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

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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.

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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
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  2. Shengyao Wu
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  3. Mingming Wang
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  4. Le Sun
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  5. Xiaojun Wang
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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)

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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

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

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