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The effect of quantum noise on two different deterministic remote state preparation of an arbitrary three-particle state protocols

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Abstract

Multi-particle quantum state deterministic remote preparation is a fundamental and important technical branch in quantum communication. Since quantum noise is unavoidable in realistic quantum communication, it is important to analyze the effect of noise on multi-particle quantum communication protocols. In this paper, we study the effects of noise, such as amplitude damping, phase damping, bit-flip and depolarizing noises, on two deterministic remote preparation of an arbitrary three-particle state protocols, which are based on two different entangled channels, namely \(\chi \) state and Brown state. The detailed mathematical analysis shows that the output states of two deterministic remote state preparation (DRSP) protocols are the same in the same noisy environment. That is to say, in the same noisy environment, the effects of noise on two DRSP protocols are the same. This conclusion proves that these two DRSP protocols will produce the same arbitrary three-particle states in the same noise channel environment, and so that these protocols are inherently convergent and can be substituted for each other in certain circumstances. In addition, this paper also takes three-particle states \(a\left| {000} \right\rangle + b{\mathrm{e}^{ic}}\left| {111} \right\rangle \) as an example and studies the relationship between the fidelity, the target state and the size of the noise factor. The results show that if the target state can be selected, an appropriate target state can effectively resist on the bit-flip noise. If the target state cannot be selected, as the increase in the size of noise factor, the fidelities of the two DRSP schemes in the amplitude damping noise and phase damping noise are always larger than those in the bit-flip noise and depolarizing noise. This conclusion indicates that two protocols have better resistance on amplitude damping and phase damping noise than the bit-flip and depolarizing noises. These findings and analyses will provide valid help in deterministic remote preparation of an arbitrary three-particle state in a noisy environment.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 61303039, 61373131), Natural Science Foundation of Shandong Province (Grant No. ZR2015FL024), Fundamental Research Funds for the Central Universities (Grant No. 2682014CX095), PAPD and CICAEET funds, Open Foundation of Jiangsu Engineering Center of Network Monitoring (Nanjing University of Information Science and Technology) (Grant No. KJR1502), and Science Foundation Ireland (SFI) under the International Strategic Cooperation Award Grant Number SFI/13/ISCA/2845.

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Appendix

Appendix

$$\begin{aligned} \rho _\chi ^D= & {} {\left( {1 - \lambda } \right) ^3}\left| T \right\rangle \left\langle T \right| + \frac{{\lambda {{\left( {1 - \lambda } \right) }^2}}}{3}\sigma _x^3\left| T \right\rangle \left\langle T \right| \sigma _x^{3\dag }\\&+ \frac{{\lambda {{\left( {1 - \lambda } \right) }^2}}}{3}\sigma _z^3\left| T \right\rangle \left\langle T \right| \sigma _z^{3\dag } + \frac{{\lambda {{\left( {1 - \lambda } \right) }^2}}}{3}\sigma _y^3\left| T \right\rangle \left\langle T \right| \sigma _y^{3\dag }\\&+ \frac{{\lambda {{\left( {1 - \lambda } \right) }^2}}}{3}\sigma _x^2\left| T \right\rangle \left\langle T \right| \sigma _x^{2\dag } + \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _x^2\sigma _x^3\left| T \right\rangle \left\langle T \right| \sigma _x^{3\dag }\sigma _x^{2\dag }\\&+ \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _x^2\sigma _z^3\left| T \right\rangle \left\langle T \right| \sigma _z^{3\dag }\sigma _x^{2\dag } + \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _x^2\sigma _y^3\left| T \right\rangle \left\langle T \right| \sigma _y^{3\dag }\sigma _x^{2\dag }\\&+ \frac{{{{\left( {1 - \lambda } \right) }^2}\lambda }}{3}\sigma _z^2\left| T \right\rangle \left\langle T \right| \sigma _z^{2\dag } + \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _z^2\sigma _x^3\left| T \right\rangle \left\langle T \right| \sigma _x^{3\dag }\sigma _z^{2\dag }\\&+ \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _z^2\sigma _z^3\left| T \right\rangle \left\langle T \right| \sigma _z^{3\dag }\sigma _z^{2\dag } + \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _z^2\sigma _y^3\left| T \right\rangle \left\langle T \right| \sigma _y^{3\dag }\sigma _z^{2\dag }\\&+ \frac{{{{\left( {1 - \lambda } \right) }^2}\lambda }}{3}\sigma _y^2\left| T \right\rangle \left\langle T \right| \sigma _y^{2\dag } + \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _y^2\sigma _x^3\left| T \right\rangle \left\langle T \right| \sigma _x^{3\dag }\sigma _y^{2\dag }\\&+ \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _y^2\sigma _z^3\left| T \right\rangle \left\langle T \right| \sigma _z^{3\dag }\sigma _y^{2\dag } + \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _y^2\sigma _y^3\left| T \right\rangle \left\langle T \right| \sigma _y^{3\dag }\sigma _y^{2\dag }\\&+ \frac{{\lambda {{\left( {1 - \lambda } \right) }^2}}}{3}\sigma _x^1\left| T \right\rangle \left\langle T \right| \sigma _x^{1\dag } + \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _x^1\sigma _x^3\left| T \right\rangle \left\langle T \right| \sigma _x^{3\dag }\sigma _x^{1\dag }\\&+ \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _x^1\sigma _z^3\left| T \right\rangle \left\langle T \right| \sigma _x^{3\dag }\sigma _z^{1\dag } + \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _x^1\sigma _y^3\left| T \right\rangle \left\langle T \right| \sigma _x^{3\dag }\sigma _y^{1\dag }\\&+ \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _x^1\sigma _x^2\left| T \right\rangle \left\langle T \right| \sigma _x^{2\dag }\sigma _x^{1\dag } + \frac{{{\lambda ^3}}}{{27}}\sigma _x^1\sigma _x^2\sigma _x^3\left| T \right\rangle \left\langle T \right| \sigma _x^{3\dag }\sigma _x^{2\dag }\sigma _x^{1\dag }\\&+ \frac{{{\lambda ^3}}}{{27}}\sigma _x^1\sigma _x^2\sigma _z^3\left| T \right\rangle \left\langle T \right| \sigma _z^{3\dag }\sigma _x^{2\dag }\sigma _x^{1\dag } + \frac{{{\lambda ^3}}}{{27}}\sigma _x^1\sigma _x^2\sigma _y^3\left| T \right\rangle \left\langle T \right| \sigma _y^{3\dag }\sigma _x^{2\dag }\sigma _x^{1\dag }\\&+ \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _x^1\sigma _z^2\left| T \right\rangle \left\langle T \right| \sigma _z^{2\dag }\sigma _x^{1\dag } + \frac{{{\lambda ^3}}}{{27}}\sigma _x^1\sigma _z^2\sigma _x^3\left| T \right\rangle \left\langle T \right| \sigma _x^{3\dag }\sigma _z^{2\dag }\sigma _x^{1\dag }\\&+ \frac{{{\lambda ^3}}}{{27}}\sigma _x^1\sigma _z^2\sigma _z^3\left| T \right\rangle \left\langle T \right| \sigma _z^{3\dag }\sigma _z^{2\dag }\sigma _x^{1\dag } + \frac{{{\lambda ^3}}}{{27}}\sigma _x^1\sigma _z^2\sigma _y^3\left| T \right\rangle \left\langle T \right| \sigma _y^{3\dag }\sigma _z^{2\dag }\sigma _x^{1\dag }\\&+ \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _x^1\sigma _y^2\left| T \right\rangle \left\langle T \right| \sigma _y^{2\dag }\sigma _x^{1\dag } + \frac{{{\lambda ^3}}}{{27}}\sigma _x^1\sigma _y^2\sigma _x^3\left| T \right\rangle \left\langle T \right| \sigma _x^{3\dag }\sigma _y^{2\dag }\sigma _x^{1\dag }\\&+ \frac{{{\lambda ^3}}}{{27}}\sigma _x^1\sigma _y^2\sigma _z^3\left| T \right\rangle \left\langle T \right| \sigma _z^{3\dag }\sigma _y^{2\dag }\sigma _x^{1\dag } + \frac{{{\lambda ^3}}}{{27}}\sigma _x^1\sigma _y^2\sigma _y^3\left| T \right\rangle \left\langle T \right| \sigma _y^{3\dag }\sigma _y^{2\dag }\sigma _x^{1\dag }\\&+ \frac{{\lambda {{\left( {1 - \lambda } \right) }^2}}}{3}\sigma _z^1\left| T \right\rangle \left\langle T \right| \sigma _z^{1\dag } + \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _z^1\sigma _x^3\left| T \right\rangle \left\langle T \right| \sigma _x^{3\dag }\sigma _z^{1\dag }\\&+ \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _z^1\sigma _z^3\left| T \right\rangle \left\langle T \right| \sigma _z^{3\dag }\sigma _z^{1\dag } + \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _z^1\sigma _y^3\left| T \right\rangle \left\langle T \right| \sigma _y^{3\dag }\sigma _z^{1\dag }\\&+ \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _z^1\sigma _x^2\left| T \right\rangle \left\langle T \right| \sigma _x^{2\dag }\sigma _z^{1\dag } + \frac{{{\lambda ^3}}}{{27}}\sigma _z^1\sigma _x^2\sigma _x^3\left| T \right\rangle \left\langle T \right| \sigma _x^{3\dag }\sigma _x^{2\dag }\sigma _z^{1\dag }\\&+ \frac{{{\lambda ^3}}}{{27}}\sigma _z^1\sigma _x^2\sigma _z^3\left| T \right\rangle \left\langle T \right| \sigma _z^{3\dag }\sigma _x^{2\dag }\sigma _z^{1\dag } + \frac{{{\lambda ^3}}}{{27}}\sigma _z^1\sigma _x^2\sigma _y^3\left| T \right\rangle \left\langle T \right| \sigma _y^{3\dag }\sigma _x^{2\dag }\sigma _z^{1\dag }\\&+ \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _z^1\sigma _z^2\left| T \right\rangle \left\langle T \right| \sigma _z^{2\dag }\sigma _z^{1\dag } + \frac{{{\lambda ^3}}}{{27}}\sigma _z^1\sigma _z^2\sigma _x^3\left| T \right\rangle \left\langle T \right| \sigma _x^{3\dag }\sigma _z^{2\dag }\sigma _z^{1\dag }\\&+ \frac{{{\lambda ^3}}}{{27}}\sigma _z^1\sigma _z^2\sigma _z^3\left| T \right\rangle \left\langle T \right| \sigma _z^{3\dag }\sigma _z^{2\dag }\sigma _z^{1\dag } + \frac{{{\lambda ^3}}}{{27}}\sigma _z^1\sigma _z^2\sigma _y^3\left| T \right\rangle \left\langle T \right| \sigma _y^{3\dag }\sigma _z^{2\dag }\sigma _z^{1\dag }\\&+ \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _z^1\sigma _y^2\left| T \right\rangle \left\langle T \right| \sigma _y^{2\dag }\sigma _z^{1\dag } + \frac{{{\lambda ^3}}}{{27}}\sigma _z^1\sigma _y^2\sigma _x^3\left| T \right\rangle \left\langle T \right| \sigma _x^{3\dag }\sigma _y^{2\dag }\sigma _z^{1\dag } \end{aligned}$$
$$\begin{aligned}&\,\,\,\,\,\quad + \frac{{{\lambda ^3}}}{{27}}\sigma _z^1\sigma _y^2\sigma _z^3\left| T \right\rangle \left\langle T \right| \sigma _z^{3\dag }\sigma _y^{2\dag }\sigma _z^{1\dag } + \frac{{{\lambda ^3}}}{{27}}\sigma _z^1\sigma _y^2\sigma _y^3\left| T \right\rangle \left\langle T \right| \sigma _y^{3\dag }\sigma _y^{2\dag }\sigma _z^{1\dag }\nonumber \\&\,\,\,\,\,\quad + \frac{{\lambda {{\left( {1 - \lambda } \right) }^2}}}{3}\sigma _y^1\left| T \right\rangle \left\langle T \right| \sigma _y^{1\dag } + \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _y^1\sigma _x^3\left| T \right\rangle \left\langle T \right| \sigma _x^{3\dag }\sigma _y^{1\dag }\nonumber \\&\,\,\,\,\,\quad + \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _y^1\sigma _z^3\left| T \right\rangle \left\langle T \right| \sigma _z^{3\dag }\sigma _y^{1\dag } + \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _y^1\sigma _y^3\left| T \right\rangle \left\langle T \right| \sigma _y^{3\dag }\sigma _y^{1\dag }\nonumber \\&\,\,\,\,\,\quad + \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _y^1\sigma _x^2\left| T \right\rangle \left\langle T \right| \sigma _x^{2\dag }\sigma _y^{1\dag } + \frac{{{\lambda ^3}}}{{27}}\sigma _y^1\sigma _x^2\sigma _x^3\left| T \right\rangle \left\langle T \right| \sigma _x^{3\dag }\sigma _x^{2\dag }\sigma _y^{1\dag }\nonumber \\&\,\,\,\,\,\quad + \frac{{{\lambda ^3}}}{{27}}\sigma _y^1\sigma _x^2\sigma _z^3\left| T \right\rangle \left\langle T \right| \sigma _z^{3\dag }\sigma _x^{2\dag }\sigma _y^{1\dag } + \frac{{{\lambda ^3}}}{{27}}\sigma _y^1\sigma _x^2\sigma _y^3\left| T \right\rangle \left\langle T \right| \sigma _y^{3\dag }\sigma _x^{2\dag }\sigma _y^{1\dag }\nonumber \\&\,\,\,\,\,\quad + \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _y^1\sigma _z^2\left| T \right\rangle \left\langle T \right| \sigma _z^{2\dag }\sigma _y^{1\dag } + \frac{{{\lambda ^3}}}{{27}}\sigma _y^1\sigma _z^2\sigma _x^3\left| T \right\rangle \left\langle T \right| \sigma _x^{3\dag }\sigma _z^{2\dag }\sigma _y^{1\dag }\nonumber \\&\,\,\,\,\,\quad + \frac{{{\lambda ^3}}}{{27}}\sigma _y^1\sigma _z^2\sigma _z^3\left| T \right\rangle \left\langle T \right| \sigma _z^{3\dag }\sigma _z^{2\dag }\sigma _y^{1\dag } + \frac{{{\lambda ^3}}}{{27}}\sigma _y^1\sigma _z^2\sigma _y^3\left| T \right\rangle \left\langle T \right| \sigma _y^{3\dag }\sigma _z^{2\dag }\sigma _y^{1\dag }\nonumber \\&\,\,\,\,\,\quad + \frac{{{\lambda ^2}\left( {1 - \lambda } \right) }}{9}\sigma _y^1\sigma _y^2\left| T \right\rangle \left\langle T \right| \sigma _z^{2\dag }\sigma _y^{1\dag } + \frac{{{\lambda ^3}}}{{27}}\sigma _y^1\sigma _y^2\sigma _x^3\left| T \right\rangle \left\langle T \right| \sigma _x^{3\dag }\sigma _y^{2\dag }\sigma _y^{1\dag }\nonumber \\&\,\,\,\,\,\quad + \frac{{{\lambda ^3}}}{{27}}\sigma _y^1\sigma _y^2\sigma _z^3\left| T \right\rangle \left\langle T \right| \sigma _z^{3\dag }\sigma _y^{2\dag }\sigma _y^{1\dag } + \frac{{{\lambda ^3}}}{{27}}\sigma _y^1\sigma _y^2\sigma _y^3\left| T \right\rangle \left\langle T \right| \sigma _y^{3\dag }\sigma _y^{2\dag }\sigma _y^{1\dag } \end{aligned}$$
(33)

Since Eq. 34 is too long and exactly the same to Eq. 33, Eq. 34 is no longer presented here.

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Sun, L., Wu, S., Qu, Z. et al. The effect of quantum noise on two different deterministic remote state preparation of an arbitrary three-particle state protocols. Quantum Inf Process 17, 283 (2018). https://doi.org/10.1007/s11128-018-2054-z

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