# Transmit and Receive Diversity in MIMO Quantum Communication for High-Fidelity Video Transmission
## 3 Methodology
### Quantum Decoding and Measurement  
At the receiver, the applied phase shift from the quantum encoder is first reversed using the inverse phase gate \(S^{\dagger}\), defined as in Equation (24).  
\[S^{\dagger}=\begin{bmatrix}1&0\\ 0&-i\end{bmatrix} \tag{24}\]  
This operation restores the original phase of the superposition state of the qubit, which is necessary to recover the transmitted information. Once the inverse phase operation is applied, the receiver can perform a measurement on the computational basis using the common projection operators defined in Equation (25).  
\[M_{0}=\ket{0}\bra{0},\quad M_{1}=\ket{1}\bra{1} \tag{25}\]  
Each measurement operator can be applied to the quantum states received \(\ket{\psi_{0}}=\frac{1}{\sqrt{2}}(\ket{0}+\ket{1})\) and \(\ket{\psi_{1}}=\frac{1}{\sqrt{2}}(\ket{0}-\ket{1})\). Then, the normalized post-measurement state can be calculated using Equation (26).  
\[\ket{\psi^{\prime}}=\frac{M_{m}\ket{\psi}}{\sqrt{\bra{\psi}M_{m}^{\dagger}M_{m }\ket{\psi}}} \tag{26}\]  
Case 1: Applying \(M_{0}\)  
For \(\ket{\psi_{0}}\):  
\[\ket{\psi^{\prime}} =\frac{\ket{0}\bra{0}\Big{(}\frac{1}{\sqrt{2}}\ket{0}+\frac{1}{ \sqrt{2}}\ket{1}\Big{)}}{\sqrt{\Big{(}\frac{1}{\sqrt{2}}\bra{0}+\frac{1}{ \sqrt{2}}\ket{1}\Big{)}(\ket{0}\bra{0}\Big{)}\Big{(}\frac{1}{\sqrt{2}}\ket{0}+ \frac{1}{\sqrt{2}}\ket{1}\Big{)}}} \tag{27}\] \[=\ket{0} \tag{28}\]For \(|\psi_{1}\rangle\):  
\[|\psi^{\prime}\rangle =\frac{|0\rangle\langle 0|\Big{(}\frac{1}{\sqrt{2}}|0\rangle-\frac{1}{ \sqrt{2}}|1\rangle\Big{)}}{\sqrt{\Big{(}\frac{1}{\sqrt{2}}\langle 0|-\frac{1}{ \sqrt{2}}\langle 1|\Big{)}(|0\rangle\langle 0|)\Big{(}\frac{1}{\sqrt{2}}|0 \rangle-\frac{1}{\sqrt{2}}|1\rangle\Big{)}}} \tag{29}\] \[=|0\rangle \tag{30}\]  
In this case (Case 1), applying \(M_{0}\) to both superposition states results in the same output \(|0\rangle\), as shown in Equations (27)-(30). Therefore, this measurement does not reveal any information about the original transmitted superposition state, making it impossible to distinguish whether the received state is \(|\psi_{0}\rangle\) or \(|\psi_{1}\rangle\).  
Case 2: Applying \(M_{1}\)  
For \(|\psi_{0}\rangle\):  
\[|\psi^{\prime}\rangle =\frac{|1\rangle\langle 1|\Big{(}\frac{1}{\sqrt{2}}|0\rangle+ \frac{1}{\sqrt{2}}|1\rangle\Big{)}}{\sqrt{\Big{(}\frac{1}{\sqrt{2}}\langle 0|+ \frac{1}{\sqrt{2}}\langle 1|\Big{)}(|1\rangle\langle 1|)\Big{(}\frac{1}{ \sqrt{2}}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle\Big{)}}} \tag{31}\] \[=|1\rangle \tag{32}\]  
For \(|\psi_{1}\rangle\):  
\[|\psi^{\prime}\rangle =\frac{|1\rangle\langle 1|\Big{(}\frac{1}{\sqrt{2}}|0\rangle- \frac{1}{\sqrt{2}}|1\rangle\Big{)}}{\sqrt{\Big{(}\frac{1}{\sqrt{2}}\langle 0|- \frac{1}{\sqrt{2}}\langle 1|\Big{)}(|1\rangle\langle 1|)\Big{(}\frac{1}{ \sqrt{2}}|0\rangle-\frac{1}{\sqrt{2}}|1\rangle\Big{)}}} \tag{33}\] \[=-|1\rangle \tag{34}\]  
Unlike \(M_{0}\), applying \(M_{1}\) reveals a phase difference in the resulting quantum state as in Equations (31)-(34). If the result is \(|1\rangle\), the original state is \(|\psi_{0}\rangle\), which corresponds to the classical bit 0. If the result is \(-|1\rangle\), the original state is \(|\psi_{1}\rangle\), which corresponds to the classical bit 1. This phase-based discrimination allows the receiver to correctly identify the transmitted superposition state. By observing the sign of the qubit state (\(+\) or \(-\)) in the post-measurement state under \(M_{1}\), the receiver can reliably extract the classical bitstream from the measured quantum states.