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Noise Effects and Perfect Controlled Remote State Preparation

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Abstract

We investigate how the efficiency is affected on the noise environments in the controlled remote state preparation protocol, where the several realistic scenarios, i.e., a part or all of the qubits are subjected to the same or different types of noise, are considered. We find that more noise or less entanglement of qubits environment lead to more efficiency in terms of average fidelity. We show that it is better way to subject the qubits in different noise channels in order to increase the fidelity of the controlled remote state preparation protocol. By using a non-maximally three-qubit pure entangled state as quantum channel, furthermore, we could realize a perfect controlled remote state preparation by choosing the right noisy environments and adjusting their relations in terms of noisy rates.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11764021, 11564018, 61765008, 11804133, 51567011), and the Research Foundation of the Education Department of Jiangxi Province (No. GJJ150339).

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Authors and Affiliations

  1. Department of physics, Jiangxi Normal University, Nanchang, 330022, China

    Yuan-hua Li, Zi-sheng Wang, Hua-qing Zhou, Hai-mei Luo, Ming-huang Sang & Yi-you Nie

  2. Key Laboratory of Optoelectronic and Telecommunication of Jiangxi province, Nanchang, 330022, China

    Yuan-hua Li, Zi-sheng Wang, Hua-qing Zhou, Hai-mei Luo, Ming-huang Sang & Yi-you Nie

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Appendix

Appendix

1.1 Bit-flip noise

The bit-flip noise changes a qubit state from ∣0〉 to ∣1〉 or from ∣1〉 to ∣0〉 with a probability p and is frequently used in the theory of quantum-error correction. The associated Kraus operators are given by

$$ {E}_1=\left(\begin{array}{cc}\sqrt{1-p}& 0\\ {}0& \sqrt{1-p}\end{array}\right),{E}_2=\left(\begin{array}{cc}0& \sqrt{p}\\ {}\sqrt{p}& 0\end{array}\right). $$
(63)

1.2 Amplitude-damping noise

The amplitude-damping noise channel allows us to describe the decay of a two-level system due to spontaneous emission of a phonon. This process is accompanied with the loss of energy and can be described by the Kraus operators,

$$ {E}_1=\left(\begin{array}{cc}1& 0\\ {}0& \sqrt{1-p}\end{array}\right),{E}_2=\left(\begin{array}{cc}0& \sqrt{p}\\ {}0& 0\end{array}\right). $$
(64)

The quantity p is regarded as a decay probability from the excited to the ground state for a two-level system.

1.3 Phase-flip noise

The phase-flip noise channel has no classical analog because it describes the loss of quantum information without loss of energy. The quantum information corresponding to the ability of a system to produce quantum interferences hence is described by the off-diagonal elements of a density matrix. Phase-flip map can be occurred in the phase kicks or scattering processes. Such a channel can be modeled by the following Kraus operators,

$$ {E}_1=\left(\begin{array}{cc}\sqrt{1-p}& 0\\ {}0& \sqrt{1-p}\end{array}\right),{E}_2=\left(\begin{array}{cc}\sqrt{p}& 0\\ {}0& -\sqrt{p}\end{array}\right). $$
(65)

1.4 Depolarizing noise

The depolarizing noise channel is a decoherent model. The Kraus operators including all possible decay ways for the depolarizing channel are given by

$$ {\displaystyle \begin{array}{l}{E}_1=\left(\begin{array}{cc}\sqrt{1-3p/4}& 0\\ {}0& \sqrt{1-3p/4}\end{array}\right),{E}_2=\left(\begin{array}{cc}0& \sqrt{p/4}\\ {}\sqrt{p/4}& 0\end{array}\right){\sigma}_x,{E}_3=\left(\begin{array}{cc}0& -i\sqrt{p/4}\\ {}i\sqrt{p/4}& 0\end{array}\right),\\ {}{E}_4=\left(\begin{array}{cc}\sqrt{p/4}& 0\\ {}0& -\sqrt{p/4}\end{array}\right).\end{array}} $$
(66)

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Li, Yh., Wang, Zs., Zhou, Hq. et al. Noise Effects and Perfect Controlled Remote State Preparation. Int J Theor Phys 58, 1172–1194 (2019). https://doi.org/10.1007/s10773-019-04010-0

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