# Multi-Access Channel Based on Quantum Detection in Wireless Optical Communication
## 2 Multi-Access Channel and Quantum Detection
### Capacity of Thermal Noise Channel  
In addition, the influence of thermal field noise on two-user detection with USD is further considered. According to the results of the detection error probability of the multi-access channel deduced in Section 2.2, the detection error probability expression between the two users with the thermal noise can be derived. We can obtain  
\[ \begin{array}{lcl}P_{e}^{\text{USD}\_2}&=P\left(1\middle|H_{0}^{0}\right)+P \left(3\middle|H_{0}^{0}\right)\\ &=\frac{P_{D}Z}{16\pi N}\int exp\left(\frac{-|\mu-\alpha|^{2}}{N}\right) \left(\left|\left\langle\mu\middle|i\alpha^{\perp}\right\rangle\right|^{2}+ \left|\left\langle\mu\middle|-i\alpha^{\perp}\right\rangle\right|^{2}\right)d^ {2}\mu\\ &=\frac{P_{D}}{16\pi N}\int\sum_{j,k=0}^{3}\left(A_{(j+3)mod4}A_{(k+3)mod4}^{*}+ A_{(j+1)mod4}A_{(k+1)mod4}^{*}\right)\times\\ & exp\left(\frac{-|\mu-\alpha|^{2}}{N}-|\mu|^{2}-|\alpha|^{2}+e^{\frac{\pi\mu}{2} }\mu^{*}\alpha+e^{-\frac{\pi\mu}{2}}\alpha^{*}\mu\right)d^{2}\mu.\end{array}  \tag{28}\]Here, \(A_{j}\) is  
\[ \begin{array}{l}A_{0}=\frac{4}{\left\|\phi_{0}\right\|^{2}}+\frac{4}{\left\| \phi_{1}\right\|^{2}}+\frac{4}{\left\|\phi_{2}\right\|^{2}}+\frac{4}{\left\| \phi_{3}\right\|^{2}},\\ A_{1}=\frac{4}{\left\|\phi_{0}\right\|^{2}}+\frac{4}{\left\|\phi_{1}\right\|^{2}} -\frac{4}{\left\|\phi_{2}\right\|^{2}}-\frac{4}{\left\|\phi_{3}\right\|^{2}}, \\ A_{2}=\frac{4}{\left\|\phi_{0}\right\|^{2}}-\frac{4}{\left\|\phi_{1}\right\|^{2 }}+\frac{4}{\left\|\phi_{2}\right\|^{2}}-\frac{4}{\left\|\phi_{3}\right\|^{2}},\\ A_{3}=\frac{4}{\left\|\phi_{0}\right\|^{2}}-\frac{4}{\left\|\phi_{1}\right\|^{2 }}-\frac{4}{\left\|\phi_{2}\right\|^{2}}+\frac{4}{\left\|\phi_{3}\right\|^{2}}.\end{array} \]  
From Equation (28), we can see that the factors affecting the error probability are signal power and noise power. Although the equation cannot be explicitly expressed in terms of the SNR, we can use \(SNR=\left|\mu\right|^{2}/N\). This facilitates our discussion of BER-SNR performance in Section 4. Further, the channel capacity of the coherent-state two-user channel measured by USD under the background of thermal field noise is  
\[ \begin{array}{l}C_{N}^{\text{USD,2}}=I(X_{0}X_{1};Y)\\ =H(Y)-H(Y|X_{0}X_{1})\\ =-\sum_{j,=0}^{4}P\big{(}y_{ij}\big{)}log\big{(}P\big{(}y_{ij}\big{)}\big{)}+ \\ \sum_{k,l=0}^{1}\sum_{i,j=0}^{4}P\big{(}x_{k}^{0}x_{l}^{1}y_{ij}\big{)}log\big{(} P\big{(}y_{ij}|x_{k}^{0}x_{l}^{1}\big{)}\big{)}.\end{array}  \tag{29}\]  
Considering the expansion of the above equation to all detection results, we obtain  
\[ \begin{array}{l}=\frac{1}{4}\sum_{i,j=0}^{4}\sum_{\begin{subarray}{c}k\in \{0,1\}\\ l\in\{2\}\end{subarray}}P(i|H_{k})P(j|H_{l})\Bigg{(}2-log\Bigg{(}\sum_{ \begin{subarray}{c}k\in\{0,1\}\\ l\in\{2\}\end{subarray}}P(i|H_{k})P(j|H_{l})\Bigg{)}\Bigg{)}+\\ \sum_{i,j=0}^{4}P(i|H_{0})P(j|H_{1})logP(i|H_{0})P(j|H_{1}).\end{array}  \tag{30}\]  
Here, \(P(i|H_{k})\) is  
\[ \begin{array}{l}P(0|H_{0})=P(1|H_{1})=P(2|H_{2})=P(3|H_{3})=\frac{P_{0}Z}{16 \pi N}\int exp\bigg{(}\frac{-\left|\mu-\alpha\right|^{2}}{N}\bigg{)}\big{|} \big{\langle}\mu\big{|}\alpha^{\perp}\big{\rangle}\big{|}^{2}d^{2}\mu\\ P(1|H_{0})=P(2|H_{1})=P(3|H_{2})=P(0|H_{3})=\frac{P_{0}Z}{16\pi N}\int exp \bigg{(}\frac{-\left|\mu-\alpha\right|^{2}}{N}\bigg{)}\big{|}\big{\langle}\mu \big{|}\alpha^{\perp}\big{\rangle}\big{|}^{2}d^{2}\mu,\\ P(2|H_{0})=P(3|H_{1})=P(0|H_{2})=P(1|H_{3})=\frac{P_{0}Z}{16\pi N}\int exp \bigg{(}\frac{-\left|\mu-\alpha\right|^{2}}{N}\bigg{)}\big{|}\big{\langle}\mu \big{|}-\alpha^{\perp}\big{\rangle}\big{|}^{2}d^{2}\mu,\\ P(3|H_{0})=P(0|H_{1})=P(1|H_{2})=P(2|H_{3})=\frac{P_{0}Z}{16\pi N}\int exp \bigg{(}\frac{-\left|\mu-\alpha\right|^{2}}{N}\bigg{)}\big{|}\big{\langle}\mu \big{|}-i\alpha^{\perp}\big{\rangle}\big{|}^{2}d^{2}\mu.\end{array} \]  
where  
\[