# Development of Quantum Private Queries Protocol on Collective-Dephasing Noise Channel
## 3 Protocol Process  
Quantum private query protocol consists of three parts: key distribution, post-processing and information acquisition. In the second stage, after post-processing, the key will be diluted, to the point that [PERSON] only knows a few keys and [PERSON] knows all keys [20]. The optimal outcome is that [PERSON] only knows one bit of information of all keys, and [PERSON] does not know where the bit that [PERSON] knows is in all keys [21; 22; 23]. Among them, step (1) to step (5) are the quantum key distribution process, step (6) to step (9) are the post-processing process, and step (10) to step (11) are the information acquisition process. The protocol process will be described in detail:  
Quantum key distribution process:  
Step1: [PERSON] creates a string of particles and sends the particles to [PERSON], each one of them is randomly placed in one of four quantum states of \(\{|0_{dp}0_{dp}\rangle,|1_{dp}1_{dp}\rangle,|0_{dp}1_{dp}\rangle,|1_{dp}0_{dp} \rangle\}\). Among them \(\{|0_{dp}0_{dp}\rangle,|1_{dp}1_{dp}\rangle\}\) represent classical bit \"0\" and \(\{|0_{dp}1_{dp}\rangle,|1_{dp}0_{dp}\rangle\}\) represent classical bit \"1\". Step2: [PERSON] receives the particles and uses two measurement base \"+\", \"-\" to measure each particle at random. Thereinto \"+\" = \(\{|\Phi_{dp}^{+}\rangle,|\Phi_{dp}^{-}\rangle\}\) and \"-\" = \(\{|\Psi_{dp}^{+}\rangle,|\Psi_{dp}^{-}\rangle\}\). These four states respectively correspond to \(\{|0_{dp}0_{dp}\rangle,|0_{dp}1_{dp}\rangle,|1_{dp}0_{dp}\rangle,|1_{dp}1_{dp} \rangle\}\). Step3: [PERSON] declares which positions the particles are in, discarding the rest. Step4: For each particle that [PERSON] gets the result of measurement, [PERSON] declares two states: one is the logical Bell state corresponding to the particle that was just sent, and the other is the random logical Bell state under the other base. [PERSON] can speculate the particle state with a half probability. The deterministic results are shown in Table 2 and the uncertain results are shown in Table 3 below. \begin{table}
\begin{tabular}{c c c c c} \hline
**Quantum States** & **Measurement** & **Possible** & **The States of** & **The Results of** \\
**Sent by [PERSON]** & **Selected by [PERSON]** & **Measurement** & **[PERSON]’s Statement** & **[PERSON]’s Measurement** \\ \hline \(|0_{dp}0_{dp}\rangle(|\[PERSON]_{dp}^{+}\rangle)\) & \(\{|\[PERSON]_{dp}^{+}\rangle,|\Phi_{dp}^{-}\rangle\}\) & \(|\[PERSON]_{dp}^{+}\rangle\) & \(\{|\[PERSON]_{dp}^{+}\rangle,|\[PERSON]_{dp}^{+}\rangle\}\) & \(|\[PERSON]_{dp}^{+}\rangle\) \\  & & & \(\{|\[PERSON]_{dp}^{+}\rangle,|\[PERSON]_{dp}^{-}\rangle\}\) & \(|\[PERSON]_{dp}^{+}\rangle\) \\  & & \(|\[PERSON]_{dp}^{-}\rangle\) & \(\{|\Phi_{dp}^{+}\rangle,|\Psi_{dp}^{-}\rangle\}\) & \(|\Phi_{dp}^{+}\rangle\) \\ \(|0_{dp}1_{dp}\rangle(|\Phi_{dp}^{-}\rangle)\) & \(\{|\Psi_{dp}^{+}\rangle,|\Psi_{dp}^{-}\rangle\}\) & \(|\Psi_{dp}^{+}\rangle\) & \(|\Phi_{dp}^{-}\rangle\) \\  & & \(|\Psi_{dp}^{-}\rangle\) & \(\{|\Phi_{dp}^{-}\rangle,|\Psi_{dp}^{+}\rangle\}\) & \(|\Psi_{dp}^{+}\rangle\) \\  & & \(|\Psi_{dp}^{-}\rangle\) & \(\{|\Phi_{dp}^{-}\rangle,|\Psi_{dp}^{+}\rangle\}\) & \(|\Psi_{dp}^{-}\rangle\) \\  & & \(|\Psi_{dp}^{-}\rangle\) & \(\{|\Psi_{dp}^{+}\rangle,|\Psi_{dp}^{+}\rangle\}\) & \(|\Psi_{dp}^{+}\rangle\) \\ \(|1_{dp}0_{dp}\rangle(|\Psi_{dp}^{+}\rangle)\) & \(\{|\Psi_{dp}^{+}\rangle,|\Psi_{dp}^{-}\rangle\}\) & \(|\Psi_{dp}^{+}\rangle\) & \(|\Psi_{dp}^{+}\rangle\) \\  & & \(\{|\Psi_{dp}^{+}\rangle,|\Psi_{dp}^{-}\rangle\}\) & \(|\Psi_{dp}^{+}\rangle\) & \(|\Psi_{dp}^{+}\rangle\) \\  & & \(\{|\Psi_{dp}^{-}\rangle,|\Phi_{dp}^{+}\rangle\}\) & \(|\Phi_{dp}^{+}\rangle\) & \(|\Psi_{dp}^{+}\rangle\) \\  & & \(\{|\Psi_{dp}^{-}\rangle,|\Phi_{dp}^{+}\rangle\}\) & \(-|\Phi_{dp}^{-}\rangle\) \\ \hline \end{tabular}
\end{table} 
Table 2: Deterministic results. Step5: [PERSON] keeps all the bits that [PERSON] had measured as the oblivious keys, in the ideal case, [PERSON] would only know a half of them, but [PERSON] does not know which bits [PERSON] got.