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Three schemes of remote information concentration based on ancilla-free phase-covariant telecloning

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Abstract

In this paper, remote information concentration is investigated which is the reverse process of the \(1\rightarrow 3\) optimal asymmetric economical phase-covariant telecloning (OAEPCT). The OAEPCT is different from the reverse process of optimal universal telecloning. It is shown that the quantum information via \(1\rightarrow 3\) OAEPCT procedure can be remotely concentrated back to a single qubit with a certain probability via several quantum channels. In these schemes, we adopt Bell measurement to measure the joint systems and use projected measurement and positive operator-valued measure to recover the original quantum state. The results shows non-maximally entangled quantum resource can be applied to information concentration.

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Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 11071178), the Research Foundation of the Education Department of Sichuan Province (Grant No. 12ZB106).

Author information

Authors and Affiliations

  1. College of Mathematics and Software Science, Sichuan Normal University, Chengdu , 610066, China

    Ming-qiang Bai, Jia-Yin Peng & Zhi-Wen Mo

  2. School of Mathematics and Information Science, Neijiang Normal University, Neijiang , 641100, China

    Jia-Yin Peng

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  1. Ming-qiang Bai
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  2. Jia-Yin Peng
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Correspondence to Ming-qiang Bai.

Appendices

Appendix 1

The symbols \(|\varsigma _i\rangle ,|\zeta _i\rangle \) and \(|\xi _i\rangle (i=1,2,3,4)\) in Eq. (4) are shown as follows

$$\begin{aligned} |\varsigma _1\rangle&= \left( \alpha |0\rangle +\beta \hbox {e}^{i\theta }|1\rangle \right) _D, |\varsigma _2\rangle =\left( \alpha |0\rangle -\beta \hbox {e}^{i\theta }|1\rangle \right) _D,\\ |\varsigma _3\rangle&= \left( \beta |1\rangle +\alpha \hbox {e}^{i\theta }|0\rangle \right) _D, |\varsigma _4\rangle =\left( \beta |1\rangle -\alpha \hbox {e}^{i\theta }|0\rangle \right) _D,\\ |\zeta _1\rangle&= |\varPhi ^+\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPsi ^+\rangle _{C3}+|\varPhi ^-\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPsi ^+\rangle _{C3} -|\varPhi ^+\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPsi ^-\rangle _{C3}\\&\quad -|\varPhi ^-\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPsi ^-\rangle _{C3},\\ |\zeta _2\rangle&= |\varPhi ^+\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPsi ^+\rangle _{C3}+|\varPhi ^-\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPsi ^+\rangle _{C3} -|\varPhi ^+\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPsi ^-\rangle _{C3}\\&\quad -|\varPhi ^-\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPsi ^-\rangle _{C3},\\ |\zeta _3\rangle&= |\varPsi ^+\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPhi ^+\rangle _{C3}+|\varPsi ^-\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPhi ^+\rangle _{C3} -|\varPsi ^+\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPhi ^-\rangle _{C3}\\&\quad -|\varPsi ^-\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPhi ^-\rangle _{C3},\\ |\zeta _4\rangle&= |\varPsi ^+\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPhi ^+\rangle _{C3}+|\varPsi ^-\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPhi ^+\rangle _{C3} -|\varPsi ^+\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPhi ^-\rangle _{C3}\\&\quad -|\varPsi ^-\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPhi ^-\rangle _{C3},\\ |\xi _1\rangle&= |\varPhi ^+\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPhi ^+\rangle _{C3}+|\varPhi ^-\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPhi ^-\rangle _{C3} -|\varPhi ^-\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPhi ^+\rangle _{C3}\\&\quad -|\varPhi ^+\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPhi ^-\rangle _{C3},\\ |\xi _2\rangle&= |\varPhi ^+\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPhi ^-\rangle _{C3}+|\varPhi ^-\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPhi ^+\rangle _{C3} -|\varPhi ^-\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPhi ^-\rangle _{C3}\\&\quad -|\varPhi ^+\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPhi ^+\rangle _{C3},\\ |\xi _3\rangle&= |\varPsi ^+\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPsi ^+\rangle _{C3}+|\varPsi ^-\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPsi ^-\rangle _{C3} -|\varPsi ^-\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPsi ^+\rangle _{C3}\\&\quad -|\varPsi ^+\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPsi ^-\rangle _{C3},\\ |\xi _4\rangle&= |\varPsi ^+\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPsi ^-\rangle _{C3}+|\varPsi ^-\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPsi ^+\rangle _{C3} -|\varPsi ^-\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPsi ^-\rangle _{C3}\\&\quad -|\varPsi ^+\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPsi ^+\rangle _{C3},\\ |\varrho _1\rangle&= |\varPsi ^+\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPhi ^+\rangle _{C3}+|\varPsi ^+\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPhi ^-\rangle _{C3} -|\varPsi ^-\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPhi ^+\rangle _{C3}\\&\quad -|\varPsi ^-\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPhi ^-\rangle _{C3},\\ |\varrho _2\rangle&= |\varPsi ^+\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPhi ^-\rangle _{C3}+|\varPsi ^+\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPhi ^+\rangle _{C3} -|\varPsi ^-\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPhi ^-\rangle _{C3} \\&\quad -|\varPsi ^-\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPhi ^+\rangle _{C3},\\ |\varrho _3\rangle&= |\varPhi ^+\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPsi ^+\rangle _{C3}+|\varPhi ^+\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPsi ^-\rangle _{C3} -|\varPhi ^-\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPsi ^+\rangle _{C3} \\&\quad -|\varPhi ^-\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPsi ^-\rangle _{C3},\\ |\varrho _4\rangle&= |\varPhi ^+\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPsi ^-\rangle _{C3}+|\varPhi ^+\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPsi ^+\rangle _{C3} -|\varPhi ^-\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPsi ^-\rangle _{C3} \\&\quad -|\varPhi ^-\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPsi ^+\rangle _{C3}. \end{aligned}$$

Appendix 2

The symbols \(|\zeta _i\rangle (i=1,2,\ldots ,24)\) in Eq.(16) are shown as follows

$$\begin{aligned} |\zeta _1\rangle&= |\varPhi ^+\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPhi ^-\rangle _{C3}-|\varPhi ^-\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPhi ^-\rangle _{C3},\\ |\zeta _2\rangle&= |\varPhi ^+\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPhi ^+\rangle _{C3}-|\varPhi ^-\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPhi ^+\rangle _{C3},\\ |\zeta _3\rangle&= |\varPhi ^-\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPhi ^+\rangle _{C3}-|\varPhi ^+\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPhi ^+\rangle _{C3},\\ |\zeta _4\rangle&= |\varPhi ^-\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPhi ^-\rangle _{C3}-|\varPhi ^+\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPhi ^-\rangle _{C3},\\ |\zeta _5\rangle&= |\varPhi ^-\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPsi ^-\rangle _{C3}-|\varPhi ^+\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPsi ^+\rangle _{C3},\\ |\zeta _6\rangle&= |\varPhi ^-\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPsi ^+\rangle _{C3}-|\varPhi ^+\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPsi ^-\rangle _{C3},\\ |\zeta _7\rangle&= |\varPhi ^+\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPsi ^+\rangle _{C3}-|\varPhi ^-\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPsi ^+\rangle _{C3},\\ |\zeta _8\rangle&= |\varPhi ^+\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPsi ^-\rangle _{C3}-|\varPhi ^-\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPsi ^-\rangle _{C3},\\ |\zeta _9\rangle&= |\varPsi ^+\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPhi ^+\rangle _{C3}-|\varPsi ^-\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPhi ^+\rangle _{C3},\\ |\zeta _{10}\rangle&= |\varPsi ^+\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPhi ^-\rangle _{C3}-|\varPsi ^-\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPhi ^-\rangle _{C3},\\ |\zeta _{11}\rangle&= |\varPsi ^-\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPhi ^-\rangle _{C3}-|\varPsi ^+\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPhi ^-\rangle _{C3},\\ |\zeta _{12}\rangle&= |\varPsi ^-\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPhi ^+\rangle _{C3}-|\varPsi ^+\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPhi ^+\rangle _{C3},\\ |\zeta _{13}\rangle&= |\varPsi ^+\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPsi ^+\rangle _{C3}-|\varPsi ^-\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPsi ^+\rangle _{C3},\\ |\zeta _{14}\rangle&= |\varPsi ^+\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPsi ^-\rangle _{C3}-|\varPsi ^-\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPsi ^-\rangle _{C3},\\ |\zeta _{15}\rangle&= |\varPsi ^-\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPsi ^-\rangle _{C3}-|\varPsi ^+\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPsi ^-\rangle _{C3},\\ |\zeta _{16}\rangle&= |\varPsi ^-\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPsi ^+\rangle _{C3}-|\varPsi ^+\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPsi ^+\rangle _{C3},\\ |\zeta _{17}\rangle&= |\varPhi ^+\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPhi ^+\rangle _{C3}+|\varPhi ^-\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPhi ^+\rangle _{C3}\\&-|\varPhi ^+\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPhi ^-\rangle _{C3}-|\varPhi ^-\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPhi ^-\rangle _{C3},\\ |\zeta _{18}\rangle&= |\varPhi ^+\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPhi ^+\rangle _{C3}+|\varPhi ^-\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPhi ^+\rangle _{C3} -|\varPhi ^+\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPhi ^-\rangle _{C3}\\&-|\varPhi ^-\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPhi ^-\rangle _{C3},\\ |\zeta _{19}\rangle&= |\varPhi ^+\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPsi ^+\rangle _{C3}+|\varPhi ^-\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPsi ^+\rangle _{C3} -|\varPhi ^+\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPsi ^-\rangle _{C3} \\&-|\varPhi ^-\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPsi ^-\rangle _{C3}, \end{aligned}$$
$$\begin{aligned} |\zeta _{20}\rangle&= |\varPhi ^+\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPsi ^+\rangle _{C3}+|\varPhi ^-\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPsi ^+\rangle _{C3} -|\varPhi ^+\rangle _{A1}|\varPhi ^-\rangle _{B2}|\varPsi ^-\rangle _{C3}\\&-|\varPhi ^-\rangle _{A1}|\varPhi ^+\rangle _{B2}|\varPsi ^-\rangle _{C3},\\ |\zeta _{21}\rangle&= |\varPsi ^+\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPhi ^-\rangle _{C3}+|\varPsi ^-\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPhi ^-\rangle _{C3} -|\varPsi ^+\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPhi ^+\rangle _{C3}\\&-|\varPsi ^-\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPhi ^+\rangle _{C3},\\ |\zeta _{22}\rangle&= |\varPsi ^-\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPhi ^-\rangle _{C3}+|\varPsi ^+\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPhi ^-\rangle _{C3} -|\varPsi ^+\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPhi ^+\rangle _{C3} \\&-|\varPsi ^-\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPhi ^+\rangle _{C3},\\ |\zeta _{23}\rangle&= |\varPsi ^+\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPsi ^+\rangle _{C3}+|\varPsi ^-\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPsi ^+\rangle _{C3} -|\varPsi ^+\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPsi ^-\rangle _{C3} \\&-|\varPsi ^-\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPsi ^-\rangle _{C3},\\ |\zeta _{24}\rangle&= |\varPsi ^+\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPsi ^+\rangle _{C3}+|\varPsi ^-\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPsi ^+\rangle _{C3} -|\varPsi ^+\rangle _{A1}|\varPsi ^+\rangle _{B2}|\varPsi ^-\rangle _{C3}\\&-|\varPsi ^-\rangle _{A1}|\varPsi ^-\rangle _{B2}|\varPsi ^-\rangle _{C3}. \end{aligned}$$

The symbols \(s_i(i=1,2,\ldots ,8)\) and \(t_i(i=1,2,3,4)\) in Eq.(17) are shown as follows

$$\begin{aligned} s_1&= b\alpha +c\gamma ,s_2=b\alpha -c\gamma ,s_3=b\beta +c\delta ,s_4=b\beta -c\delta ,\\ s_5&= b\gamma +c\alpha ,s_6=b\gamma -c\alpha ,s_7=b\delta +c\beta ,s_8=b\delta -c\beta ,\\ t_1&= a\beta , t_2=a\alpha , t_3=a\delta , t_4=a\gamma . \end{aligned}$$

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Bai, Mq., Peng, JY. & Mo, ZW. Three schemes of remote information concentration based on ancilla-free phase-covariant telecloning. Quantum Inf Process 13, 1067–1083 (2014). https://doi.org/10.1007/s11128-013-0712-8

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