# Quantum Privacy Query Protocol Based on GHZ-like States
## 3 Quantum Privacy Query Based on the GHZ-like State
### Quantum Privacy Query  
**Step 1. ** First, [PERSON] generates N GHZ-like states and enough non-orthogonal states. These non-orthogonal states are randomly in one of these four states: \(\{|0\rangle,|1\rangle,|+\rangle,|-\rangle\}\). The GHZ-like states are shown below:  
\[|\text{GHZ-like}\rangle_{123}=\frac{1}{2}(|000\rangle+|011\rangle+|110\rangle+|10 1\rangle)_{123} \tag{7}\]  
The footer 123 represents particle 1, particle 2 and particle 3, respectively. We can also represent the GHZ-like states as a sequence C:  
\[C:C_{1}(1),C_{1}(2),C_{1}(3);C_{2}(1),C_{2}(2),C_{2}(3);\ldots;C_{n}(1),C_{n}(2),C_{n}(3) \tag{8}\]  
**Step 2. ** First, [PERSON] generates and announces a binary number \(R\) of \(r_{1}r_{2}r_{3}\ldots r_{n}\), where \(r_{i}(1\leq i\leq n)\) is \(C_{i}(1)\oplus C_{i}(2)\oplus C_{i}(3)\). [PERSON] randomly inserts enough decoy photons \(\{|0\rangle,|1\rangle,|+\rangle,|-\rangle\}\) into the sequence of particles, each randomly in one of four states. [PERSON] then retains particle 1 in sequence \(C\), assigns particle 2 to [PERSON], and assigns particle 3 to [PERSON]. **Step 3. ** When all the particles are released, [PERSON] and [PERSON] publicly announce which particle positions were not successfully received. Then, the three participants will remove the particles from this position together. [PERSON] and [PERSON] will not lie during this step because this step does not reveal any private information, and lying does not benefit [PERSON] and [PERSON] in any way. Then, in order to check whether the entangled particles are safely distributed and the particle releases are listened to, non-orthogonal particles are used for security checking. This is performed as follows: firstly, in Step 2, [PERSON] randomly inserts enough decoy photons \(\{|0\rangle,|1\rangle,|+\rangle,|-\rangle\}\) when distributing the particles of sequence \(C\), and each particle is randomly in one of the four states. Secondly, [PERSON] tells [PERSON] and [PERSON] about the positions of the decoy photons inserted in sequence \(C\) and the prepared measurement bases. Then, [PERSON] and [PERSON] use [PERSON]'s information to measure these decoy photons and tell [PERSON] about the measurement results of these decoy photons. Finally, [PERSON] determines whether there is a listener or not by comparing the initial state of the decoy photon inserted at the beginning with the measurement results of [PERSON] and [PERSON]. If the error rate is higher than the threshold, there is a listener, and the protocol is terminated, returning to Step 2 and starting again; otherwise, the subsequent protocol continues. **Step 4. ** Next, [PERSON], [PERSON], and [PERSON] measure the sequence of particles in their hands by the binary number R: \(r_{1}r_{2}r_{3}\ldots r_{n}\), where \(r_{i}(1\leq i\leq n)\), as shown in Table 2, and record the measurement. At the same time, the particle measured by [PERSON] is called sequence S1; similarly, the particle retained by the server is called sequence S2; and the particle owned by [PERSON] is called sequence S3. **Step 5. ** At this point, [PERSON] chooses a GHZ-like state at random, joining [PERSON] to choose the \(i\)th GHZ-like state, and then he secretly tells [PERSON] about this bit and his measurements about this bit. **Step 6. ** As shown in Table 3, by [PERSON] secretly telling [PERSON] the above information, [PERSON] can infer the bits of the relative location [PERSON]. Firstly, when the \(i\)th bit of the binary number R is 0, the measurement base used is the Z measurement base. If [PERSON] secretly tells [PERSON] it is \(|1\rangle\), [PERSON] and [PERSON] have opposite results, so [PERSON] can infer [PERSON]'s measurement result according to her own measurement result. If [PERSON] secretly tells [PERSON] it is \(|0\rangle\), [PERSON] and [PERSON] have the same results, so [PERSON] can infer [PERSON]'s measurement result according to her own measurement result. Secondly, when the \(i\)th bit of the binary number R is 1, then the measurement base used is the X measurement base.