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Deterministic Controlled Remote State Preparation of Real-Parameter Multi-Qubit States via Maximal Slice States

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Abstract

By exploiting three-qubit entangled states and appropriate measurement basis, we propose efficient protocols for deterministic controlled remote state preparation of arbitrary real-parameter multi-qubit states, in which the maximal slice states are used as quantum channel. The successful probability of our schemes can reach up to 100% by using multi-qubit mutually orthogonal measurement basis without the introduction of auxiliary particles. Based on the implementation schemes for preparing arbitrary two- and three-qubit states with real parameters, we have derived the controlled remote state preparation protocols for arbitrary real-parameter multi-qubit states.

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Acknowledgements

This work is supported by the Program for National Natural Science Foundation of China (Grant No. 61803382), Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2018JQ6020) and China Postdoctoral Science Foundation Funded Project (Project No. 2018M643869).

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Authors and Affiliations

  1. Information and Navigation College, Air Force Engineering University, Xi’an, Shanxi, 710000, People’s Republic of China

    Kaihang Zhou, Lei Shi, Yang Xue, Chao Huang, Zhiqiang Ma & Jiahua Wei

  2. Equipment Management and UAV Engineering College, Air Force Engineering University, Xi’an, Shanxi, 710000, People’s Republic of China

    Bingbing Luo

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  1. Kaihang Zhou
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  2. Lei Shi
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  3. Bingbing Luo
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  4. Yang Xue
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Correspondence to Kaihang Zhou.

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Appendices

Appendix: A

In Step 2 of Section 3, we present the state of particles (B1,C1,B2,C2,B3,C3) when the measurement result of particles (A1,A2,A3) is |Γ1〉. For the sake of the completeness, the state of particles (B1,C1,B2,C2,B3,C3) in terms of other measurement results |Γk〉 (k = 0, 2, 3,⋯ , 7) are supplemented as follows

$$ \begin{array}{@{}rcl@{}} |\psi_{0}\rangle_{B_{1}C_{1}B_{2}C_{2}B_{3}C_{3}}&=&\langle{\varGamma}_{0}|\psi\rangle_{total} \\ &=&\sum\limits_{|m_{C_{i}}\rangle=|x^{\pm}_{i}\rangle_{C_{i}}}\left( \frac{1}{\sqrt{2}}\right)^{3}P_{x_{1}}P_{x_{2}}P_{x_{3}}\otimes|m_{C_{1}}\rangle\otimes|m_{C_{2}}\rangle\otimes|m_{C_{3}}\rangle \\ &&\otimes\left( a_{0}|000\rangle+(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{1}|001\rangle+(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}a_{2}|010\rangle\right. \\ &&+(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{3}|011\rangle+(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}a_{4}|100\rangle \\ &&+(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{5}|101\rangle +(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}a_{6}|110\rangle \\ &&+\left.(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{7}|111\rangle\right)_{B_{1}B_{2}B_{3}} \end{array} $$
$$ \begin{array}{@{}rcl@{}} |\psi_{2}\rangle_{B_{1}C_{1}B_{2}C_{2}B_{3}C_{3}}&=&\langle{\varGamma}_{2}|\psi\rangle_{total} \\ &=&\sum\limits_{|m_{C_{i}}\rangle=|x^{\pm}_{i}\rangle_{C_{i}}}\left( \frac{1}{\sqrt{2}}\right)^{3}P_{x_{1}}P_{x_{2}}P_{x_{3}}\otimes|m_{C_{1}}\rangle\otimes|m_{C_{2}}\rangle\otimes|m_{C_{3}}\rangle \\ &&\otimes\left( a_{2}|000\rangle-(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{3}|001\rangle-(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}a_{0}|010\rangle\right. \\ &&+(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{1}|011\rangle+(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}a_{6}|100\rangle \\ &&+(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{7}|101\rangle -(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}a_{4}|110\rangle \\ &&-\left.(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{5}|111\rangle\right)_{B_{1}B_{2}B_{3}} \end{array} $$
$$ \begin{array}{@{}rcl@{}} |\psi_{3}\rangle_{B_{1}C_{1}B_{2}C_{2}B_{3}C_{3}}&=&\langle{\varGamma}_{3}|\psi\rangle_{total} \\ &=&\sum\limits_{|m_{C_{i}}\rangle=|x^{\pm}_{i}\rangle_{C_{i}}}\left( \frac{1}{\sqrt{2}}\right)^{3}P_{x_{1}}P_{x_{2}}P_{x_{3}}\otimes|m_{C_{1}}\rangle\otimes|m_{C_{2}}\rangle\otimes|m_{C_{3}}\rangle \\ &&\otimes\left( a_{3}|000\rangle+(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{2}|001\rangle-(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}a_{1}|010\rangle\right. \\ &&-(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{0}|011\rangle+(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}a_{7}|100\rangle \\ &&-(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{6}|101\rangle +(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}a_{5}|110\rangle \\ &&-\left.(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{4}|111\rangle\right)_{B_{1}B_{2}B_{3}} \end{array} $$
$$ \begin{array}{@{}rcl@{}} |\psi_{4}\rangle_{B_{1}C_{1}B_{2}C_{2}B_{3}C_{3}}&=&\langle{\varGamma}_{4}|\psi\rangle_{total} \\ &=&\sum\limits_{|m_{C_{i}}\rangle=|x^{\pm}_{i}\rangle_{C_{i}}}\left( \frac{1}{\sqrt{2}}\right)^{3}P_{x_{1}}P_{x_{2}}P_{x_{3}}\otimes|m_{C_{1}}\rangle\otimes|m_{C_{2}}\rangle\otimes|m_{C_{3}}\rangle \\ &&\otimes\left( a_{4}|000\rangle-(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{5}|001\rangle-(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}a_{6}|010\rangle\right. \\ &&-(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{7}|011\rangle-(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}a_{0}|100\rangle \\ &&+(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{1}|101\rangle +(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}a_{2}|110\rangle \\ &&+\left.(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{3}|111\rangle\right)_{B_{1}B_{2}B_{3}} \end{array} $$
$$ \begin{array}{@{}rcl@{}} |\psi_{5}\rangle_{B_{1}C_{1}B_{2}C_{2}B_{3}C_{3}}&=&\langle{\varGamma}_{5}|\psi\rangle_{total} \\ &=&\sum\limits_{|m_{C_{i}}\rangle=|x^{\pm}_{i}\rangle_{C_{i}}}\left( \frac{1}{\sqrt{2}}\right)^{3}P_{x_{1}}P_{x_{2}}P_{x_{3}}\otimes|m_{C_{1}}\rangle\otimes|m_{C_{2}}\rangle\otimes|m_{C_{3}}\rangle \\ &&\otimes\left( a_{5}|000\rangle+(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{4}|001\rangle-(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}a_{7}|010\rangle\right. \\ &&+(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{6}|011\rangle-(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}a_{1}|100\rangle \\ &&-(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{0}|101\rangle -(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}a_{3}|110\rangle \\ &&+\left.(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{2}|111\rangle\right)_{B_{1}B_{2}B_{3}} \end{array} $$
$$ \begin{array}{@{}rcl@{}} |\psi_{6}\rangle_{B_{1}C_{1}B_{2}C_{2}B_{3}C_{3}}&=&\langle{\varGamma}_{6}|\psi\rangle_{total} \\ &=&{\sum}_{|m_{C_{i}}\rangle=|x^{\pm}_{i}\rangle_{C_{i}}}\left( \frac{1}{\sqrt{2}}\right)^{3}P_{x_{1}}P_{x_{2}}P_{x_{3}}\otimes|m_{C_{1}}\rangle\otimes|m_{C_{2}}\rangle\otimes|m_{C_{3}}\rangle\\ &&\otimes\left( a_{6}|000\rangle+(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{7}|001\rangle+(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}a_{4}|010\rangle\right. \\ &&-(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{5}|011\rangle-(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}a_{2}|100\rangle \\ &&+(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{3}|101\rangle -(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}a_{0}|110\rangle \\ &&-\left.(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{1}|111\rangle\right)_{B_{1}B_{2}B_{3}} \end{array} $$
$$ \begin{array}{@{}rcl@{}} |\psi_{7}\rangle_{B_{1}C_{1}B_{2}C_{2}B_{3}C_{3}}&=&\langle{\varGamma}_{7}|\psi\rangle_{total} \\ &=&{\sum}_{|m_{C_{i}}\rangle=|x^{\pm}_{i}\rangle_{C_{i}}}\left( \frac{1}{\sqrt{2}}\right)^{3}P_{x_{1}}P_{x_{2}}P_{x_{3}}\otimes|m_{C_{1}}\rangle\otimes|m_{C_{2}}\rangle\otimes|m_{C_{3}}\rangle \\ &&\otimes\left( a_{7}|000\rangle-(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{6}|001\rangle+(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}a_{5}|010\rangle\right. \\ &&+(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{4}|011\rangle-(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}a_{3}|100\rangle \\ &&-(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{2}|101\rangle +(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}a_{1}|110\rangle \\ &&-\left.(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}a_{0}|111\rangle\right)_{B_{1}B_{2}B_{3}} \end{array} $$

Appendix: B

When the measurement result of particles \((A_{1},A_{2},A_{3})\) is \(|{\varGamma }_{1}\rangle \) in Step 3 of Section 3, we have presented the unitary transformation \(U_{{\varGamma }_{1}}\) that Bob needs to perform. For the other measurement results \(|{\varGamma }_{k}\rangle ~(k=0,2,3,\cdots ,7)\), the unitary transformations \(U_{{\varGamma }_{k}}\) on particles \((B_{1},B_{2},B_{3})\) are expressed as follows

$$ \begin{array}{@{}rcl@{}} U_{{\varGamma}_{0}}&=&|001\rangle\langle000|+(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|001\rangle\langle001|+(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}|010\rangle\langle010| \\ &&+(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|011\rangle\langle011|+(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}|100\rangle\langle100| \\ &&+(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|101\rangle\langle101| +(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}|110\rangle\langle110| \\ &&+(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|111\rangle\langle111| \end{array} $$
$$ \begin{array}{@{}rcl@{}} U_{{\varGamma}_{2}}&=&|010\rangle\langle000|-(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|011\rangle\langle001|-(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}|000\rangle\langle010| \\ &&+(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|001\rangle\langle011|+(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}|110\rangle\langle100| \\ &&+(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|111\rangle\langle101| -(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}|100\rangle\langle110| \\ &&-(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|101\rangle\langle111| \end{array} $$
$$ \begin{array}{@{}rcl@{}} U_{{\varGamma}_{3}}&=&|011\rangle\langle000|+(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|010\rangle\langle001|-(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}|001\rangle\langle010| \\ &&-(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|000\rangle\langle011|+(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}|111\rangle\langle100| \\ &&-(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|110\rangle\langle101| +(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}|101\rangle\langle110| \\ &&-(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|100\rangle\langle111| \end{array} $$
$$ \begin{array}{@{}rcl@{}} U_{{\varGamma}_{4}}&=&|100\rangle\langle000|-(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|101\rangle\langle001|-(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}|110\rangle\langle010| \\ &&-(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|111\rangle\langle011|-(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}|000\rangle\langle100| \\ &&+(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|001\rangle\langle101| +(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}|010\rangle\langle110| \\ &&+(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|011\rangle\langle111| \end{array} $$
$$ \begin{array}{@{}rcl@{}} U_{{\varGamma}_{5}}&=&|101\rangle\langle000|+(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|100\rangle\langle001|-(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}|111\rangle\langle010| \\ &&+(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|110\rangle\langle011|-(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}|001\rangle\langle100| \\ &&-(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|000\rangle\langle101| -(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}|011\rangle\langle110| \\ &&+(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|010\rangle\langle111| \end{array} $$
$$ \begin{array}{@{}rcl@{}} U_{{\varGamma}_{6}}&=&|110\rangle\langle000|+(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|111\rangle\langle001|+(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}|100\rangle\langle010| \\ &&-(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|101\rangle\langle011|-(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}|010\rangle\langle100| \\ &&+(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|011\rangle\langle101| -(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}|000\rangle\langle110| \\ &&-(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|001\rangle\langle111| \end{array} $$
$$ \begin{array}{@{}rcl@{}} U_{{\varGamma}_{7}}&=&|111\rangle\langle000|-(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|110\rangle\langle001|+(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}|101\rangle\langle010| \\ &&+(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|100\rangle\langle011|-(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}|011\rangle\langle100| \\ &&-(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|010\rangle\langle101| +(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}|001\rangle\langle110| \\ &&-(-1)^{\langle x_{1}^{-}|m_{C_{1}}\rangle}(-1)^{\langle x_{2}^{-}|m_{C_{2}}\rangle}(-1)^{\langle x_{3}^{-}|m_{C_{3}}\rangle}|000\rangle\langle111| \end{array} $$

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Zhou, K., Shi, L., Luo, B. et al. Deterministic Controlled Remote State Preparation of Real-Parameter Multi-Qubit States via Maximal Slice States. Int J Theor Phys 58, 4079–4092 (2019). https://doi.org/10.1007/s10773-019-04274-6

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