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Controlled bidirectional remote preparation of three-qubit state

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Abstract

We present a novel scheme for controlled bidirectional remote state preparation by using thirteen-qubit entangled state as the quantum channel, where both Alice and Bob transfer an arbitrary three-qubit state to each other simultaneously via the control of Charlie. Firstly, in the ideal environment, we consider our scheme in two cases that the coefficients of prepared state are real and complex, respectively. The corresponding measurement bases are devised. Secondly, we discuss our scheme in four types of noisy environment (bit-flip, phase-flip, amplitude-damping and phase-damping noisy environments) and calculate the corresponding fidelities of the output state. Finally, the efficiency of our scheme is calculated and some discussions are given.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant Nos. 61671087, 61272514, 61170272, 61502048, 61373131, 61309029), the National Development Foundation for Cryptological Research (Grant No. MMJJ201401012), the Fok Ying Tung Education Foundation (Grant No. 131067), Open Foundation of Guizhou Provincial Key Laboratory of Public Big Data (2017BDKFJJ007) and Program for New Century Excellent Talents in University (Grant No. NCET-13-0681).

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Correspondence to Xiu-Bo Chen.

Appendix

Appendix

Here, the whole corresponding local unitary operators \(U_{357}\) and \(U_{8ac}\) are given when the coefficients of the prepared state are real. These operators are executed by Alice and Bob on qubits 357 and 8ac, respectively. The I, X, iY, Z are Pauli operations.

A’s result

B’s result

C’s result

\(U_{357}\)

\(U_{8ac}\)

\(|M_{0}\rangle _{9bd}\)

\(|N_{0}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes I_{5}\otimes I_{7}\)

\(I_{8}\otimes I_{a}\otimes I_{c}\)

\(|M_{0}\rangle _{9bd}\)

\(|N_{1}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes I_{5}\otimes I_{7}\)

\(I_{8}\otimes I_{a}\otimes iY_{c}\)

\(|M_{0}\rangle _{9bd}\)

\(|N_{2}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes I_{5}\otimes I_{7}\)

\(I_{8}\otimes -iY_{a}\otimes Z_{c}\)

\(|M_{0}\rangle _{9bd}\)

\(|N_{3}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes I_{5}\otimes I_{7}\)

\(I_{8}\otimes iY_{a}\otimes X_{c}\)

\(|M_{0}\rangle _{9bd}\)

\(|N_{4}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes I_{5}\otimes I_{7}\)

\(iY_{8}\otimes Z_{a}\otimes Z_{c}\)

\(|M_{0}\rangle _{9bd}\)

\(|N_{5}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes I_{5}\otimes I_{7}\)

\(iY_{8}\otimes I_{a}\otimes X_{c}\)

\(|M_{0}\rangle _{9bd}\)

\(|N_{6}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes I_{5}\otimes I_{7}\)

\(iY_{8}\otimes X_{a}\otimes Z_{c}\)

\(|M_{0}\rangle _{9bd}\)

\(|N_{7}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes I_{5}\otimes I_{7}\)

\(X_{8}\otimes iY_{a}\otimes X_{c}\)

\(|M_{1}\rangle _{9bd}\)

\(|N_{0}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes I_{5}\otimes iY_{7}\)

\(I_{8}\otimes I_{a}\otimes I_{c}\)

\(|M_{1}\rangle _{9bd}\)

\(|N_{1}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes I_{5}\otimes iY_{7}\)

\(I_{8}\otimes I_{a}\otimes iY_{c}\)

\(|M_{1}\rangle _{9bd}\)

\(|N_{2}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes I_{5}\otimes iY_{7}\)

\(I_{8}\otimes -iY_{a}\otimes Z_{c}\)

\(|M_{1}\rangle _{9bd}\)

\(|N_{3}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes I_{5}\otimes iY_{7}\)

\(I_{8}\otimes iY_{a}\otimes X_{c}\)

\(|M_{1}\rangle _{9bd}\)

\(|N_{4}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes I_{5}\otimes iY_{7}\)

\(iY_{8}\otimes Z_{a}\otimes Z_{c}\)

\(|M_{1}\rangle _{9bd}\)

\(|N_{5}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes I_{5}\otimes iY_{7}\)

\(iY_{8}\otimes I_{a}\otimes X_{c}\)

\(|M_{1}\rangle _{9bd}\)

\(|N_{6}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes I_{5}\otimes iY_{7}\)

\(iY_{8}\otimes X_{a}\otimes Z_{c}\)

\(|M_{1}\rangle _{9bd}\)

\(|N_{7}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes I_{5}\otimes iY_{7}\)

\(X_{8}\otimes iY_{a}\otimes X_{c}\)

\(|M_{2}\rangle _{9bd}\)

\(|N_{0}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes -iY_{5}\otimes Z_{7}\)

\(I_{8}\otimes I_{a}\otimes I_{c}\)

\(|M_{2}\rangle _{9bd}\)

\(|N_{1}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes -iY_{5}\otimes Z_{7}\)

\(I_{8}\otimes I_{a}\otimes iY_{c}\)

\(|M_{2}\rangle _{9bd}\)

\(|N_{2}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes -iY_{5}\otimes Z_{7}\)

\(I_{8}\otimes -iY_{a}\otimes Z_{c}\)

\(|M_{2}\rangle _{9bd}\)

\(|N_{3}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes -iY_{5}\otimes Z_{7}\)

\(I_{8}\otimes iY_{a}\otimes X_{c}\)

\(|M_{2}\rangle _{9bd}\)

\(|N_{4}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes -iY_{5}\otimes Z_{7}\)

\(iY_{8}\otimes Z_{a}\otimes Z_{c}\)

\(|M_{2}\rangle _{9bd}\)

\(|N_{5}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes -iY_{5}\otimes Z_{7}\)

\(iY_{8}\otimes I_{a}\otimes X_{c}\)

\(|M_{2}\rangle _{9bd}\)

\(|N_{6}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes -iY_{5}\otimes Z_{7}\)

\(iY_{8}\otimes X_{a}\otimes Z_{c}\)

\(|M_{2}\rangle _{9bd}\)

\(|N_{7}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes -iY_{5}\otimes Z_{7}\)

\(X_{8}\otimes iY_{a}\otimes X_{c}\)

\(|M_{3}\rangle _{9bd}\)

\(|N_{0}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes iY_{5}\otimes X_{7}\)

\(I_{8}\otimes I_{a}\otimes I_{c}\)

\(|M_{3}\rangle _{9bd}\)

\(|N_{1}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes iY_{5}\otimes X_{7}\)

\(I_{8}\otimes I_{a}\otimes iY_{c}\)

\(|M_{3}\rangle _{9bd}\)

\(|N_{2}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes iY_{5}\otimes X_{7}\)

\(I_{8}\otimes -iY_{a}\otimes Z_{c}\)

\(|M_{3}\rangle _{9bd}\)

\(|N_{3}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes iY_{5}\otimes X_{7}\)

\(I_{8}\otimes iY_{a}\otimes X_{c}\)

\(|M_{3}\rangle _{9bd}\)

\(|N_{4}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes iY_{5}\otimes X_{7}\)

\(iY_{8}\otimes Z_{a}\otimes Z_{c}\)

\(|M_{3}\rangle _{9bd}\)

\(|N_{5}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes iY_{5}\otimes X_{7}\)

\(iY_{8}\otimes I_{a}\otimes X_{c}\)

\(|M_{3}\rangle _{9bd}\)

\(|N_{6}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes iY_{5}\otimes X_{7}\)

\(iY_{8}\otimes X_{a}\otimes Z_{c}\)

\(|M_{3}\rangle _{9bd}\)

\(|N_{7}\rangle _{246}\)

\(|0\rangle _{1}\)

\(I_{3}\otimes iY_{5}\otimes X_{7}\)

\(X_{8}\otimes iY_{a}\otimes X_{c}\)

\(|M_{4}\rangle _{9bd}\)

\(|N_{0}\rangle _{246}\)

\(|0\rangle _{1}\)

\(iY_{3}\otimes Z_{5}\otimes Z_{7}\)

\(I_{8}\otimes I_{a}\otimes I_{c}\)

\(|M_{4}\rangle _{9bd}\)

\(|N_{1}\rangle _{246}\)

\(|0\rangle _{1}\)

\(iY_{3}\otimes Z_{5}\otimes Z_{7}\)

\(I_{8}\otimes I_{a}\otimes iY_{c}\)

\(|M_{4}\rangle _{9bd}\)

\(|N_{2}\rangle _{246}\)

\(|0\rangle _{1}\)

\(iY_{3}\otimes Z_{5}\otimes Z_{7}\)

\(I_{8}\otimes -iY_{a}\otimes Z_{c}\)

\(|M_{4}\rangle _{9bd}\)

\(|N_{3}\rangle _{246}\)

\(|0\rangle _{1}\)

\(iY_{3}\otimes Z_{5}\otimes Z_{7}\)

\(I_{8}\otimes iY_{a}\otimes X_{c}\)

\(|M_{4}\rangle _{9bd}\)

\(|N_{4}\rangle _{246}\)

\(|0\rangle _{1}\)

\(iY_{3}\otimes Z_{5}\otimes Z_{7}\)

\(iY_{8}\otimes Z_{a}\otimes Z_{c}\)

\(|M_{4}\rangle _{9bd}\)

\(|N_{5}\rangle _{246}\)

\(|0\rangle _{1}\)

\(iY_{3}\otimes Z_{5}\otimes Z_{7}\)

\(iY_{8}\otimes I_{a}\otimes X_{c}\)

\(|M_{4}\rangle _{9bd}\)

\(|N_{6}\rangle _{246}\)

\(|0\rangle _{1}\)

\(iY_{3}\otimes Z_{5}\otimes Z_{7}\)

\(iY_{8}\otimes X_{a}\otimes Z_{c}\)

\(|M_{4}\rangle _{9bd}\)

\(|N_{7}\rangle _{246}\)

\(|0\rangle _{1}\)

\(iY_{3}\otimes Z_{5}\otimes Z_{7}\)

\(X_{8}\otimes iY_{a}\otimes X_{c}\)

\(|M_{5}\rangle _{9bd}\)

\(|N_{0}\rangle _{246}\)

\(|0\rangle _{1}\)

\(iY_{3}\otimes I_{5}\otimes X_{7}\)

\(I_{8}\otimes I_{a}\otimes I_{c}\)

\(|M_{5}\rangle _{9bd}\)

\(|N_{1}\rangle _{246}\)

\(|0\rangle _{1}\)

\(iY_{3}\otimes I_{5}\otimes X_{7}\)

\(I_{8}\otimes I_{a}\otimes iY_{c}\)

\(|M_{5}\rangle _{9bd}\)

\(|N_{2}\rangle _{246}\)

\(|0\rangle _{1}\)

\(iY_{3}\otimes I_{5}\otimes X_{7}\)

\(I_{8}\otimes -iY_{a}\otimes Z_{c}\)

\(|M_{5}\rangle _{9bd}\)

\(|N_{3}\rangle _{246}\)

\(|0\rangle _{1}\)

\(iY_{3}\otimes I_{5}\otimes X_{7}\)

\(I_{8}\otimes iY_{a}\otimes X_{c}\)

\(|M_{5}\rangle _{9bd}\)

\(|N_{4}\rangle _{246}\)

\(|0\rangle _{1}\)

\(iY_{3}\otimes I_{5}\otimes X_{7}\)

\(iY_{8}\otimes Z_{a}\otimes Z_{c}\)

\(|M_{5}\rangle _{9bd}\)

\(|N_{5}\rangle _{246}\)

\(|0\rangle _{1}\)

\(iY_{3}\otimes I_{5}\otimes X_{7}\)

\(iY_{8}\otimes I_{a}\otimes X_{c}\)

\(|M_{5}\rangle _{9bd}\)

\(|N_{6}\rangle _{246}\)

\(|0\rangle _{1}\)

\(iY_{3}\otimes I_{5}\otimes X_{7}\)

\(iY_{8}\otimes X_{a}\otimes Z_{c}\)

\(|M_{5}\rangle _{9bd}\)

\(|N_{7}\rangle _{246}\)

\(|0\rangle _{1}\)

\(iY_{3}\otimes I_{5}\otimes X_{7}\)

\(X_{8}\otimes iY_{a}\otimes X_{c}\)

\(|M_{6}\rangle _{9bd}\)

\(|N_{0}\rangle _{246}\)

\(|0\rangle _{1}\)

\(iY_{3}\otimes X_{5}\otimes Z_{7}\)

\(I_{8}\otimes I_{a}\otimes I_{c}\)

\(|M_{6}\rangle _{9bd}\)

\(|N_{1}\rangle _{246}\)

\(|0\rangle _{1}\)

\(iY_{3}\otimes X_{5}\otimes Z_{7}\)

\(I_{8}\otimes I_{a}\otimes iY_{c}\)

\(|M_{6}\rangle _{9bd}\)

\(|N_{2}\rangle _{246}\)

\(|0\rangle _{1}\)

\(iY_{3}\otimes X_{5}\otimes Z_{7}\)

\(I_{8}\otimes -iY_{a}\otimes Z_{c}\)

\(|M_{6}\rangle _{9bd}\)

\(|N_{3}\rangle _{246}\)

\(|0\rangle _{1}\)

\(iY_{3}\otimes X_{5}\otimes Z_{7}\)

\(I_{8}\otimes iY_{a}\otimes X_{c}\)

\(|M_{6}\rangle _{9bd}\)

\(|N_{4}\rangle _{246}\)

\(|0\rangle _{1}\)

\(iY_{3}\otimes X_{5}\otimes Z_{7}\)

\(iY_{8}\otimes Z_{a}\otimes Z_{c}\)

\(|M_{6}\rangle _{9bd}\)

\(|N_{5}\rangle _{246}\)

\(|0\rangle _{1}\)

\(iY_{3}\otimes X_{5}\otimes Z_{7}\)

\(iY_{8}\otimes I_{a}\otimes X_{c}\)

\(|M_{6}\rangle _{9bd}\)

\(|N_{6}\rangle _{246}\)

\(|0\rangle _{1}\)

\(iY_{3}\otimes X_{5}\otimes Z_{7}\)

\(iY_{8}\otimes X_{a}\otimes Z_{c}\)

\(|M_{6}\rangle _{9bd}\)

\(|N_{7}\rangle _{246}\)

\(|0\rangle _{1}\)

\(iY_{3}\otimes X_{5}\otimes Z_{7}\)

\(X_{8}\otimes iY_{a}\otimes X_{c}\)

\(|M_{7}\rangle _{9bd}\)

\(|N_{0}\rangle _{246}\)

\(|0\rangle _{1}\)

\(X_{3}\otimes iY_{5}\otimes X_{7}\)

\(I_{8}\otimes I_{a}\otimes I_{c}\)

\(|M_{7}\rangle _{9bd}\)

\(|N_{1}\rangle _{246}\)

\(|0\rangle _{1}\)

\(X_{3}\otimes iY_{5}\otimes X_{7}\)

\(I_{8}\otimes I_{a}\otimes iY_{c}\)

\(|M_{7}\rangle _{9bd}\)

\(|N_{2}\rangle _{246}\)

\(|0\rangle _{1}\)

\(X_{3}\otimes iY_{5}\otimes X_{7}\)

\(I_{8}\otimes -iY_{a}\otimes Z_{c}\)

\(|M_{7}\rangle _{9bd}\)

\(|N_{3}\rangle _{246}\)

\(|0\rangle _{1}\)

\(X_{3}\otimes iY_{5}\otimes X_{7}\)

\(I_{8}\otimes iY_{a}\otimes X_{c}\)

\(|M_{7}\rangle _{9bd}\)

\(|N_{4}\rangle _{246}\)

\(|0\rangle _{1}\)

\(X_{3}\otimes iY_{5}\otimes X_{7}\)

\(iY_{8}\otimes Z_{a}\otimes Z_{c}\)

\(|M_{7}\rangle _{9bd}\)

\(|N_{5}\rangle _{246}\)

\(|0\rangle _{1}\)

\(X_{3}\otimes iY_{5}\otimes X_{7}\)

\(iY_{8}\otimes I_{a}\otimes X_{c}\)

\(|M_{7}\rangle _{9bd}\)

\(|N_{6}\rangle _{246}\)

\(|0\rangle _{1}\)

\(X_{3}\otimes iY_{5}\otimes X_{7}\)

\(iY_{8}\otimes X_{a}\otimes Z_{c}\)

\(|M_{7}\rangle _{9bd}\)

\(|N_{7}\rangle _{246}\)

\(|0\rangle _{1}\)

\(X_{3}\otimes iY_{5}\otimes X_{7}\)

\(X_{8}\otimes iY_{a}\otimes X_{c}\)

\(|M_{0}\rangle _{9bd}\)

\(|N_{0}\rangle _{246}\)

\(|1\rangle _{1}\)

\(-iY_{3}\otimes -iY_{5}\otimes iY_{7}\)

\(-iY_{8}\otimes -iY_{a}\otimes iY_{c}\)

\(|M_{0}\rangle _{9bd}\)

\(|N_{1}\rangle _{246}\)

\(|1\rangle _{1}\)

\(-iY_{3}\otimes -iY_{5}\otimes iY_{7}\)

\(-iY_{8}\otimes -iY_{a}\otimes I_{c}\)

\(|M_{0}\rangle _{9bd}\)

\(|N_{2}\rangle _{246}\)

\(|1\rangle _{1}\)

\(-iY_{3}\otimes -iY_{5}\otimes iY_{7}\)

\(iY_{8}\otimes I_{a}\otimes X_{c}\)

\(|M_{0}\rangle _{9bd}\)

\(|N_{3}\rangle _{246}\)

\(|1\rangle _{1}\)

\(-iY_{3}\otimes -iY_{5}\otimes iY_{7}\)

\(iY_{8}\otimes I_{a}\otimes Z_{c}\)

\(|M_{0}\rangle _{9bd}\)

\(|N_{4}\rangle _{246}\)

\(|1\rangle _{1}\)

\(-iY_{3}\otimes -iY_{5}\otimes iY_{7}\)

\(I_{8}\otimes X_{a}\otimes X_{c}\)

\(|M_{0}\rangle _{9bd}\)

\(|N_{5}\rangle _{246}\)

\(|1\rangle _{1}\)

\(-iY_{3}\otimes -iY_{5}\otimes iY_{7}\)

\(I_{8}\otimes iY_{a}\otimes Z_{c}\)

\(|M_{0}\rangle _{9bd}\)

\(|N_{6}\rangle _{246}\)

\(|1\rangle _{1}\)

\(-iY_{3}\otimes -iY_{5}\otimes iY_{7}\)

\(I_{8}\otimes Z_{a}\otimes X_{c}\)

\(|M_{0}\rangle _{9bd}\)

\(|N_{7}\rangle _{246}\)

\(|1\rangle _{1}\)

\(-iY_{3}\otimes -iY_{5}\otimes iY_{7}\)

\(I_{8}\otimes -Z_{a}\otimes Z_{c}\)

\(|M_{1}\rangle _{9bd}\)

\(|N_{0}\rangle _{246}\)

\(|1\rangle _{1}\)

\(-iY_{3}\otimes -iY_{5}\otimes I_{7}\)

\(-iY_{8}\otimes -iY_{a}\otimes iY_{c}\)

\(|M_{1}\rangle _{9bd}\)

\(|N_{1}\rangle _{246}\)

\(|1\rangle _{1}\)

\(-iY_{3}\otimes -iY_{5}\otimes I_{7}\)

\(-iY_{8}\otimes -iY_{a}\otimes I_{c}\)

\(|M_{1}\rangle _{9bd}\)

\(|N_{2}\rangle _{246}\)

\(|1\rangle _{1}\)

\(-iY_{3}\otimes -iY_{5}\otimes I_{7}\)

\(iY_{8}\otimes I_{a}\otimes X_{c}\)

\(|M_{1}\rangle _{9bd}\)

\(|N_{3}\rangle _{246}\)

\(|1\rangle _{1}\)

\(-iY_{3}\otimes -iY_{5}\otimes I_{7}\)

\(iY_{8}\otimes I_{a}\otimes Z_{c}\)

\(|M_{1}\rangle _{9bd}\)

\(|N_{4}\rangle _{246}\)

\(|1\rangle _{1}\)

\(-iY_{3}\otimes -iY_{5}\otimes I_{7}\)

\(I_{8}\otimes X_{a}\otimes X_{c}\)

\(|M_{1}\rangle _{9bd}\)

\(|N_{5}\rangle _{246}\)

\(|1\rangle _{1}\)

\(-iY_{3}\otimes -iY_{5}\otimes I_{7}\)

\(I_{8}\otimes iY_{a}\otimes Z_{c}\)

\(|M_{1}\rangle _{9bd}\)

\(|N_{6}\rangle _{246}\)

\(|1\rangle _{1}\)

\(-iY_{3}\otimes -iY_{5}\otimes I_{7}\)

\(I_{8}\otimes Z_{a}\otimes X_{c}\)

\(|M_{1}\rangle _{9bd}\)

\(|N_{7}\rangle _{246}\)

\(|1\rangle _{1}\)

\(-iY_{3}\otimes -iY_{5}\otimes I_{7}\)

\(I_{8}\otimes -Z_{a}\otimes Z_{c}\)

\(|M_{2}\rangle _{9bd}\)

\(|N_{0}\rangle _{246}\)

\(|1\rangle _{1}\)

\(iY_{3}\otimes I_{5}\otimes X_{7}\)

\(-iY_{8}\otimes -iY_{a}\otimes iY_{c}\)

\(|M_{2}\rangle _{9bd}\)

\(|N_{1}\rangle _{246}\)

\(|1\rangle _{1}\)

\(iY_{3}\otimes I_{5}\otimes X_{7}\)

\(-iY_{8}\otimes -iY_{a}\otimes I_{c}\)

\(|M_{2}\rangle _{9bd}\)

\(|N_{2}\rangle _{246}\)

\(|1\rangle _{1}\)

\(iY_{3}\otimes I_{5}\otimes X_{7}\)

\(iY_{8}\otimes I_{a}\otimes X_{c}\)

\(|M_{2}\rangle _{9bd}\)

\(|N_{3}\rangle _{246}\)

\(|1\rangle _{1}\)

\(iY_{3}\otimes I_{5}\otimes X_{7}\)

\(iY_{8}\otimes I_{a}\otimes Z_{c}\)

\(|M_{2}\rangle _{9bd}\)

\(|N_{4}\rangle _{246}\)

\(|1\rangle _{1}\)

\(iY_{3}\otimes I_{5}\otimes X_{7}\)

\(I_{8}\otimes X_{a}\otimes X_{c}\)

\(|M_{2}\rangle _{9bd}\)

\(|N_{5}\rangle _{246}\)

\(|1\rangle _{1}\)

\(iY_{3}\otimes I_{5}\otimes X_{7}\)

\(I_{8}\otimes iY_{a}\otimes Z_{c}\)

\(|M_{2}\rangle _{9bd}\)

\(|N_{6}\rangle _{246}\)

\(|1\rangle _{1}\)

\(iY_{3}\otimes I_{5}\otimes X_{7}\)

\(I_{8}\otimes Z_{a}\otimes X_{c}\)

\(|M_{2}\rangle _{9bd}\)

\(|N_{7}\rangle _{246}\)

\(|1\rangle _{1}\)

\(iY_{3}\otimes I_{5}\otimes X_{7}\)

\(I_{8}\otimes -Z_{a}\otimes Z_{c}\)

\(|M_{3}\rangle _{9bd}\)

\(|N_{0}\rangle _{246}\)

\(|1\rangle _{1}\)

\(iY_{3}\otimes I_{5}\otimes Z_{7}\)

\(-iY_{8}\otimes -iY_{a}\otimes iY_{c}\)

\(|M_{3}\rangle _{9bd}\)

\(|N_{1}\rangle _{246}\)

\(|1\rangle _{1}\)

\(iY_{3}\otimes I_{5}\otimes Z_{7}\)

\(-iY_{8}\otimes -iY_{a}\otimes I_{c}\)

\(|M_{3}\rangle _{9bd}\)

\(|N_{2}\rangle _{246}\)

\(|1\rangle _{1}\)

\(iY_{3}\otimes I_{5}\otimes Z_{7}\)

\(iY_{8}\otimes I_{a}\otimes X_{c}\)

\(|M_{3}\rangle _{9bd}\)

\(|N_{3}\rangle _{246}\)

\(|1\rangle _{1}\)

\(iY_{3}\otimes I_{5}\otimes Z_{7}\)

\(iY_{8}\otimes I_{a}\otimes Z_{c}\)

\(|M_{3}\rangle _{9bd}\)

\(|N_{4}\rangle _{246}\)

\(|1\rangle _{1}\)

\(iY_{3}\otimes I_{5}\otimes Z_{7}\)

\(I_{8}\otimes X_{a}\otimes X_{c}\)

\(|M_{3}\rangle _{9bd}\)

\(|N_{5}\rangle _{246}\)

\(|1\rangle _{1}\)

\(iY_{3}\otimes I_{5}\otimes Z_{7}\)

\(I_{8}\otimes iY_{a}\otimes Z_{c}\)

\(|M_{3}\rangle _{9bd}\)

\(|N_{6}\rangle _{246}\)

\(|1\rangle _{1}\)

\(iY_{3}\otimes I_{5}\otimes Z_{7}\)

\(I_{8}\otimes Z_{a}\otimes X_{c}\)

\(|M_{3}\rangle _{9bd}\)

\(|N_{7}\rangle _{246}\)

\(|1\rangle _{1}\)

\(iY_{3}\otimes I_{5}\otimes Z_{7}\)

\(I_{8}\otimes -Z_{a}\otimes Z_{c}\)

\(|M_{4}\rangle _{9bd}\)

\(|N_{0}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes X_{5}\otimes X_{7}\)

\(-iY_{8}\otimes -iY_{a}\otimes iY_{c}\)

\(|M_{4}\rangle _{9bd}\)

\(|N_{1}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes X_{5}\otimes X_{7}\)

\(-iY_{8}\otimes -iY_{a}\otimes I_{c}\)

\(|M_{4}\rangle _{9bd}\)

\(|N_{2}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes X_{5}\otimes X_{7}\)

\(iY_{8}\otimes I_{a}\otimes X_{c}\)

\(|M_{4}\rangle _{9bd}\)

\(|N_{3}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes X_{5}\otimes X_{7}\)

\(iY_{8}\otimes I_{a}\otimes Z_{c}\)

\(|M_{4}\rangle _{9bd}\)

\(|N_{4}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes X_{5}\otimes X_{7}\)

\(I_{8}\otimes X_{a}\otimes X_{c}\)

\(|M_{4}\rangle _{9bd}\)

\(|N_{5}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes X_{5}\otimes X_{7}\)

\(I_{8}\otimes iY_{a}\otimes Z_{c}\)

\(|M_{4}\rangle _{9bd}\)

\(|N_{6}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes X_{5}\otimes X_{7}\)

\(I_{8}\otimes Z_{a}\otimes X_{c}\)

\(|M_{4}\rangle _{9bd}\)

\(|N_{7}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes X_{5}\otimes X_{7}\)

\(I_{8}\otimes -Z_{a}\otimes Z_{c}\)

\(|M_{5}\rangle _{9bd}\)

\(|N_{0}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes iY_{5}\otimes Z_{7}\)

\(-iY_{8}\otimes -iY_{a}\otimes iY_{c}\)

\(|M_{5}\rangle _{9bd}\)

\(|N_{1}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes iY_{5}\otimes Z_{7}\)

\(-iY_{8}\otimes -iY_{a}\otimes I_{c}\)

\(|M_{5}\rangle _{9bd}\)

\(|N_{2}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes iY_{5}\otimes Z_{7}\)

\(iY_{8}\otimes I_{a}\otimes X_{c}\)

\(|M_{5}\rangle _{9bd}\)

\(|N_{3}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes iY_{5}\otimes Z_{7}\)

\(iY_{8}\otimes I_{a}\otimes Z_{c}\)

\(|M_{5}\rangle _{9bd}\)

\(|N_{4}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes iY_{5}\otimes Z_{7}\)

\(I_{8}\otimes X_{a}\otimes X_{c}\)

\(|M_{5}\rangle _{9bd}\)

\(|N_{5}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes iY_{5}\otimes Z_{7}\)

\(I_{8}\otimes iY_{a}\otimes Z_{c}\)

\(|M_{5}\rangle _{9bd}\)

\(|N_{6}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes iY_{5}\otimes Z_{7}\)

\(I_{8}\otimes Z_{a}\otimes X_{c}\)

\(|M_{5}\rangle _{9bd}\)

\(|N_{7}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes iY_{5}\otimes Z_{7}\)

\(I_{8}\otimes -Z_{a}\otimes Z_{c}\)

\(|M_{6}\rangle _{9bd}\)

\(|N_{0}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes Z_{5}\otimes X_{7}\)

\(-iY_{8}\otimes -iY_{a}\otimes iY_{c}\)

\(|M_{6}\rangle _{9bd}\)

\(|N_{1}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes Z_{5}\otimes X_{7}\)

\(-iY_{8}\otimes -iY_{a}\otimes I_{c}\)

\(|M_{6}\rangle _{9bd}\)

\(|N_{2}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes Z_{5}\otimes X_{7}\)

\(iY_{8}\otimes I_{a}\otimes X_{c}\)

\(|M_{6}\rangle _{9bd}\)

\(|N_{3}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes Z_{5}\otimes X_{7}\)

\(iY_{8}\otimes I_{a}\otimes Z_{c}\)

\(|M_{6}\rangle _{9bd}\)

\(|N_{4}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes Z_{5}\otimes X_{7}\)

\(I_{8}\otimes X_{a}\otimes X_{c}\)

\(|M_{6}\rangle _{9bd}\)

\(|N_{5}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes Z_{5}\otimes X_{7}\)

\(I_{8}\otimes iY_{a}\otimes Z_{c}\)

\(|M_{6}\rangle _{9bd}\)

\(|N_{6}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes Z_{5}\otimes X_{7}\)

\(I_{8}\otimes Z_{a}\otimes X_{c}\)

\(|M_{6}\rangle _{9bd}\)

\(|N_{7}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes Z_{5}\otimes X_{7}\)

\(I_{8}\otimes -Z_{a}\otimes Z_{c}\)

\(|M_{7}\rangle _{9bd}\)

\(|N_{0}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes -Z_{5}\otimes Z_{7}\)

\(-iY_{8}\otimes -iY_{a}\otimes iY_{c}\)

\(|M_{7}\rangle _{9bd}\)

\(|N_{1}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes -Z_{5}\otimes Z_{7}\)

\(-iY_{8}\otimes -iY_{a}\otimes I_{c}\)

\(|M_{7}\rangle _{9bd}\)

\(|N_{2}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes -Z_{5}\otimes Z_{7}\)

\(iY_{8}\otimes I_{a}\otimes X_{c}\)

\(|M_{7}\rangle _{9bd}\)

\(|N_{3}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes -Z_{5}\otimes Z_{7}\)

\(iY_{8}\otimes I_{a}\otimes Z_{c}\)

\(|M_{7}\rangle _{9bd}\)

\(|N_{4}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes -Z_{5}\otimes Z_{7}\)

\(I_{8}\otimes X_{a}\otimes X_{c}\)

\(|M_{7}\rangle _{9bd}\)

\(|N_{5}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes -Z_{5}\otimes Z_{7}\)

\(I_{8}\otimes iY_{a}\otimes Z_{c}\)

\(|M_{7}\rangle _{9bd}\)

\(|N_{6}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes -Z_{5}\otimes Z_{7}\)

\(I_{8}\otimes Z_{a}\otimes X_{c}\)

\(|M_{7}\rangle _{9bd}\)

\(|N_{7}\rangle _{246}\)

\(|1\rangle _{1}\)

\(I_{3}\otimes -Z_{5}\otimes Z_{7}\)

\(I_{8}\otimes -Z_{a}\otimes Z_{c}\)

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Chen, XB., Sun, YR., Xu, G. et al. Controlled bidirectional remote preparation of three-qubit state. Quantum Inf Process 16, 244 (2017). https://doi.org/10.1007/s11128-017-1690-z

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  • DOI: https://doi.org/10.1007/s11128-017-1690-z

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