right] \times \left[ {{{\left( {{t_1} + {t_2}} \right) }^ * }\left\langle {00} \right| + {{\left( {{t_0} - {t_3}} \right) }^ * }\left\langle {01} \right| } \right. \\&\left. { +\, {{\left( {{t_0} + {t_3}} \right) }^ * }\left\langle {10} \right| + {{\left( {{t_1} - {t_2}} \right) }^ * }\left\langle {11} \right| } \right] + \frac{{{\lambda ^2}}}{9}\left[ {\left( {{t_0} - {t_3}} \right) \left| {00} \right\rangle + \left( {{t_1} + {t_2}} \right) \left| {01} \right\rangle } \right. \\&\left. { +\, \left( {{t_1} - {t_2}} \right) \left| {10} \right\rangle + \left( {{t_0} + {t_3}} \right) \left| {11} \right\rangle } \right] \times \left[ {{{\left( {{t_0} - {t_3}} \right) }^ * }\left\langle {00} \right| + {{\left( {{t_1} + {t_2}} \right) }^ * }\left\langle {01} \right| } \right. \\&\left. { +\, {{\left( {{t_1} - {t_2}} \right) }^ * }\left\langle {10} \right| + {{\left( {{t_0} + {t_3}} \right) }^ * }\left\langle {11} \right| } \right] + \frac{{\lambda \left( {1 - \lambda } \right) }}{3}\left[ {\left( {{t_0} + {t_3}} \right) \left| {00} \right\rangle } \right. \end{aligned}$$
$$\begin{aligned}&\quad \left. { +\, \left( {{t_2} - {t_1}} \right) \left| {01} \right\rangle - \left( {{t_1} + {t_2}} \right) \left| {10} \right\rangle + \left( {{t_0} - {t_3}} \right) \left| {11} \right\rangle } \right] \times \left[ {{{\left( {{t_0} + {t_3}} \right) }^ * }\left\langle {00} \right| } \right. \nonumber \\&\quad \left. { +\, {{\left( {{t_2} - {t_1}} \right) }^ * }\left\langle {01} \right| - {{\left( {{t_1} + {t_2}} \right) }^ * }\left\langle {10} \right| + {{\left( {{t_0} - {t_3}} \right) }^ * }\left\langle {11} \right| } \right] + \frac{{{\lambda ^2}}}{9}\left[ {\left( {{t_2} - {t_1}} \right) \left| {00} \right\rangle } \right. \nonumber \\&\quad \left. { +\, \left( {{t_0} + {t_3}} \right) \left| {01} \right\rangle + \left( {{t_0} - {t_3}} \right) \left| {10} \right\rangle - \left( {{t_1} + {t_2}} \right) \left| {11} \right\rangle } \right] \times \left[ {{{\left( {{t_2} - {t_1}} \right) }^ * }\left\langle {00} \right| } \right. \nonumber \\&\quad \left. { +\, {{\left( {{t_0} + {t_3}} \right) }^ * }\left\langle {01} \right| + {{\left( {{t_0} - {t_3}} \right) }^ * }\left\langle {10} \right| - {{\left( {{t_1} + {t_2}} \right) }^ * }\left\langle {11} \right| } \right] + \frac{{{\lambda ^2}}}{9}\left[ {\left( {{t_0} + {t_3}} \right) \left| {00} \right\rangle } \right. \nonumber \\&\quad \left. { +\, \left( {{t_1} - {t_2}} \right) \left| {01} \right\rangle - \left( {{t_1} + {t_2}} \right) \left| {10} \right\rangle + \left( {{t_3} - {t_0}} \right) \left| {11} \right\rangle } \right] \times \left[ {{{\left( {{t_0} + {t_3}} \right) }^ * }\left\langle {00} \right| } \right. \nonumber \\&\quad \left. { +\, {{\left( {{t_1} - {t_2}} \right) }^ * }\left\langle {01} \right| - {{\left( {{t_1} + {t_2}} \right) }^ * }\left\langle {10} \right| + {{\left( {{t_3} - {t_0}} \right) }^ * }\left\langle {11} \right| } \right] + \frac{{{\lambda ^2}}}{9}\left[ {\left( {{t_1} - {t_2}} \right) \left| {00} \right\rangle } \right. \nonumber \\&\quad \left. { +\, \left( {{t_0} + {t_3}} \right) \left| {01} \right\rangle + \left( {{t_3} - {t_0}} \right) \left| {10} \right\rangle - \left( {{t_1} + {t_2}} \right) \left| {11} \right\rangle } \right] \times \left[ {{{\left( {{t_1} - {t_2}} \right) }^ * }\left\langle {00} \right| } \right. \nonumber \\&\quad \left. { +\, {{\left( {{t_0} + {t_3}} \right) }^ * }\left\langle {01} \right| + {{\left( {{t_3} - {t_0}} \right) }^ * }\left\langle {10} \right| - {{\left( {{t_1} + {t_2}} \right) }^ * }\left\langle {11} \right| } \right] \nonumber \\&\quad +\, \frac{{\lambda \left( {1 - \lambda } \right) }}{3}\left[ { - \left( {{t_1} + {t_2}} \right) \left| {00} \right\rangle + \left( {{t_0} - {t_3}} \right) \left| {01} \right\rangle + \left( {{t_0} + {t_3}} \right) \left| {10} \right\rangle } \right. \nonumber \\&\quad \left. { +\, \left( {{t_2} - {t_1}} \right) \left| {11} \right\rangle } \right] \times \left[ { - {{\left( {{t_1} + {t_2}} \right) }^ * }\left\langle {00} \right| + {{\left( {{t_0} - {t_3}} \right) }^ * }\left\langle {01} \right| + {{\left( {{t_0} + {t_3}} \right) }^ * }\left\langle {10} \right| } \right. \nonumber \\&\quad \left. { +\, {{\left( {{t_2} - {t_1}} \right) }^ * }\left\langle {11} \right| } \right] + \frac{{{\lambda ^2}}}{9}\left[ {\left( {{t_0} - {t_3}} \right) \left| {00} \right\rangle - \left( {{t_1} + {t_2}} \right) \left| {01} \right\rangle + \left( {{t_2} - {t_1}} \right) \left| {10} \right\rangle } \right. \nonumber \\&\quad \left. { +\, \left( {{t_0} + {t_3}} \right) \left| {11} \right\rangle } \right] \times \left[ {{{\left( {{t_0} - {t_3}} \right) }^ * }\left\langle {00} \right| - {{\left( {{t_1} + {t_2}} \right) }^ * }\left\langle {01} \right| + {{\left( {{t_2} - {t_1}} \right) }^ * }\left\langle {10} \right| } \right. \nonumber \\&\quad \left. { +\, {{\left( {{t_0} + {t_3}} \right) }^ * }\left\langle {11} \right| } \right] + \frac{{{\lambda ^2}}}{9}\left[ { - \left( {{t_1} + {t_2}} \right) \left| {00} \right\rangle + \left( {{t_3} - {t_0}} \right) \left| {01} \right\rangle + \left( {{t_0} + {t_3}} \right) \left| {10} \right\rangle } \right. \nonumber \\&\quad \left. { +\, \left( {{t_1} - {t_2}} \right) \left| {11} \right\rangle } \right] \times \left[ { - {{\left( {{t_1} + {t_2}} \right) }^ * }\left\langle {00} \right| + {{\left( {{t_3} - {t_0}} \right) }^ * }\left\langle {01} \right| + {{\left( {{t_0} + {t_3}} \right) }^ * }\left\langle {10} \right| } \right. \nonumber \\&\quad \left. { +\, {{\left( {{t_1} - {t_2}} \right) }^ * }\left\langle {11} \right| } \right] + \frac{{{\lambda ^2}}}{9}\left[ {\left( {{t_0} - {t_3}} \right) \left| {00} \right\rangle + \left( {{t_1} + {t_2}} \right) \left| {01} \right\rangle + \left( {{t_2} - {t_1}} \right) \left| {10} \right\rangle } \right. \nonumber \\&\quad \left. { -\, \left( {{t_0} + {t_3}} \right) \left| {11} \right\rangle } \right] \times \left[ {{{\left( {{t_0} - {t_3}} \right) }^ * }\left\langle {00} \right| + {{\left( {{t_1} + {t_2}} \right) }^ * }\left\langle {01} \right| + {{\left( {{t_2} - {t_1}} \right) }^ * }\left\langle {10} \right| } \right. \nonumber \\&\quad \left. {\left. { -\, {{\left( {{t_0} + {t_3}} \right) }^ * }\left\langle {11} \right| } \right] } \right\} \end{aligned}$$
(45)
$$\begin{aligned} \rho _{\mathrm{out}}^{BD} = \,&\frac{1}{2}\left\{ {{{\left( {1 - \lambda } \right) }^2}\left[ {\left( {{b_1} + {b_2}} \right) \left| {00} \right\rangle + \left( {{b_0} + {b_3}} \right) \left| {01} \right\rangle + \left( {{b_3} - {b_0}} \right) \left| {10} \right\rangle } \right. } \right. \\&\quad \left. { +\, \left( {{b_1} - {b_2}} \right) \left| {11} \right\rangle } \right] \times \left[ {{{\left( {{b_1} + {b_2}} \right) }^ * }\left\langle {00} \right| + {{\left( {{b_0} + {b_3}} \right) }^ * }\left\langle {01} \right| + {{\left( {{b_3} - {b_0}} \right) }^ * }\left\langle {10} \right| } \right. \\&\quad \left. { +\, {{\left( {{b_1} - {b_2}} \right) }^ * }\left\langle {11} \right| } \right] + \frac{{\lambda \left( {1 - \lambda } \right) }}{3}\left[ {\left( {{b_0} + {b_3}} \right) \left| {00} \right\rangle + \left( {{b_1} + {b_2}} \right) \left| {01} \right\rangle } \right. \\&\quad \left. { +\, \left( {{b_1} - {b_2}} \right) \left| {10} \right\rangle + \left( {{b_3} - {b_0}} \right) \left| {11} \right\rangle } \right] \times \left[ {{{\left( {{b_0} + {b_3}} \right) }^ * }\left\langle {00} \right| + {{\left( {{b_1} + {b_2}} \right) }^ * }\left\langle {01} \right| } \right. \\&\quad \left. { +\, {{\left( {{b_1} - {b_2}} \right) }^ * }\left\langle {10} \right| + {{\left( {{b_3} - {b_0}} \right) }^ * }\left\langle {11} \right| } \right] + \frac{{\lambda \left( {1 - \lambda } \right) }}{3}\left[ {\left( {{b_1} + {b_2}} \right) \left| {00} \right\rangle } \right. \\&\quad \left. { -\, \left( {{b_0} + {b_3}} \right) \left| {01} \right\rangle + \left( {{b_3} - {b_0}} \right) \left| {10} \right\rangle + \left( {{b_2} - {b_1}} \right) \left| {11} \right\rangle } \right] \times \left[ {{{\left( {{b_1} + {b_2}} \right) }^ * }\left\langle {00} \right| } \right. \\&\quad \left. { -\, {{\left( {{b_0} + {b_3}} \right) }^ * }\left\langle {01} \right| + {{\left( {{b_3} - {b_0}} \right) }^ * }\left\langle {10} \right| + {{\left( {{b_2} - {b_1}} \right) }^ * }\left\langle {11} \right| } \right] \\&\quad + \frac{{\lambda \left( {1 - \lambda } \right) }}{3}\left[ { - \left( {{b_0} + {b_3}} \right) \left| {00} \right\rangle + \left( {{b_1} + {b_2}} \right) \left| {01} \right\rangle + \left( {{b_2} - {b_1}} \right) \left| {10} \right\rangle } \right. \\&\quad \left. { +\, \left( {{b_3} - {b_0}} \right) \left| {11} \right\rangle } \right] \times \left[ { - {{\left( {{b_0} + {b_3}} \right) }^ * }\left\langle {00} \right| + {{\left( {{b_1} + {b_2}} \right) }^ * }\left\langle {01} \right| + {{\left( {{b_2} - {b_1}} \right) }^ * }\left\langle {10} \right| } \right. \\&\quad \left. { + {{\left( {{b_3} - {b_0}} \right) }^ * }\left\langle {11} \right| } \right] + \frac{{\lambda \left( {1 - \lambda } \right) }}{3}\left[ {\left( {{b_3} - {b_0}} \right) \left| {00} \right\rangle + \left( {{b_1} - {b_2}} \right) \left| {01} \right\rangle } \right. \\&\quad \left. { +\, \left( {{b_1} + {b_2}} \right) \left| {10} \right\rangle + \left( {{b_0} + {b_3}} \right) \left| {11} \right\rangle } \right] \times \left[ {{{\left( {{b_3} - {b_0}} \right) }^ * }\left\langle {00} \right| + {{\left( {{b_1} - {b_2}} \right) }^ * }\left\langle {01} \right| } \right. \\&\quad \left. { +\, {{\left( {{b_1} + {b_2}} \right) }^ * }\left\langle {10} \right| + {{\left( {{b_0} + {b_3}} \right) }^ * }\left\langle {11} \right| } \right] + \frac{{{\lambda ^2}}}{9}\left[ {\left( {{b_1} - {b_2}} \right) \left| {00} \right\rangle + \left( {{b_3} - {b_0}} \right) \left| {01} \right\rangle } \right. \\&\quad \left. { +\, \left( {{b_0} + {b_3}} \right) \left| {10} \right\rangle + \left( {{b_1} + {b_2}} \right) \left| {11} \right\rangle } \right] \times \left[ {{{\left( {{b_1} - {b_2}} \right) }^ * }\left\langle {00} \right| + {{\left( {{b_3} - {b_0}} \right) }^ * }\left\langle {01} \right| } \right. \\&\quad \left. { +\, {{\left( {{b_0} + {b_3}} \right) }^ * }\left\langle {10} \right| + {{\left( {{b_1} + {b_2}} \right) }^ * }\left\langle {11} \right| } \right] + \frac{{{\lambda ^2}}}{9}\left[ {\left( {{b_3} - {b_0}} \right) \left| {00} \right\rangle + \left( {{b_2} - {b_1}} \right) \left| {01} \right\rangle } \right. \\&\quad \left. { +\, \left( {{b_1} + {b_2}} \right) \left| {10} \right\rangle - \left( {{b_0} + {b_3}} \right) \left| {11} \right\rangle } \right] \times \left[ {{{\left( {{b_3} - {b_0}} \right) }^ * }\left\langle {00} \right| + {{\left( {{b_2} - {b_1}} \right) }^ * }\left\langle {01} \right| } \right. \\&\quad \left. { +\, {{\left( {{b_1} + {b_2}} \right) }^ * }\left\langle {10} \right| - {{\left( {{b_0} + {b_3}} \right) }^ * }\left\langle {11} \right| } \right] + \frac{{{\lambda ^2}}}{9}\left[ {\left( {{b_2} - {b_1}} \right) \left| {00} \right\rangle + \left( {{b_3} - {b_0}} \right) \left| {01} \right\rangle } \right. \\&\quad \left. { -\, \left( {{b_0} + {b_3}} \right) \left| {10} \right\rangle + \left( {{b_1} + {b_2}} \right) \left| {11} \right\rangle } \right] \times \left[ {{{\left( {{b_2} - {b_1}} \right) }^ * }\left\langle {00} \right| + {{\left( {{b_3} - {b_0}} \right) }^ * }\left\langle {01} \right| } \right. \\&\quad \left. { -\, {{\left( {{b_0} + {b_3}} \right) }^ * }\left\langle {10} \right| + {{\left( {{b_1} + {b_2}} \right) }^ * }\left\langle {11} \right| } \right] + \frac{{\lambda \left( {1 - \lambda } \right) }}{3}\left[ {\left( {{b_1} + {b_2}} \right) \left| {00} \right\rangle } \right. \end{aligned}$$
$$\begin{aligned}&\qquad \left. { +\, \left( {{b_0} + {b_3}} \right) \left| {01} \right\rangle + \left( {{b_0} - {b_3}} \right) \left| {10} \right\rangle + \left( {{b_2} - {b_1}} \right) \left| {11} \right\rangle } \right] \times \left[ {{{\left( {{b_1} + {b_2}} \right) }^ * }\left\langle {00} \right| } \right. \nonumber \\&\qquad \left. { +\, {{\left( {{b_0} + {b_3}} \right) }^ * }\left\langle {01} \right| + {{\left( {{b_0} - {b_3}} \right) }^ * }\left\langle {10} \right| + {{\left( {{b_2} - {b_1}} \right) }^ * }\left\langle {11} \right| } \right] + \frac{{{\lambda ^2}}}{9}\left[ {\left( {{b_0} + {b_3}} \right) \left| {00} \right\rangle } \right. \nonumber \\&\qquad \left. { +\, \left( {{b_1} + {b_2}} \right) \left| {01} \right\rangle + \left( {{b_2} - {b_1}} \right) \left| {10} \right\rangle + \left( {{b_0} - {b_3}} \right) \left| {11} \right\rangle } \right] \times \left[ {{{\left( {{b_0} + {b_3}} \right) }^ * }\left\langle {00} \right| } \right. \nonumber \\&\qquad \left. { +\, {{\left( {{b_1} + {b_2}} \right) }^ * }\left\langle {01} \right| + {{\left( {{b_2} - {b_1}} \right) }^ * }\left\langle {10} \right| + {{\left( {{b_0} - {b_3}} \right) }^ * }\left\langle {11} \right| } \right] + \frac{{{\lambda ^2}}}{9}\left[ {\left( {{b_1} + {b_2}} \right) \left| {00} \right\rangle } \right. \nonumber \\&\qquad \left. { -\, \left( {{b_0} + {b_3}} \right) \left| {01} \right\rangle + \left( {{b_0} - {b_3}} \right) \left| {10} \right\rangle + \left( {{b_1} - {b_2}} \right) \left| {11} \right\rangle } \right] \times \left[ {{{\left( {{b_1} + {b_2}} \right) }^ * }\left\langle {00} \right| } \right. \nonumber \\&\qquad \left. { -\, {{\left( {{b_0} + {b_3}} \right) }^ * }\left\langle {01} \right| + {{\left( {{b_0} - {b_3}} \right) }^ * }\left\langle {10} \right| + {{\left( {{b_1} - {b_2}} \right) }^ * }\left\langle {11} \right| } \right] \nonumber \\&\qquad +\, \frac{{{\lambda ^2}}}{9}\left[ { - \left( {{b_0} + {b_3}} \right) \left| {00} \right\rangle + \left( {{b_1} + {b_2}} \right) \left| {01} \right\rangle + \left( {{b_1} - {b_2}} \right) \left| {10} \right\rangle + \left( {{b_0} - {b_3}} \right) \left| {11} \right\rangle } \right] \nonumber \\&\qquad \times \, \left[ { - {{\left( {{b_0} + {b_3}} \right) }^ * }\left\langle {00} \right| + {{\left( {{b_1} + {b_2}} \right) }^ * }\left\langle {01} \right| + {{\left( {{b_1} - {b_2}} \right) }^ * }\left\langle {10} \right| + {{\left( {{b_0} - {b_3}} \right) }^ * }\left\langle {11} \right| } \right] \nonumber \\&\qquad +\, \frac{{\lambda \left( {1 - \lambda } \right) }}{3}\left[ {\left( {{b_0} - {b_3}} \right) \left| {00} \right\rangle + \left( {{b_2} - {b_1}} \right) \left| {01} \right\rangle + \left( {{b_1} + {b_2}} \right) \left| {10} \right\rangle + \left( {{b_0} + {b_3}} \right) \left| {11} \right\rangle } \right] \nonumber \\&\qquad \times \, \left[ {{{\left( {{b_0} - {b_3}} \right) }^ * }\left\langle {00} \right| + {{\left( {{b_2} - {b_1}} \right) }^ * }\left\langle {01} \right| + {{\left( {{b_1} + {b_2}} \right) }^ * }\left\langle {10} \right| + {{\left( {{b_0} + {b_3}} \right) }^ * }\left\langle {11} \right| } \right] \nonumber \\&\qquad +\, \frac{{{\lambda ^2}}}{9}\left[ {\left( {{b_2} - {b_1}} \right) \left| {00} \right\rangle + \left( {{b_0} - {b_3}} \right) \left| {01} \right\rangle + \left( {{b_0} + {b_3}} \right) \left| {10} \right\rangle + \left( {{b_1} + {b_2}} \right) \left| {11} \right\rangle } \right] \nonumber \\&\qquad \times \, \left[ {{{\left( {{b_2} - {b_1}} \right) }^ * }\left\langle {00} \right| + {{\left( {{b_0} - {b_3}} \right) }^ * }\left\langle {01} \right| + {{\left( {{b_0} + {b_3}} \right) }^ * }\left\langle {10} \right| + {{\left( {{b_1} + {b_2}} \right) }^ * }\left\langle {11} \right| } \right] \nonumber \\&\qquad +\, \frac{{{\lambda ^2}}}{9}\left[ {\left( {{b_0} - {b_3}} \right) \left| {00} \right\rangle + \left( {{b_1} - {b_2}} \right) \left| {01} \right\rangle + \left( {{b_1} + {b_2}} \right) \left| {10} \right\rangle - \left( {{b_0} + {b_3}} \right) \left| {11} \right\rangle } \right] \nonumber \\&\qquad \times \, \left[ {{{\left( {{b_0} - {b_3}} \right) }^ * }\left\langle {00} \right| + {{\left( {{b_1} - {b_2}} \right) }^ * }\left\langle {01} \right| + {{\left( {{b_1} + {b_2}} \right) }^ * }\left\langle {10} \right| - {{\left( {{b_0} + {b_3}} \right) }^ * }\left\langle {11} \right| } \right] \nonumber \\&\qquad +\, \frac{{{\lambda ^2}}}{9}\left[ {\left( {{b_1} - {b_2}} \right) \left| {00} \right\rangle + \left( {{b_0} - {b_3}} \right) \left| {01} \right\rangle - \left( {{b_0} + {b_3}} \right) \left| {10} \right\rangle + \left( {{b_1} + {b_2}} \right) \left| {11} \right\rangle } \right] \nonumber \\&\qquad \left. { \times \, \left[ {{{\left( {{b_1} - {b_2}} \right) }^ * }\left\langle {00} \right| + {{\left( {{b_0} - {b_3}} \right) }^ * }\left\langle {01} \right| - {{\left( {{b_0} + {b_3}} \right) }^ * }\left\langle {10} \right| + {{\left( {{b_1} + {b_2}} \right) }^ * }\left\langle {11} \right| } \right] } \right\} \end{aligned}$$
(46)
$$\begin{aligned} \rho _{\mathrm{out}}^{GD} =&{\left( {1 - \lambda } \right) ^2}\left( {{a_0}\left| {00} \right\rangle + {a_1}{e^{i{\theta _1}}}\left| {01} \right\rangle + {a_2}{e^{i{\theta _2}}}\left| {10} \right\rangle + {a_3}{e^{i{\theta _3}}}\left| {11} \right\rangle } \right) \times \left( {{a_0}\left\langle {00} \right| } \right. \\&\quad \left. { +\, {a_1}{e^{ - i{\theta _1}}}\left\langle {01} \right| + {a_2}{e^{ - i{\theta _2}}}\left\langle {10} \right| + {a_3}{e^{ - i{\theta _3}}}\left\langle {11} \right| } \right) + \frac{{\lambda \left( {1 - \lambda } \right) }}{3}\left( {{a_0}\left| {01} \right\rangle } \right. \\&\quad \left. { +\, {a_1}{e^{i{\theta _1}}}\left| {00} \right\rangle + {a_2}{e^{i{\theta _2}}}\left| {11} \right\rangle + {a_3}{e^{i{\theta _3}}}\left| {10} \right\rangle } \right) \times \left( {{a_0}\left\langle {01} \right| + {a_1}{e^{ - i{\theta _1}}}\left\langle {00} \right| } \right. \\&\quad \left. { +\, {a_2}{e^{ - i{\theta _2}}}\left\langle {11} \right| + {a_3}{e^{ - i{\theta _3}}}\left\langle {10} \right| } \right) + \frac{{\lambda \left( {1 - \lambda } \right) }}{3}\left( {{a_0}\left| {00} \right\rangle - {a_1}{e^{i{\theta _1}}}\left| {01} \right\rangle } \right. \\&\quad \left. { +\, {a_2}{e^{i{\theta _2}}}\left| {10} \right\rangle - {a_3}{e^{i{\theta _3}}}\left| {11} \right\rangle } \right) \times \left( {{a_0}\left\langle {00} \right| - {a_1}{e^{ - i{\theta _1}}}\left\langle {01} \right| + {a_2}{e^{ - i{\theta _2}}}\left\langle {10} \right| } \right. \\&\quad \left. { -\, {a_3}{e^{ - i{\theta _3}}}\left\langle {11} \right| } \right) + \frac{{\lambda \left( {1 - \lambda } \right) }}{3}\left( {{a_0}\left| {01} \right\rangle - {a_1}{e^{i{\theta _1}}}\left| {00} \right\rangle + {a_2}{e^{i{\theta _2}}}\left| {11} \right\rangle } \right. \\&\quad \left. { -\, {a_3}{e^{i{\theta _3}}}\left| {10} \right\rangle } \right) \times \left( {{a_0}\left\langle {01} \right| - {a_1}{e^{ - i{\theta _1}}}\left\langle {00} \right| + {a_2}{e^{ - i{\theta _2}}}\left\langle {11} \right| - {a_3}{e^{ - i{\theta _3}}}\left\langle {10} \right| } \right) \\&\quad +\, \frac{{\lambda \left( {1 - \lambda } \right) }}{3}\left( {{a_0}\left| {10} \right\rangle + {a_1}{e^{i{\theta _1}}}\left| {11} \right\rangle + {a_2}{e^{i{\theta _2}}}\left| {00} \right\rangle + {a_3}{e^{i{\theta _3}}}\left| {01} \right\rangle } \right) \\&\quad \times \, \left( {{a_0}\left\langle {10} \right| + {a_1}{e^{ - i{\theta _1}}}\left\langle {11} \right| + {a_2}{e^{ - i{\theta _2}}}\left\langle {00} \right| + {a_3}{e^{ - i{\theta _3}}}\left\langle {01} \right| } \right) + \frac{{{\lambda ^2}}}{9}\left( {{a_0}\left| {11} \right\rangle } \right. \\&\quad \left. { +\, {a_1}{e^{i{\theta _1}}}\left| {10} \right\rangle + {a_2}{e^{i{\theta _2}}}\left| {01} \right\rangle + {a_3}{e^{i{\theta _3}}}\left| {00} \right\rangle } \right) \times \left( {{a_0}\left\langle {11} \right| + {a_1}{e^{ - i{\theta _1}}}\left\langle {10} \right| } \right. \\&\quad \left. { +\, {a_2}{e^{ - i{\theta _2}}}\left\langle {01} \right| + {a_3}{e^{ - i{\theta _3}}}\left\langle {00} \right| } \right) + \frac{{{\lambda ^2}}}{9}\left( {{a_0}\left| {10} \right\rangle - {a_1}{e^{i{\theta _1}}}\left| {11} \right\rangle + {a_2}{e^{i{\theta _2}}}\left| {00} \right\rangle } \right. \\&\quad \left. { -\, {a_3}{e^{i{\theta _3}}}\left| {01} \right\rangle } \right) \times \left( {{a_0}\left\langle {10} \right| - {a_1}{e^{ - i{\theta _1}}}\left\langle {11} \right| + {a_2}{e^{ - i{\theta _2}}}\left\langle {00} \right| - {a_3}{e^{ - i{\theta _3}}}\left\langle {01} \right| } \right) \\&\quad +\, \frac{{{\lambda ^2}}}{9}\left( {{a_0}\left| {11} \right\rangle - {a_1}{e^{i{\theta _1}}}\left| {10} \right\rangle + {a_2}{e^{i{\theta _2}}}\left| {01} \right\rangle - {a_3}{e^{i{\theta _3}}}\left| {00} \right\rangle } \right) \times \left( {{a_0}\left| {11} \right\rangle } \right. \\&\quad \left. { -\, {a_1}{e^{ - i{\theta _1}}}\left| {10} \right\rangle + {a_2}{e^{ - i{\theta _2}}}\left| {01} \right\rangle - {a_3}{e^{ - i{\theta _3}}}\left| {00} \right\rangle } \right) + \frac{{\lambda \left( {1 - \lambda } \right) }}{3}\left( {{a_0}\left| {00} \right\rangle } \right. \end{aligned}$$
$$\begin{aligned}&\quad \left. { +\, {a_1}{e^{i{\theta _1}}}\left| {01} \right\rangle - {a_2}{e^{i{\theta _2}}}\left| {10} \right\rangle - {a_3}{e^{i{\theta _3}}}\left| {11} \right\rangle } \right) \times \left( {{a_0}\left\langle {00} \right| + {a_1}{e^{ - i{\theta _1}}}\left\langle {01} \right| } \right. \nonumber \\&\quad \left. { -\, {a_2}{e^{ - i{\theta _2}}}\left\langle {10} \right| - {a_3}{e^{ - i{\theta _3}}}\left\langle {11} \right| } \right) + \frac{{{\lambda ^2}}}{9}\left( {{a_0}\left| {01} \right\rangle + {a_1}{e^{i{\theta _1}}}\left| {00} \right\rangle - {a_2}{e^{i{\theta _2}}}\left| {11} \right\rangle } \right. \nonumber \\&\quad \left. { -\, {a_3}{e^{i{\theta _3}}}\left| {10} \right\rangle } \right) \times \left( {{a_0}\left\langle {01} \right| + {a_1}{e^{ - i{\theta _1}}}\left\langle {00} \right| - {a_2}{e^{ - i{\theta _2}}}\left\langle {11} \right| - {a_3}{e^{ - i{\theta _3}}}\left\langle {10} \right| } \right) \nonumber \\&\quad +\, \frac{{{\lambda ^2}}}{9}\left( {{a_0}\left| {00} \right\rangle - {a_1}{e^{i{\theta _1}}}\left| {01} \right\rangle - {a_2}{e^{i{\theta _2}}}\left| {10} \right\rangle + {a_3}{e^{i{\theta _3}}}\left| {11} \right\rangle } \right) \times \left( {{a_0}\left| {00} \right\rangle } \right. \nonumber \\&\quad \left. { -\, {a_1}{e^{ - i{\theta _1}}}\left| {01} \right\rangle - {a_2}{e^{ - i{\theta _2}}}\left| {10} \right\rangle + {a_3}{e^{ - i{\theta _3}}}\left| {11} \right\rangle } \right) + \frac{{{\lambda ^2}}}{9}\left( {{a_0}\left| {01} \right\rangle - {a_1}{e^{i{\theta _1}}}\left| {00} \right\rangle } \right. \nonumber \\&\quad \left. { -\, {a_2}{e^{i{\theta _2}}}\left| {11} \right\rangle + {a_3}{e^{i{\theta _3}}}\left| {10} \right\rangle } \right) \times \left( {{a_0}\left\langle {01} \right| - {a_1}{e^{ - i{\theta _1}}}\left\langle {00} \right| - {a_2}{e^{ - i{\theta _2}}}\left\langle {11} \right| } \right. \nonumber \\&\quad \left. { +\, {a_3}{e^{ - i{\theta _3}}}\left\langle {10} \right| } \right) + \frac{{\lambda \left( {1 - \lambda } \right) }}{3}\left( {{a_0}\left| {10} \right\rangle + {a_1}{e^{i{\theta _1}}}\left| {11} \right\rangle - {a_2}{e^{i{\theta _2}}}\left| {00} \right\rangle } \right. \nonumber \\&\quad \left. { -\, {a_3}{e^{i{\theta _3}}}\left| {01} \right\rangle } \right) \times \left( {{a_0}\left\langle {10} \right| + {a_1}{e^{ - i{\theta _1}}}\left\langle {11} \right| - {a_2}{e^{ - i{\theta _2}}}\left\langle {00} \right| - {a_3}{e^{ - i{\theta _3}}}\left\langle {01} \right| } \right) \nonumber \\&\quad +\, \frac{{{\lambda ^2}}}{9}\left( {{a_0}\left| {11} \right\rangle + {a_1}{e^{i{\theta _1}}}\left| {10} \right\rangle - {a_2}{e^{i{\theta _2}}}\left| {01} \right\rangle - {a_3}{e^{i{\theta _3}}}\left| {00} \right\rangle } \right) \times \left( {{a_0}\left\langle {11} \right| } \right. \nonumber \\&\quad \left. { +\, {a_1}{e^{ - i{\theta _1}}}\left\langle {10} \right| - {a_2}{e^{ - i{\theta _2}}}\left\langle {01} \right| - {a_3}{e^{ - i{\theta _3}}}\left\langle {00} \right| } \right) + \frac{{{\lambda ^2}}}{9}\left( {{a_0}\left| {10} \right\rangle - {a_1}{e^{i{\theta _1}}}\left| {11} \right\rangle } \right. \nonumber \\&\quad \left. { -\, {a_2}{e^{i{\theta _2}}}\left| {00} \right\rangle + {a_3}{e^{i{\theta _3}}}\left| {01} \right\rangle } \right) \times \left( {{a_0}\left\langle {10} \right| - {a_1}{e^{ - i{\theta _1}}}\left\langle {11} \right| - {a_2}{e^{ - i{\theta _2}}}\left\langle {00} \right| } \right. \nonumber \\&\quad \left. { +\, {a_3}{e^{ - i{\theta _3}}}\left\langle {01} \right| } \right) + \frac{{{\lambda ^2}}}{9}\left( {{a_0}\left| {11} \right\rangle - {a_1}{e^{i{\theta _1}}}\left| {10} \right\rangle - {a_2}{e^{i{\theta _2}}}\left| {01} \right\rangle + {a_3}{e^{i{\theta _3}}}\left| {00} \right\rangle } \right) \nonumber \\&\quad \times \, \left( {{a_0}\left\langle {11} \right| - {a_1}{e^{ - i{\theta _1}}}\left\langle {10} \right| - {a_2}{e^{ - i{\theta _2}}}\left\langle {01} \right| + {a_3}{e^{ - i{\theta _3}}}\left\langle {00} \right| } \right) \end{aligned}$$
(47)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qu, Z., Wu, S., Wang, M. et al. Effect of quantum noise on deterministic remote state preparation of an arbitrary two-particle state via various quantum entangled channels. Quantum Inf Process 16, 306 (2017). https://doi.org/10.1007/s11128-017-1759-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-017-1759-8

Keywords

Search

Navigation