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Bidirectional Quantum Teleportation with GHZ States and EPR Pairs via Entanglement Swapping

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Abstract

A novel bidirectional quantum teleportation schema with GHZ states and EPR pairs via entanglement swapping is proposed. Two GHZ states and two pure EPR pairs build the quantum physical system. Within this quantum system, two entangled GHZ states serve for the quantum channel, and via the entanglement swapping the new entanglements are generated from the involved qbuits after the measurements are performed on both sides. CNOT operation, single qubit measurement and the appropriate unitary transformation are executed on each side, the pure EPR states are simultaneously teleported to each other. The presented protocol bears the economical and efficient advantage compared with the conventional protocols.

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Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 61672279).

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Authors and Affiliations

  1. School of Computer Science & Technology, Nanjing TECH University, Nanjing, 211816, China

    Zhenlong Du, Xiaoli Li & Xuejun Liu

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  1. Zhenlong Du
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  2. Xiaoli Li
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  3. Xuejun Liu
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Correspondence to Zhenlong Du.

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Appendix

Appendix

1.1 \(|\varphi _{0}\rangle \rightarrow |\varphi _{1}\rangle \) transition

The quantum system state transits from \(\varphi _{0}\rightarrow \varphi _{1}\) via CNOT unitary matrix \(\textit {U}^{p_{1}}_{CN}\) and \(\textit {U}^{q_{1}}_{CN}\). The transition \(\textit {U}^{p_{1}}_{CN}\textit {U}^{q_{1}}_{CN}|\varphi _{0}\rangle \rightarrow |\varphi _{1}\rangle \) is represented as follows.

$$ \begin{array}{ll} &\textit{U}^{p_{1}}_{CN}\textit{U}^{q_{1}}_{CN}{|\varphi_{0}\rangle}_{A_{1}B_{1}B_{2}A_{2}A_{3}B_{3}p_{1}p_{2}q_{1}q_{2}} = \\ &\textit{U}^{p_{1}}_{CN}\textit{U}^{q_{1}}_{CN} \\ &\frac{1}{2}\{\left( |000000\rangle+|000111\rangle+|111000\rangle+|111111\rangle\right)_{A_{1}B_{1}B_{2}A_{2}A_{3}B_{3}}\alpha_{0}\alpha_{1}|0000\rangle_{p_{1}p_{2}q_{1}q_{2}} \\ & +\left( |000000\rangle+|000111\rangle+|111000\rangle+|111111\rangle\right)_{A_{1}B_{1}B_{2}A_{2}A_{3}B_{3}}\alpha_{0}\beta_{1}|0011\rangle_{p_{1}p_{2}q_{1}q_{2}}\\ & +\left( |000000\rangle+|000111\rangle+|111000\rangle+|111111\rangle\right)_{A_{1}B_{1}B_{2}A_{2}A_{3}B_{3}}\beta_{0}\alpha_{1}|1100\rangle_{p_{1}p_{2}q_{1}q_{2}} \\ & +\left( |000000\rangle+|000111\rangle+|111000\rangle+|111111\rangle\right)_{A_{1}B_{1}B_{2}A_{2}A_{3}B_{3}}\beta_{0}\beta_{1}|1111\rangle_{p_{1}p_{2}q_{1}q_{2}}\}\\ &=\\ &{|\varphi_{1}\rangle}_{A_{1}B_{1}B_{2}A_{2}A_{3}B_{3}p_{1}p_{2}q_{1}q_{2}} = \\ &\frac{1}{2}\{\left( |000000\rangle+|000111\rangle+|111000\rangle+|111111\rangle\right)_{A_{1}B_{1}B_{2}A_{2}A_{3}B_{3}}\alpha_{0}\alpha_{1}|0000\rangle_{p_{1}p_{2}q_{1}q_{2}}\\ & +\left( |000001\rangle+|000110\rangle+|111001\rangle+|111110\rangle\right)_{A_{1}B_{1}B_{2}A_{2}A_{3}B_{3}}\alpha_{0}\beta_{1}|0011\rangle_{p_{1}p_{2}q_{1}q_{2}} \\ &+\left( |100000\rangle+|100111\rangle+|011000\rangle+|011111\rangle\right)_{A_{1}B_{1}B_{2}A_{2}A_{3}B_{3}}\beta_{0}\alpha_{1}|1100\rangle_{p_{1}p_{2}q_{1}q_{2}}\\ &+\left( |100001\rangle+|100110\rangle+|011001\rangle+|011110\rangle\right)_{A_{1}B_{1}B_{2}A_{2}A_{3}B_{3}}\beta_{0}\beta_{1}|1111\rangle_{p_{1}p_{2}q_{1}q_{2}}\}\\ &=\frac{1}{2}(\varPsi_{0}\rangle_{A_{1}A_{2}A_{3}}|\varPsi_{0}\rangle_{B_{1}B_{2}B_{3}}+ |\varPsi_{1}\rangle_{A_{1}A_{2}A_{3}}|\varPsi_{1}\rangle_{B_{1}B_{2}B_{3}}+|\varPsi_{2}\rangle_{A_{1}A_{2}A_{3}}|\varPsi_{2}\rangle_{B_{1}B_{2}B_{3}}\\ &\quad +|\varPsi_{3}\rangle_{A_{1}A_{2}A_{3}}|\varPsi_{3}\rangle_{B_{1}B_{2}B_{3}})\alpha_{0}\alpha_{1}|0000\rangle_{p_{1}p_{2}q_{1}q_{2}} \\ &+\frac{1}{2}(|\varPsi_{0}\rangle_{A_{1}A_{2}A_{3}}|\varPsi_{6}\rangle_{B_{1}B_{2}B_{3}}-|\varPsi_{0}\rangle_{A_{1}A_{2}A_{3}}|\varPsi_{7}\rangle_{B_{1}B_{2}B_{3}}+|\varPsi_{2}\rangle_{A_{1}A_{2}A_{3}}|\varPsi_{0}\rangle_{B_{1}B_{2}B_{3}}\\ &\quad -|\varPsi_{2}\rangle_{A_{1}A_{2}A_{3}}|\varPsi_{1}\rangle_{B_{1}B_{2}B_{3}})\alpha_{0}\beta_{1}|0011\rangle_{p_{1}p_{2}q_{1}q_{2}}\\ &+\frac{1}{2}(|\varPsi_{2}\rangle_{A_{1}A_{2}A_{3}}|\varPsi_{0}\rangle_{B_{1}B_{2}B_{3}}+|\varPsi_{3}\rangle_{A_{1}A_{2}A_{3}}|\varPsi_{1}\rangle_{B_{1}B_{2}B_{3}} +|\varPsi_{0}\rangle_{A_{1}A_{2}A_{3}}|\varPsi_{6}\rangle_{B_{1}B_{2}B_{3}}\\ &\quad +|\varPsi_{1}\rangle_{A_{1}A_{2}A_{3}}|\varPsi_{7}\rangle_{B_{1}B_{2}B_{3}} )\beta_{0}\alpha_{1}|1100\rangle_{p_{1}p_{2}q_{1}q_{2}} \end{array} $$

1.2 The Measurement Results and Corresponding System Collapsed States

The measurement results on Alice’ and Bob’s end are separately given in the first column and second one. The collapsed states of qubits B1, B2, A2, A3, p2 and q2 are listed in the third column.

Table 3 The measurement results and the corresponding collapsed states
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Du, Z., Li, X. & Liu, X. Bidirectional Quantum Teleportation with GHZ States and EPR Pairs via Entanglement Swapping. Int J Theor Phys 59, 622–631 (2020). https://doi.org/10.1007/s10773-019-04355-6

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