# Multi-Party Controlled Semi-Quantum Dialogue Protocol Based on Hyperentangled Bell States
## 2 MCSQD Protocol  
### Preliminary Preparations  
#### 2.1.1 Preparation of Quantum States  
In the proposed MCSQD protocol, the controller [PERSON]\({}_{1}\) and [PERSON] possess full quantum capabilities. In contrast, participant [PERSON] is classical, meaning he can only perform the following classical operations:  
1. Measure and prepare qubits exclusively in the Z-basis;
2. Reflect qubits to other participants without any disturbance;
3. Reorder qubits using different delay lines.  
[PERSON]\({}_{1}\) is responsible for preparing \(N\) identical hyperentangled photon pairs in the polarization-spatial mode:  
\[\ket{\Phi}_{AB}^{P_{S}}=\ket{\eta}_{AB}^{P}\otimes\ket{\xi}_{AB}^{S} \tag{1}\]  
where the subscripts A and B denote two distinct photons in the system, and the superscripts P and S represent the polarization DOF and the spatial DOF, respectively. This can be specifically expressed as:  
\[\ket{\varphi^{\pm}}^{P}=\frac{1}{\sqrt{2}}(\ket{\mathrm{HH}}\pm\ket{\mathrm{ VV}})^{P}\]  
\[\ket{\psi^{\pm}}^{P}=\frac{1}{\sqrt{2}}(\ket{\mathrm{HV}}\pm\ket{\mathrm{VH}})^{P}\]  
\[\ket{\varphi^{\pm}}^{S}=\frac{1}{\sqrt{2}}(\ket{a_{1}b_{1}}\pm\ket{a_{2}b_{2}} )^{S}\]  
\[\ket{\psi^{\pm}}^{S}=\frac{1}{\sqrt{2}}(\ket{a_{1}b_{2}}\pm\ket{a_{2}b_{1}})^{S} \tag{2}\]Here, H and V represent the horizontal and vertical polarization states of the hyperentangled photon pair, while \(a_{1},a_{2}(b_{1},b_{2})\) denote the distinct spatial modes of photon A(B). In 2005, the [PERSON] team [26] successfully prepared hyperentangled two-photon states in both polarization and spatial DOF by exciting a \(\beta\)-barium borate (BBO) crystal with femtosecond pump pulses. Subsequent advancements in quantum optical technology have enabled full discrimination of hyperentangled states [27; 28], with extensive applications across quantum information processing domains including entanglement purification, concentration protocols, and Bell-state analysis.  
During the measurement process, several measurement bases are utilized, including \(Z_{P}\), \(X_{P}\), and \(Z_{M}=Z_{P}\otimes Z_{S}\). Specifically, for the polarization DOF:  
\[Z_{P}=\{|\mathrm{H}\rangle,|\mathrm{V}\rangle\}\]  
\[X_{P}=\{|+_{P}\rangle=\frac{1}{\sqrt{2}}(|\mathrm{H}\rangle+|\mathrm{V} \rangle),|-_{P}\rangle=\frac{1}{\sqrt{2}}(|\mathrm{H}\rangle-|\mathrm{V} \rangle)\} \tag{3}\]  
In the spatial-mode DOF, \(Z_{S}\in\{Z_{SA},Z_{SB}\}\), where:  
\[Z_{SA}=\{|a_{1}\rangle,|a_{2}\rangle\}\]  
\[Z_{SB}=\{|b_{1}\rangle,|b_{2}\rangle\} \tag{4}\]  
To address the question of how to select \(Z_{S}\) for measurement, [PERSON] (or [PERSON]) will choose an appropriate measurement basis based on the photon sequence in their possession. For instance, if [PERSON] wishes to determine the quantum state of particle B in the spatial-mode DOF within \(S_{B}\), he will select the \(Z_{SB}\) measurement basis.  
Compared to most previous SQD protocols, the proposed protocol introduces significant improvements in the encoding method by adopting an adaptive Huffman compression coding algorithm. This approach constructs a Huffman tree to assign values to each character, markedly optimizing the storage space complexity for classical parties. The classical party leverages an \"immediate measurement and transmission\" mechanism, enabling it to complete encoding and decoding operations without temporarily storing qubits. Utilizing Huffman compression coding for information transmission greatly reduces the storage space required on the classical side, enhancing communication efficiency in semi-quantum secure dialogue.