# Multi-Access Channel Based on Quantum Detection in Wireless Optical Communication
## 2 Multi-Access Channel and Quantum Detection
### Multi-User Detection Based on Quantum Measurement  
The task of multi-user detection requires us to distinguish between different signals from different users. For this, there are many existing quantum multi-user detection methods available, such as optimal Bayesian criterion detection and least mean square detection. As far as the symmetric coherent states in the model are concerned, under the condition of equal prior probability, the least mean square detection is consistent with the optimal Bayesian criterion detection. In the physical sense, they are all MED measurements using orthogonal projection measurements. Therefore, after the measurement, the probability of an inter-user detection error is  
\[P_{e}^{\text{MED}}=\sum_{j=0}^{K-1}\sum_{ \begin{array}{c}k=0,k\
eq j\\ k\
eq j+K\end{array} }^{2K-1}\ \ \Big{[}P\Big{(}k\Big{|}H_{0}^{j}\Big{)}P\Big{(}H_{0}^{j} \Big{)}+P\Big{(}k\Big{|}H_{1}^{j}\Big{)}P\Big{(}H_{1}^{j}\Big{)}\Big{]}. \tag{4}\]  
Because the prior probability of the signal set sent by all users is assumed to be equal, so  
\[P\Big{(}H_{0}^{j}\Big{)}=P\Big{(}H_{1}^{j}\Big{)}=\frac{1}{2K}. \tag{5}\]  
The above equation is simplified to  
\[P_{e}^{\text{MED}}=\frac{1}{2K}\sum_{j=0}^{K-1}\sum_{ \begin{array}{c}k=0,k \
eq j\\ k\
eq j+K\end{array} }^{2K-1}\ \ \Big{[}P\Big{(}k\Big{|}H_{0}^{j}\Big{)}+P\Big{(}k \Big{|}H_{1}^{j}\Big{)}\Big{]}. \tag{6}\]  
The ultimate purpose of the above detection method is to minimize the average detection error probability. However, for each individual detection process, there is always a certain probability of error in detection, resulting in ambiguous results. Therefore, these detection methods are vague detections. On the other hand, let us say we take the USD measurement. Then, it is necessary to apply the generalized quantum measurement principle to detect all user signal sets, and to construct a common POVM operator uniformly. Its purpose is to obtain clear test results. Figure 3: Average user interference, \(r\), for multi-user coherent-state signals. Below, we give the construction of the POVM operator measured using USD. First, according to the coherent state's definition, the following properties are obtained. \[\big{|}\alpha_{j+1}\big{\rangle}=U\big{|}\alpha_{j}\big{\rangle}=U^{j}\big{|} \alpha_{0}\big{\rangle}, \tag{7a}\] \[|\alpha_{0}\rangle=U\big{|}\alpha_{2K-1}\big{\rangle},\] (7b) \[U^{2K}=I. \tag{7c}\]  
It can be verified that the form of the unitary operator in the above equation is  
\[U=P_{H}e^{\frac{\pi i\hat{n}}{K}}P_{H}. \tag{8}\]  
Here, \(\hat{n}=a^{+}a\) is the photon number operator, and its eigenstate is the photon number state. According to the forms of Equations (7) and (8), we further construct the eigenstates of the operator \(U\) and use them as a set of canonical orthonormal bases of the subspace H, namely  
\[U=\sum_{k=0}^{2K-1}e^{\frac{\pi i\hat{n}}{K}}|\gamma_{k}\rangle\langle\gamma_{ k}|. \tag{9}\]  
where \(\langle\gamma_{k}|\gamma_{k^{\prime}}\rangle=\delta_{kk^{\prime}}\); it is easy to know that \(|\gamma_{k}\rangle\) is orthonormal. Since there are linearly independent coherent states between \(\big{|}\alpha_{j}\big{\rangle}\), we have  
\[\big{|}\alpha_{j}\big{\rangle}=\sum_{k=0}^{2K-1}c_{k}e^{\frac{\pi i\hat{n}j}{ K}}|\gamma_{k}\rangle. \tag{10}\]  
The corresponding reciprocal state is given by  
\[\big{|}\alpha_{j}^{\perp}\big{\rangle}=Z^{-\frac{1}{2}}\sum_{k=0}^{2K-1}c_{k} ^{\ast-1}e^{\frac{\pi i\hat{n}j}{K}}|\gamma_{k}\rangle. \tag{11}\]  
In it,  
\[Z=\sum_{k=0}^{2K-1}|c_{k}|^{-2}. \tag{12}\]  
According to the symmetry of the signal state, the POVM operators \(\{E_{0},\cdots,E_{2K-1},E_{F}\}\) with the number of \(2K+1\) can be given by the following equations.