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Probabilistic authenticated quantum dialogue

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Abstract

This work proposes a probabilistic authenticated quantum dialogue (PAQD) based on Bell states with the following notable features. (1) In our proposed scheme, the dialogue is encoded in a probabilistic way, i.e., the same messages can be encoded into different quantum states, whereas in the state-of-the-art authenticated quantum dialogue (AQD), the dialogue is encoded in a deterministic way; (2) the pre-shared secret key between two communicants can be reused without any security loophole; (3) each dialogue in the proposed PAQD can be exchanged within only one-step quantum communication and one-step classical communication. However, in the state-of-the-art AQD protocols, both communicants have to run a QKD protocol for each dialogue and each dialogue requires multiple quantum as well as classical communicational steps; (4) nevertheless, the proposed scheme can resist the man-in-the-middle attack, the modification attack, and even other well-known attacks.

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Acknowledgments

We would like to thank all the anonymous referees and the associate editor Michael Frey for their valuable suggestions. We also would like to thank the Ministry of Science and Technology of Republic of China for financial support of this research under Contract No. MOST 104-2221-E-006-102 -.

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Correspondence to Tzonelih Hwang.

Appendices

Appendix 1: The logical qubits immune to collective noises

1.1 The logical qubits immune to collective-dephasing noise

The four logical Bell states can be defined as follows:

$$\begin{aligned} \left| \Phi _{dp}^{+}\right\rangle= & {} \frac{1}{\sqrt{2}}\left( \left| 0_{dp}\right\rangle \left| 0_{dp}\right\rangle +\left| 1_{dp}\right\rangle \left| 1_{dp}\right\rangle \right) \nonumber \\= & {} \frac{1}{\sqrt{2}}\left( \left| 01\right\rangle \left| 01\right\rangle +\left| 10\right\rangle \left| 10\right\rangle \right) \nonumber \\ \left| \Phi _{dp}^{-}\right\rangle= & {} \frac{1}{\sqrt{2}}\left( \left| 0_{dp}\right\rangle \left| 0_{dp}\right\rangle -\left| 1_{dp}\right\rangle \left| 1_{dp}\right\rangle \right) \nonumber \\= & {} \frac{1}{\sqrt{2}}\left( \left| 01\right\rangle \left| 01\right\rangle -\left| 10\right\rangle \left| 10\right\rangle \right) \nonumber \\ \left| \Psi _{dp}^{+}\right\rangle= & {} \frac{1}{\sqrt{2}}\left( \left| 0_{dp}\right\rangle \left| 1_{dp}\right\rangle +\left| 1_{dp}\right\rangle \left| 0_{dp}\right\rangle \right) \\= & {} \frac{1}{\sqrt{2}}\left( \left| 01\right\rangle \left| 10\right\rangle +\left| 10\right\rangle \left| 01\right\rangle \right) \nonumber \\ \left| \Psi _{dp}^{-}\right\rangle= & {} \frac{1}{\sqrt{2}}\left( \left| 0_{dp}\right\rangle \left| 1_{dp}\right\rangle -\left| 1_{dp}\right\rangle \left| 0_{dp}\right\rangle \right) \nonumber \\= & {} \frac{1}{\sqrt{2}}\left( \left| 01\right\rangle \left| 10\right\rangle -\left| 10\right\rangle \left| 01\right\rangle \right) \nonumber \end{aligned}$$
(1)

Four unitary operations, \(\Omega _{I}\), \(\Omega _{z}\), \(\Omega _{x}\), and \(\Omega _{y}\), used to transform a logical Bell state into another (see also Table 2), can be defined as follows:

$$\begin{aligned} \begin{array}{ccccc} &{} &{} \Omega _{I} &{} = &{} I_{1}\otimes I_{2}\\ &{} &{} \Omega _{z} &{} = &{} U_{z1}\otimes I_{2}\\ &{} &{} \Omega _{x} &{} = &{} U_{x1}\otimes U_{x2}\\ &{} &{} \Omega _{y} &{} = &{} U_{y1}\otimes U_{x2} \end{array} \end{aligned}$$
(2)

where \(I=\left| 0\right\rangle \left\langle 0\right| +\left| 1\right\rangle \left\langle 1\right| \), \(U_{z}=\left| 0\right\rangle \left\langle 0\right| -\left| 1\right\rangle \left\langle 1\right| \),\(U_{x}=\left| 1\right\rangle \left\langle 0\right| +\left| 0\right\rangle \left\langle 1\right| \), and \(U_{y}=\left| 0\right\rangle \left\langle 1\right| -\left| 1\right\rangle \left\langle 0\right| \).

Table 2 The transformation of logical Bell states in the collective-dephasing noise

1.2 The logical qubits immune to collective-rotation noise

The logical Bell states under collective-rotation noise can be defined as follows:

$$\begin{aligned} \left| \Phi _{r}^{+}\right\rangle= & {} \frac{1}{\sqrt{2}}\left( \left| 0_{r}\right\rangle \left| 0_{r}\right\rangle +\left| 1_{r}\right\rangle \left| 1_{r}\right\rangle \right) \nonumber \\= & {} \frac{1}{\sqrt{2}}\left( \left| \Phi ^{+}\right\rangle \left| \Phi ^{+}\right\rangle +\left| \Psi ^{-}\right\rangle \left| \Psi ^{-}\right\rangle \right) \nonumber \\ \left| \Phi _{r}^{-}\right\rangle= & {} \frac{1}{\sqrt{2}}\left( \left| 0_{r}\right\rangle \left| 0_{r}\right\rangle -\left| 1_{r}\right\rangle \left| 1_{r}\right\rangle \right) \nonumber \\= & {} \frac{1}{\sqrt{2}}\left( \left| \Phi ^{+}\right\rangle \left| \Phi ^{+}\right\rangle -\left| \Psi ^{-}\right\rangle \left| \Psi ^{-}\right\rangle \right) \\ \left| \Psi _{r}^{+}\right\rangle= & {} \frac{1}{\sqrt{2}}\left( \left| 0_{r}\right\rangle \left| 1_{r}\right\rangle +\left| 1_{r}\right\rangle \left| 0_{r}\right\rangle \right) \nonumber \\= & {} \frac{1}{\sqrt{2}}\left( \left| \Phi ^{+}\right\rangle \left| \Psi ^{-}\right\rangle +\left| \Psi ^{-}\right\rangle \left| \Phi ^{+}\right\rangle \right) \nonumber \\ \left| \Psi _{r}^{-}\right\rangle= & {} \frac{1}{\sqrt{2}}\left( \left| 0_{r}\right\rangle \left| 1_{r}\right\rangle -\left| 1_{r}\right\rangle \left| 0_{r}\right\rangle \right) \nonumber \\= & {} \frac{1}{\sqrt{2}}\left( \left| \Phi ^{+}\right\rangle \left| \Psi ^{-}\right\rangle -\left| \Psi ^{-}\right\rangle \left| \Phi ^{+}\right\rangle \right) \nonumber \end{aligned}$$
(3)

Four unitary operations, \(\Theta _{I}\), \(\Theta _{z}\), \(\Theta _{x}\), and \(\Theta _{y}\), used to transform a logical Bell state into another (see also Table 3), can also be defined as follows:

$$\begin{aligned} \begin{array}{ccccc} &{} &{} \Theta _{I} &{} = &{} I_{1}\otimes I_{2}\\ &{} &{} \Theta _{z} &{} = &{} U_{z1}\otimes U_{z2}\\ &{} &{} \Theta _{x} &{} = &{} U_{z1}\otimes U_{x2}\\ &{} &{} \Theta _{y} &{} = &{} I_{1}\otimes U_{y2} \end{array} \end{aligned}$$
(4)
Table 3 The transformation of logical Bell states in the collective-rotation noises

Appendix 2: The quantum states after performing CE attack

Suppose Eve’s CNOT operations are define as \(U_{E}\).

1.1 Case 1

$$\begin{aligned} U_{E}= & {} CNOT(b_{3},E_{3})CNOT(a_{3},E_{3})CNOT(b_{2},E_{2})\\&\times CNOT(a_{2},E_{2}) CNOT(b_{1},E_{1})CNOT(a_{1},E_{1})\\&U_{E}\left| \Phi ^{+}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{+}\right\rangle _{a_{2}b_{2}}\left| \Phi ^{+}\right\rangle _{a_{3}b_{3}}\left| 0\right\rangle _{E_{1}}\left| 0\right\rangle _{E_{2}}\left| 0\right\rangle _{E_{3}}\\= & {} \left| \Phi ^{+}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{+}\right\rangle _{a_{2}b_{2}}\left| \Phi ^{+}\right\rangle _{a_{3}b_{3}}\left| 0\right\rangle _{E_{1}}\left| 0\right\rangle _{E_{2}}\left| 0\right\rangle _{E_{3}} \end{aligned}$$

1.2 Case 2

$$\begin{aligned} U_{E}= & {} CNOT(b_{2},E_{3})CNOT(a_{3},E_{3})CNOT(b_{3},E_{2})CNOT(a_{2},E_{2})CNOT(b_{1},E_{1})CNOT(a_{1},E_{1})\\&U_{E}\left| \Phi ^{+}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{+}\right\rangle _{a_{2}b_{2}}\left| \Phi ^{+}\right\rangle _{a_{3}b_{3}}\left| 0\right\rangle _{E_{1}}\left| 0\right\rangle _{E_{2}}\left| 0\right\rangle _{E_{3}}\\= & {} \left| \Phi ^{+}\right\rangle _{a_{1}b_{1}}\left| 0\right\rangle _{E_{1}}\left( \frac{1}{2}\left( \left| 0000\right\rangle +\left| 1111\right\rangle \right) _{a_{2}b_{2}a_{3}b_{3}}\left| 0\right\rangle _{E_{2}}\left| 0\right\rangle _{E_{3}}\right. \\&\left. +\,\frac{1}{2}\left( \left| 0011\right\rangle +\left| 1100\right\rangle \right) _{a_{2}b_{2}a_{3}b_{3}}\left| 1\right\rangle _{E_{2}}\left| 1\right\rangle _{E_{3}}\right) \\= & {} \left| \Phi ^{+}\right\rangle _{a_{1}b_{1}}\left| 0\right\rangle _{E_{1}}\left( \begin{array}{l} \frac{1}{2}\left( \left| \Phi ^{+}\right\rangle _{a_{2}b_{2}}\left| \Phi ^{+}\right\rangle _{a_{3}b_{3}}+\left| \Phi ^{-}\right\rangle _{a_{2}b_{2}}\left| \Phi ^{-}\right\rangle _{a_{3}b_{3}}\right) \left| 0\right\rangle _{E_{2}}\left| 0\right\rangle _{E_{3}}+\\ \frac{1}{2}\left( \left| \Phi ^{+}\right\rangle _{a_{2}b_{2}}\left| \Phi ^{+}\right\rangle _{a_{3}b_{3}}-\left| \Phi ^{-}\right\rangle _{a_{2}b_{2}}\left| \Phi ^{-}\right\rangle _{a_{3}b_{3}}\right) \left| 1\right\rangle _{E_{2}}\left| 1\right\rangle _{E_{3}}\end{array}\right) \\= & {} \frac{1}{2}\left( \begin{array}{l} \left| \Phi ^{+}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{+}\right\rangle _{a_{2}b_{2}}\left| \Phi ^{+}\right\rangle _{a_{3}b_{3}}\left| 0\right\rangle _{E_{1}}\left| 0\right\rangle _{E_{2}}\left| 0\right\rangle _{E_{3}}+\left| \Phi ^{+}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{-}\right\rangle _{a_{2}b_{2}}\left| \Phi ^{-}\right\rangle _{a_{3}b_{3}}\left| 0\right\rangle _{E_{1}}\left| 0\right\rangle _{E_{2}}\left| 0\right\rangle _{E_{3}}+\\ \left| \Phi ^{+}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{+}\right\rangle _{a_{2}b_{2}}\left| \Phi ^{+}\right\rangle _{a_{3}b_{3}}\left| 0\right\rangle _{E_{1}}\left| 1\right\rangle _{E_{2}}\left| 1\right\rangle _{E_{3}}+\left| \Phi ^{+}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{-}\right\rangle _{a_{2}b_{2}}\left| \Phi ^{-}\right\rangle _{a_{3}b_{3}}\left| 0\right\rangle _{E_{1}}\left| 1\right\rangle _{E_{2}}\left| 1\right\rangle _{E_{3}}\end{array}\right) \end{aligned}$$

1.3 Case 3

$$\begin{aligned} U_{E}= & {} CNOT(b_{2},E_{3})CNOT(a_{3},E_{3})CNOT(b_{1},E_{2})CNOT(a_{2},E_{2})CNOT(b_{2},E_{1})CNOT(a_{1},E_{1})\\&U_{E}\left| \Phi ^{+}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{+}\right\rangle _{a_{2}b_{2}}\left| \Phi ^{+}\right\rangle _{a_{3}b_{3}}\left| 0\right\rangle _{E_{1}}\left| 0\right\rangle _{E_{2}}\left| 0\right\rangle _{E_{3}}\\= & {} \left( \frac{1}{2}\left( \left| 0000\right\rangle +\left| 1111\right\rangle \right) _{a_{1}b_{1}a_{2}b_{2}}\left| 0\right\rangle _{E_{1}}\left| 0\right\rangle _{E_{2}}\right. \\&\left. +\,\frac{1}{2}\left( \left| 0011\right\rangle +\left| 1100\right\rangle \right) _{a_{1}b_{1}a_{2}b_{2}}\left| 1\right\rangle _{E_{1}}\left| 1\right\rangle _{E_{2}}\right) \left| \Phi ^{+}\right\rangle _{a_{3}b_{3}}\left| 0\right\rangle _{E_{3}}\\= & {} \left( \begin{array}{l} \frac{1}{2}\left( \left| \Phi ^{+}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{+}\right\rangle _{a_{2}b_{2}}+\left| \Phi ^{-}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{-}\right\rangle _{a_{2}b_{2}}\right) \left| 0\right\rangle _{E_{1}}\left| 0\right\rangle _{E_{2}}+\\ \frac{1}{2}\left( \left| \Phi ^{+}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{+}\right\rangle _{a_{2}b_{2}}-\left| \Phi ^{-}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{-}\right\rangle _{a_{2}b_{2}}\right) \left| 1\right\rangle _{E_{1}}\left| 1\right\rangle _{E_{2}}\end{array}\right) \left| \Phi ^{+}\right\rangle _{a_{3}b_{3}}\left| 0\right\rangle _{E_{3}}\\= & {} \frac{1}{2}\left( \begin{array}{l} \left| \Phi ^{+}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{+}\right\rangle _{a_{2}b_{2}}\left| \Phi ^{+}\right\rangle _{a_{3}b_{3}}\left| 0\right\rangle _{E_{1}}\left| 0\right\rangle _{E_{2}}\left| 0\right\rangle _{E_{3}}+\left| \Phi ^{-}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{-}\right\rangle _{a_{2}b_{2}}\left| \Phi ^{+}\right\rangle _{a_{3}b_{3}}\left| 0\right\rangle _{E_{1}}\left| 0\right\rangle _{E_{2}}\left| 0\right\rangle _{E_{3}}+\\ \left| \Phi ^{+}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{+}\right\rangle _{a_{2}b_{2}}\left| \Phi ^{+}\right\rangle _{a_{3}b_{3}}\left| 1\right\rangle _{E_{1}}\left| 1\right\rangle _{E_{2}}\left| 0\right\rangle _{E_{3}}+\left| \Phi ^{-}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{-}\right\rangle _{a_{2}b_{2}}\left| \Phi ^{+}\right\rangle _{a_{3}b_{3}}\left| 1\right\rangle _{E_{1}}\left| 1\right\rangle _{E_{2}}\left| 0\right\rangle _{E_{3}}\end{array}\right) \end{aligned}$$

1.4 Case 4

$$\begin{aligned} U_{E}= & {} CNOT(b_{1},E_{3})CNOT(a_{3},E_{3})CNOT(b_{2},E_{2})CNOT(a_{2},E_{2})CNOT(b_{3},E_{1})CNOT(a_{1},E_{1})\\&U_{E}\left| \Phi ^{+}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{+}\right\rangle _{a_{2}b_{2}}\left| \Phi ^{+}\right\rangle _{a_{3}b_{3}}\left| 0\right\rangle _{E_{1}}\left| 0\right\rangle _{E_{2}}\left| 0\right\rangle _{E_{3}}\\= & {} \left( \frac{1}{2}\left( \left| 0000\right\rangle +\left| 1111\right\rangle \right) _{a_{1}b_{1}a_{3}b_{3}}\left| 0\right\rangle _{E_{1}}\left| 0\right\rangle _{E_{3}}\right. \\&\left. +\,\frac{1}{2}\left( \left| 0011\right\rangle +\left| 1100\right\rangle \right) _{a_{1}b_{1}a_{3}b_{3}}\left| 1\right\rangle _{E_{1}}\left| 1\right\rangle _{E_{3}}\right) \left| \Phi ^{+}\right\rangle _{a_{2}b_{2}}\left| 0\right\rangle _{E_{2}}\\= & {} \left( \begin{array}{l} \frac{1}{2}\left( \left| \Phi ^{+}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{+}\right\rangle _{a_{3}b_{3}}+\left| \Phi ^{-}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{-}\right\rangle _{a_{3}b_{3}}\right) \left| 0\right\rangle _{E_{1}}\left| 0\right\rangle _{E_{3}}+\\ \frac{1}{2}\left( \left| \Phi ^{+}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{+}\right\rangle _{a_{3}b_{3}}-\left| \Phi ^{-}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{-}\right\rangle _{a_{3}b_{3}}\right) \left| 1\right\rangle _{E_{1}}\left| 1\right\rangle _{E_{3}}\end{array}\right) \left| \Phi ^{+}\right\rangle _{a_{2}b_{2}}\left| 0\right\rangle _{E_{2}}\\= & {} \frac{1}{2}\left( \begin{array}{l} \left| \Phi ^{+}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{+}\right\rangle _{a_{2}b_{2}}\left| \Phi ^{+}\right\rangle _{a_{3}b_{3}}\left| 0\right\rangle _{E_{1}}\left| 0\right\rangle _{E_{2}}\left| 0\right\rangle _{E_{3}}+\left| \Phi ^{-}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{+}\right\rangle _{a_{2}b_{2}}\left| \Phi ^{-}\right\rangle _{a_{3}b_{3}}\left| 0\right\rangle _{E_{1}}\left| 0\right\rangle _{E_{2}}\left| 0\right\rangle _{E_{3}}+\\ \left| \Phi ^{+}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{+}\right\rangle _{a_{2}b_{2}}\left| \Phi ^{+}\right\rangle _{a_{3}b_{3}}\left| 1\right\rangle _{E_{1}}\left| 0\right\rangle _{E_{2}}\left| 1\right\rangle _{E_{3}}+\left| \Phi ^{-}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{+}\right\rangle _{a_{2}b_{2}}\left| \Phi ^{-}\right\rangle _{a_{3}b_{3}}\left| 1\right\rangle _{E_{1}}\left| 0\right\rangle _{E_{2}}\left| 1\right\rangle _{E_{3}}\end{array}\right) \end{aligned}$$

1.5 Case 5

$$\begin{aligned} U_{E}= & {} CNOT(b_{1},E_{3})CNOT(a_{3},E_{3})CNOT(b_{3},E_{2})CNOT(a_{2},E_{2})CNOT(b_{2},E_{1})CNOT(a_{1},E_{1})\\&U_{E}\left| \Phi ^{+}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{+}\right\rangle _{a_{2}b_{2}}\left| \Phi ^{+}\right\rangle _{a_{3}b_{3}}\left| 0\right\rangle _{E_{1}}\left| 0\right\rangle _{E_{2}}\left| 0\right\rangle _{E_{3}}\\= & {} \frac{1}{2\sqrt{2}}\left[ \begin{array}{l} \left( \left| 000000\right\rangle +\left| 111111\right\rangle \right) \left| 000\right\rangle +\left( \left| 001100\right\rangle +\left| 110011\right\rangle \right) \left| 110\right\rangle +\\ \left( \left| 110000\right\rangle +\left| 001111\right\rangle \right) \left| 101\right\rangle +\left( \left| 111100\right\rangle +\left| 000011\right\rangle \right) \left| 011\right\rangle \end{array}\right] _{a_{1}b_{1}a_{2}b_{2}a_{3}b_{3}E_{1}E_{2}E_{3}}\\= & {} \frac{1}{4}\left[ \begin{array}{l} \left( \left| \Phi ^{+}\right\rangle \left| \Phi ^{+}\right\rangle \left| \Phi ^{+}\right\rangle +\left| \Phi ^{+}\right\rangle \left| \Phi ^{-}\right\rangle \left| \Phi ^{-}\right\rangle +\left| \Phi ^{-}\right\rangle \left| \Phi ^{+}\right\rangle \left| \Phi ^{-}\right\rangle +\left| \Phi ^{-}\right\rangle \left| \Phi ^{-}\right\rangle \left| \Phi ^{+}\right\rangle \right) \left| 000\right\rangle +\\ \left( \left| \Phi ^{+}\right\rangle \left| \Phi ^{+}\right\rangle \left| \Phi ^{+}\right\rangle -\left| \Phi ^{+}\right\rangle \left| \Phi ^{-}\right\rangle \left| \Phi ^{-}\right\rangle +\left| \Phi ^{-}\right\rangle \left| \Phi ^{+}\right\rangle \left| \Phi ^{-}\right\rangle -\left| \Phi ^{-}\right\rangle \left| \Phi ^{-}\right\rangle \left| \Phi ^{+}\right\rangle \right) \left| 110\right\rangle +\\ \left( \left| \Phi ^{+}\right\rangle \left| \Phi ^{+}\right\rangle \left| \Phi ^{+}\right\rangle +\left| \Phi ^{+}\right\rangle \left| \Phi ^{-}\right\rangle \left| \Phi ^{-}\right\rangle -\left| \Phi ^{-}\right\rangle \left| \Phi ^{+}\right\rangle \left| \Phi ^{-}\right\rangle -\left| \Phi ^{-}\right\rangle \left| \Phi ^{-}\right\rangle \left| \Phi ^{+}\right\rangle \right) \left| 101\right\rangle +\\ \left( \left| \Phi ^{+}\right\rangle \left| \Phi ^{+}\right\rangle \left| \Phi ^{+}\right\rangle -\left| \Phi ^{+}\right\rangle \left| \Phi ^{-}\right\rangle \left| \Phi ^{-}\right\rangle -\left| \Phi ^{-}\right\rangle \left| \Phi ^{+}\right\rangle \left| \Phi ^{-}\right\rangle +\left| \Phi ^{-}\right\rangle \left| \Phi ^{-}\right\rangle \left| \Phi ^{+}\right\rangle \right) \left| 011\right\rangle \end{array}\right] _{a_{1}b_{1}a_{2}b_{2}a_{3}b_{3}E_{1}E_{2}E_{3}} \end{aligned}$$

1.6 Case 6

$$\begin{aligned} U_{E}= & {} CNOT(b_{2},E_{3})CNOT(a_{3},E_{3})CNOT(b_{1},E_{2})CNOT(a_{2},E_{2})CNOT(b_{3},E_{1})CNOT(a_{1},E_{1})\\&U_{E}\left| \Phi ^{+}\right\rangle _{a_{1}b_{1}}\left| \Phi ^{+}\right\rangle _{a_{2}b_{2}}\left| \Phi ^{+}\right\rangle _{a_{3}b_{3}}\left| 0\right\rangle _{E_{1}}\left| 0\right\rangle _{E_{2}}\left| 0\right\rangle _{E_{3}}\\= & {} \frac{1}{2\sqrt{2}}\left[ \begin{array}{l} \left( \left| 000000\right\rangle +\left| 111111\right\rangle \right) \left| 000\right\rangle +\left( \left| 001100\right\rangle +\left| 110011\right\rangle \right) \left| 011\right\rangle +\\ \left( \left| 110000\right\rangle +\left| 001111\right\rangle \right) \left| 110\right\rangle +\left( \left| 111100\right\rangle +\left| 000011\right\rangle \right) \left| 101\right\rangle \end{array}\right] _{a_{1}b_{1}a_{2}b_{2}a_{3}b_{3}E_{1}E_{2}E_{3}}\\= & {} \frac{1}{4}\left[ \begin{array}{l} \left( \left| \Phi ^{+}\right\rangle \left| \Phi ^{+}\right\rangle \left| \Phi ^{+}\right\rangle +\left| \Phi ^{+}\right\rangle \left| \Phi ^{-}\right\rangle \left| \Phi ^{-}\right\rangle +\left| \Phi ^{-}\right\rangle \left| \Phi ^{+}\right\rangle \left| \Phi ^{-}\right\rangle +\left| \Phi ^{-}\right\rangle \left| \Phi ^{-}\right\rangle \left| \Phi ^{+}\right\rangle \right) \left| 000\right\rangle +\\ \left( \left| \Phi ^{+}\right\rangle \left| \Phi ^{+}\right\rangle \left| \Phi ^{+}\right\rangle -\left| \Phi ^{+}\right\rangle \left| \Phi ^{-}\right\rangle \left| \Phi ^{-}\right\rangle +\left| \Phi ^{-}\right\rangle \left| \Phi ^{+}\right\rangle \left| \Phi ^{-}\right\rangle -\left| \Phi ^{-}\right\rangle \left| \Phi ^{-}\right\rangle \left| \Phi ^{+}\right\rangle \right) \left| 011\right\rangle +\\ \left( \left| \Phi ^{+}\right\rangle \left| \Phi ^{+}\right\rangle \left| \Phi ^{+}\right\rangle +\left| \Phi ^{+}\right\rangle \left| \Phi ^{-}\right\rangle \left| \Phi ^{-}\right\rangle -\left| \Phi ^{-}\right\rangle \left| \Phi ^{+}\right\rangle \left| \Phi ^{-}\right\rangle -\left| \Phi ^{-}\right\rangle \left| \Phi ^{-}\right\rangle \left| \Phi ^{+}\right\rangle \right) \left| 110\right\rangle +\\ \left( \left| \Phi ^{+}\right\rangle \left| \Phi ^{+}\right\rangle \left| \Phi ^{+}\right\rangle -\left| \Phi ^{+}\right\rangle \left| \Phi ^{-}\right\rangle \left| \Phi ^{-}\right\rangle -\left| \Phi ^{-}\right\rangle \left| \Phi ^{+}\right\rangle \left| \Phi ^{-}\right\rangle +\left| \Phi ^{-}\right\rangle \left| \Phi ^{-}\right\rangle \left| \Phi ^{+}\right\rangle \right) \left| 101\right\rangle \end{array}\right] _{a_{1}b_{1}a_{2}b_{2}a_{3}b_{3}E_{1}E_{2}E_{3}} \end{aligned}$$

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Hwang, T., Luo, YP. Probabilistic authenticated quantum dialogue. Quantum Inf Process 14, 4631–4650 (2015). https://doi.org/10.1007/s11128-015-1140-8

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