# Multi-Access Channel Based on Quantum Detection in Wireless Optical Communication
## 2 Multi-Access Channel and Quantum Detection
### Two-User Channel Capacities Based on MED and USD  
Next, we take the two-user channel as the research object. The channel model of two users based on symmetrical coherent-state signals is shown in Figure 4.  
Two users have a set of coherent-state signals modulated by BPSK, respectively,  
\[\text{User 1}: \begin{cases}H_{0}^{0}&|\alpha_{0}\rangle=|\alpha\rangle\\ H_{1}^{0}&|\alpha_{2}\rangle=|-\alpha\rangle\\ \text{User 2}: \begin{cases}H_{0}^{0}&|\alpha_{1}\rangle=|i\alpha\rangle\\ H_{1}^{1}&|\alpha_{3}\rangle=|-i\alpha\rangle\end{cases} \]  
According to the analysis in the previous section, from the conclusions of [34; 35] about MED measurement, the measurement probability of the two-user MED measurement falling on each result can be calculated as follows:  
\[P\big{(}k\big{|}H_{0}^{0}\big{)}=\tfrac{1}{4}\times\Bigg{(}\sum\limits_{m,n=0} ^{3}\frac{\text{\emph{exp}}\Big{(}(e^{\frac{m\pi}{2}-1}\Big{)}|\alpha|^{2}+ \frac{m\pi(|m-i\rangle\text{mod}4|_{1}}{2}\Big{)}\Big{)}}{2\sqrt{1+e^{-|\alpha |^{2}+(|\alpha|^{2}+\frac{m\pi}{2})i}+e^{-|\alpha|^{2}+\pi i|^{2}+\pi i|^{2}-(| \alpha|^{2}-\frac{3m\pi}{2})i}}\Big{)}}^{2}, \tag{19}\]  
The above equation represents the probability that user 1 transmits the signal state \(|\alpha\rangle\), and the measurement result of the receiver is \(k\). According to the prior probability equality and symmetry of the transmitted signal, the more general expression of measurement probability is deduced as  
\[P\Big{(}k\Big{|}H_{l}^{j}\Big{)}=P\Big{(}(k-j-2l)mod4\Big{|}H_{0}^{0}\Big{)}. \tag{20}\]  
Figure 4: Two-user channel based on coherent-state signals. Among it, \(k\in\{0,1,2,3\},j,l\in\{0,1\}\). Therefore, from Equations (6) and (20), it can be known that the inter-user detection error probability measured by the two-user MED is  
\[ \begin{array}{c}P_{e}^{\text{MED},2}=\frac{1}{4}\sum\limits_{j=0}^{1}\sum \limits_{\begin{array}{c}k=0,k\
eq j\\ k\
eq j+2\end{array}}^{3}\Big{[}P\Big{(}k\Big{|}H_{0}^{j}\Big{)}+P\Big{(}k \Big{|}H_{1}^{j}\Big{)}\Big{]}\\ \qquad\qquad\qquad\qquad\qquad k\
eq j+2\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad=P\big{(}1\big{|}H_{0}^{0}\big{)}+P \big{(}3\big{|}H_{0}^{0}\big{)}.\end{array}  \tag{21}\]  
On the other hand, consider the result of Equation (18). Under equal conditions, perfect error-free detection between users can be achieved with ideal USD measurements. This shows that in the problem of two-user communication, USD measurement will have a better user discrimination performance than MED measurement. As far as the channel capacity of the two-user communication is concerned, the MED measurement will theoretically obtain the maximum capacity of the coherent-state two-user channel. Here, finding the maximum channel capacity is equivalent to finding the Von Neumann Entropy of the coherent-state signal. The mixed state of the transmitted signal is  
\[\rho=\frac{1}{4}\sum\limits_{j=0}^{3}\lvert\alpha_{j}\rangle\langle\alpha_{j}\rvert. \tag{22}\]  
From Equation (22), using the MED measurement, the maximum capacity of the coherent-state two-user channel is found to be  
\[ \begin{array}{c}C^{\text{MED},2}=S(\rho)\\ =S\bigg{(}\frac{1}{4}\sum\limits_{j=0}^{3}\lvert\alpha_{j}\rangle\langle\alpha _{j}\rvert\bigg{)}\\ =\frac{1}{4}\sum\limits_{j=0}^{3}f_{j}(\alpha)\big{(}2-log\big{(}f_{j}(\alpha) \big{)}\big{)}.\end{array}  \tag{23}\]  
In it,  
\[f_{j}(\alpha)=\Big{(}1+e^{-\lvert\alpha\rvert^{2}+(\lvert\alpha\rvert^{2}+ \frac{\alpha\beta}{2})j}+e^{-2\lvert\alpha\rvert^{2}+\pi\beta}+e^{-\lvert \alpha\rvert^{2}-(\lvert\alpha\rvert^{2}-\frac{3\alpha j}{2})j}\Big{)}. \tag{24}\]  
In addition, if the channel capacity of the coherent-state two-user channel for the USD case is theoretically investigated, it can be inferred that the maximum channel capacity that can be achieved by the USD measurement will be slightly lower than that of the MED measurement. To this end, the following will give the maximum channel capacity expression of the two-user USD measurement based on the result of the BPSK signal detection.