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Secure simultaneous dense coding using \(\chi \)-type entangled state

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Abstract

A violation of a Bell inequality can formally be expressed as a witness for quantum nonlocality. A new four-qubit Bell inequality is optimally violated by \(\chi \)-type entangled state, but not by four-qubit GHZ, W and cluster state. Hence, \(\chi \)-type entangled state is a good candidate to implement simultaneous dense coding. In this paper, we propose a simultaneous dense coding protocol with genuine four-particle entangled state, \(\chi \)-type entangled state, in which two receivers can simultaneously obtain their respective classical information sent by a sender. The double controlled-NOT operator, which is used as the locking operator, play a crucial role in our protocol. The security of simultaneous dense coding is analyzed against the intercept-resend attack. The preparation of \(\chi \)-type entangled state has been studied in various physics systems, and it has been experimentally generated with nearly deterministic scheme and high generation rate. Thus, our protocol is feasible with the current experimental technology.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 11671284), Sichuan Provincial Natural Science Foundation of China (Grant Nos. 2015JY0002, 2017JY0197) and the Research Foundation of the Education Department of Sichuan Province (Grant No. 15ZA0032).

Author information

Authors and Affiliations

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

    Xue Yang, Ming-qiang Bai, Zhi-cui Zuo & Zhi-wen Mo

  2. Institute of Intelligent Information and Quantum Information, Sichuan Normal University, Chengdu, 610066, China

    Xue Yang, Ming-qiang Bai, Zhi-cui Zuo & Zhi-wen Mo

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  1. Xue Yang
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  2. Ming-qiang Bai
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  3. Zhi-cui Zuo
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  4. Zhi-wen Mo
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Corresponding author

Correspondence to Zhi-wen Mo.

Appendix

Appendix

In this appendix, we present a detailed proof of theorem mentioned in Sect. 3. In following theorem, we can see that the reduced density matrix in subsystem \(A_1B\) and in subsystem \(A_2C\) are the same, which are independent of \(b_1,b_2,b_3,b_4,c_1,c_2,c_3,c_4\). That means Bob and Charlie can know nothing about their respective information that Alice wants to send to them unless they collaborate.

Theorem

For each \(b_1,b_2,b_3,b_4,c_1,c_2,c_3,c_4\in \{0,1\}\), \(\rho _{3214}=\rho _{3^{\prime }2^{\prime }1^{\prime }4^{\prime }}=\frac{I}{16}\), where \(\rho _{3214}\) and \(\rho _{3^{\prime }2^{\prime }1^{\prime }4^{\prime }}\) are the reduced density matrices in subsystems \(A_1B\) and \(A_2C\), after step (2) (but before step (3)).

Proof

Case 1 When \(i=j\), or \(i=00, j=01\), or \(i=01, j=00\), or \(i=10, j=11\), or \(i=11, j=10\), after step (2), the state of the composite system becomes

$$\begin{aligned} \begin{aligned} |\psi (2)\rangle&=\mathrm{DCNOT}_{2,2^{\prime }}\mathrm{DCNOT}_{3,3^{\prime }}|\phi (b_1b_2b_3b_4)\rangle _{3214}\otimes |\phi (c_1c_2c_3c_4)\rangle _{3^{\prime }2^{\prime }1^{\prime }4^{\prime }}\\&=\frac{1}{8}\sum _{p=0}^1\sum _{q=0}^1\sum _{p^{\prime }=0}^1\sum _{q^{\prime }=0}^1(-\,1)^{f(b_1b_2b_3b_4pq)+f(c_1c_2c_3c_4p^{\prime }q^{\prime })}\\&\quad \left( |c_2\oplus p^{\prime }\oplus 1\rangle _3|c_4\oplus q^{\prime }\oplus 1\rangle _2|p\rangle _1|q\rangle _4\otimes |b_2\oplus p\oplus c_2\oplus p^{\prime }\rangle _{3^{\prime }}|b_4\oplus q\oplus c_4 \right. \left. \oplus q^{\prime }\rangle _{2^{\prime }}|p^{\prime }\rangle _{1^{\prime }}|q^{\prime }\rangle _{4^{\prime }}\right. \\&\left. \quad +\,(-\,1)^{q^{\prime }}|c_2\oplus p^{\prime }\rangle _3|c_4\oplus q^{\prime }\rangle _2|p\rangle _1|q\rangle _4\otimes |b_2\oplus p\oplus c_2\oplus p^{\prime }\right. \left. \oplus 1\rangle _{3^{\prime }}|b_4\oplus q\oplus c_4\oplus q^{\prime }\oplus 1\rangle _{2^{\prime }}|p^{\prime }\rangle _{1^{\prime }}|q^{\prime }\rangle _{4^{\prime }}\right. \\&\left. \quad +\,(-\,1)^q|c_2\oplus p^{\prime }\oplus 1\rangle _3|c_4\oplus q^{\prime }\oplus 1\rangle _2|p\rangle _1|q\rangle _4\otimes |b_2\oplus p\oplus c_2\right. \left. \oplus p^{\prime }\oplus 1\rangle _{3^{\prime }}|b_4\oplus q\oplus c_4\oplus q^{\prime }\oplus 1\rangle _{2^{\prime }}|p^{\prime }\rangle _{1^{\prime }}|q^{\prime }\rangle _{4^{\prime }}\right. \\&\left. \quad +\,(-\,1)^{q+q^{\prime }}|c_2\oplus p^{\prime }\rangle _3|c_4\oplus q^{\prime }\rangle _2|p\rangle _1|q\rangle _4\otimes |b_2\oplus p\oplus c_2\oplus p^{\prime }\rangle _{3^{\prime }}|b_4\right. \left. \oplus q\oplus c_4\oplus q^{\prime }\rangle _{2^{\prime }}|p^{\prime }\rangle _{1^{\prime }}|q^{\prime }\rangle _{4^{\prime }}\right) . \end{aligned} \end{aligned}$$
(14)

The reduced density matrix in subsystem \(A_1B\) is

$$\begin{aligned} \begin{aligned} \rho _{A_1B}&=\rho _{3214}\\&=Tr_{3^{\prime }2^{\prime }1^{\prime }4^{\prime }}(|\psi (2)\rangle \langle \psi (2)|)\\&=\frac{1}{64}\sum _{p=0}^1\sum _{q=0}^1\sum _{p^{\prime }=0}^1\sum _{q^{\prime }=0}^1 \left( |c_2\oplus p^{\prime }\oplus 1\rangle _3|c_4\oplus q^{\prime }\oplus 1\rangle _2|p\rangle _1|q\rangle _4\langle q|{_1}\langle p|{_2}\langle c_4\right. \left. \oplus q^{\prime }\oplus 1|{_3}\langle c_2\oplus p^{\prime }\oplus 1|\right. \\&\left. \quad +\,|c_2\oplus p^{\prime }\rangle _3|c_4\oplus q^{\prime }\rangle _2|p\rangle _1|q\rangle _4 \langle q|{_1}\langle p|{_2}\langle c_4\oplus q^{\prime }|{_3}\langle c_2\oplus p^{\prime }|\right. \\&\left. \quad +\,|c_2\oplus p^{\prime }\oplus 1\rangle _3|c_4\oplus q^{\prime }\oplus 1\rangle _2|p\rangle _1|q\rangle _4\langle q|{_1}\langle p|{_2}\langle c_4\oplus q^{\prime }\oplus 1|{_3}\langle c_2\oplus p^{\prime }\oplus 1|\right. \\&\left. \quad +\, |c_2\oplus p^{\prime }\rangle _3|c_4\oplus q^{\prime }\rangle _2|p\rangle _1|q\rangle _4 \langle q|{_1}\langle p|{_2}\langle c_4\oplus q^{\prime }|{_3}\langle c_2\oplus p^{\prime }|\right) \\&=\frac{I}{16}. \end{aligned} \end{aligned}$$
(15)

Similarly, the reduced density matrix in subsystem \(A_2C\) is also \(\frac{I}{16}\).

Case 2 When any one of these conditions of \(i=j\), or \(i=00, j=01\), or \(i=01, j=00\), or \(i=10, j=11\), or \(i=11, j=10\) is not satisfied, after step (2), the state of the composite system becomes

$$\begin{aligned} \begin{aligned} |\psi (2)\rangle&=\mathrm{DCNOT}_{2,2^{\prime }}\mathrm{DCNOT}_{3,3^{\prime }}|\phi (b_1b_2b_3b_4)\rangle _{3214}\otimes |\phi (c_1c_2c_3c_4)\rangle _{3^{\prime }2^{\prime }1^{\prime }4^{\prime }}\\&=\frac{1}{8}\sum _{p=0}^1\sum _{q=0}^1\sum _{p^{\prime }=0}^1\sum _{q^{\prime }=0}^1(-1)^{f(b_1b_2b_3b_4pq)+f(c_1c_2c_3c_4p^{\prime }q^{\prime })}\\&\,\quad \left( |c_2\oplus p^{\prime }\oplus 1\rangle _3|c_4\oplus q^{\prime }\oplus 1\rangle _2|p\rangle _1|q\rangle _4\otimes |b_2\oplus p\oplus c_2\oplus p^{\prime }\rangle _{3^{\prime }}|b_4\oplus q\right. \left. \oplus c_4\oplus q^{\prime }\rangle _{2^{\prime }}|p^{\prime }\rangle _{1^{\prime }}|q^{\prime }\rangle _{4^{\prime }}\right. \\&\left. \quad +\, (-\,1)^{q^{\prime }+1}|c_2\oplus p^{\prime }\rangle _3|c_4\oplus q^{\prime }\rangle _2|p\rangle _1|q\rangle _4\otimes |b_2\oplus p\oplus c_2\oplus p^{\prime }\oplus 1\rangle _{3^{\prime }}|b_4\oplus q\right. \left. \oplus c_4\oplus q^{\prime }\oplus 1\rangle _{2^{\prime }}|p^{\prime }\rangle _{1^{\prime }}|q^{\prime }\rangle _{4^{\prime }}\right. \\&\left. \quad +\, (-\,1)^{q+1}|c_2\oplus p^{\prime }\oplus 1\rangle _3|c_4\oplus q^{\prime }\oplus 1\rangle _2|p\rangle _1|q\rangle _4\otimes |b_2\oplus p\oplus c_2\right. \left. \oplus p^{\prime }\oplus 1\rangle _{3^{\prime }}|b_4\oplus q\oplus c_4\oplus q^{\prime }\oplus 1\rangle _{2^{\prime }}|p^{\prime }\rangle _{1^{\prime }}|q^{\prime }\rangle _{4^{\prime }}\right. \\&\left. \quad +\,(-\,1)^{q+q^{\prime }}|c_2\oplus p^{\prime }\rangle _3|c_4\oplus q^{\prime }\rangle _2|p\rangle _1|q\rangle _4\otimes |b_2\oplus p\oplus c_2\right. \left. \oplus p^{\prime }\rangle _{3^{\prime }}|b_4\oplus q\oplus c_4\oplus q^{\prime }\rangle _{2^{\prime }}|p^{\prime }\rangle _{1^{\prime }}|q^{\prime }\rangle _{4^{\prime }}\right) . \end{aligned} \end{aligned}$$
(16)

The reduced density matrix in subsystem \(A_1B\) is

$$\begin{aligned} \begin{aligned} \rho _{A_1B}&=\rho _{3214}\\&=Tr_{3'2'1'4'}(|\psi (2)\rangle \langle \psi (2)|)\\&=\frac{1}{64}\sum _{p=0}^1\sum _{q=0}^1\sum _{p'=0}^1\sum _{q'=0}^1\left( |c_2\oplus p'\oplus 1\rangle _3|c_4\right. \left. \oplus q'\oplus 1\rangle _2|p\rangle _1|q\rangle _4\langle q|{_1}\langle p|{_2}\langle c_4\oplus q'\oplus 1|{_3}\langle c_2\oplus p'\oplus 1|\right. \\&\left. \quad +\, |c_2\oplus p'\rangle _3|c_4\oplus q'\rangle _2|p\rangle _1|q\rangle _4 \langle q|{_1}\langle p|{_2}\langle c_4\oplus q'|{_3}\langle c_2\oplus p'|\right. \\&\left. \quad +\, |c_2\oplus p'\oplus 1\rangle _3|c_4\oplus q'\oplus 1\rangle _2|p\rangle _1|q\rangle _4\langle q|{_1}\langle p|{_2}\langle c_4\oplus q'\oplus 1|{_3}\langle c_2\oplus p'\oplus 1|\right. \\&\left. \quad +\,|c_2\oplus p'\rangle _3|c_4\oplus q'\rangle _2|p\rangle _1|q\rangle _4 \langle q|{_1}\langle p|{_2}\langle c_4\oplus q'|{_3}\langle c_2\oplus p'|\right) \\&=\frac{I}{16}. \end{aligned} \end{aligned}$$
(17)

Similarly, the reduced density matrix in subsystem \(A_2C\) is also \(\frac{I}{16}\). \(\square \)

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Yang, X., Bai, Mq., Zuo, Zc. et al. Secure simultaneous dense coding using \(\chi \)-type entangled state. Quantum Inf Process 17, 261 (2018). https://doi.org/10.1007/s11128-018-2022-7

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