# 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
First, according to Equations (16) and (17), the probability that the USD measurement of two users falls on each result is  
\[P\Big{(}k\Big{|}H_{l}^{j}\Big{)}=0,(k\
eq j+2l), \tag{25a}\] \[P\Big{(}j+2l\Big{|}H_{l}^{j}\Big{)}=tr\Big{(}E_{j+2l}\big{|}\alpha_{j+2l} \Big{)}\Big{\langle}\alpha_{j+2l}\Big{|}\Big{)}=P_{D},\] (25b) \[P\Big{(}4\Big{|}H_{l}^{j}\Big{)}=1-P\Big{(}j+2l\Big{|}H_{l}^{j}\Big{)}. \tag{25c}\]  
where \(k\in\{0,1,2,3\},j,l\in\{0,1\}\), and \(P_{D}\) is defined as  
\[P_{D}=\frac{1}{4}\times min\Big{(}\parallel\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\  
\[ \begin{array}{l}\left\|\left|\phi_{0}\right\rangle\right.\right\|=\sqrt{\left( \langle\alpha|+\langle i\alpha|+\langle-\alpha|+\langle-i\alpha|\rangle(|\alpha \rangle+|i\alpha\rangle+|-\alpha\rangle+|-i\alpha\rangle\right)}=\\ \hskip 14.226378 pt2\sqrt{1+e^{-|\alpha|^{2}+i|\alpha|^{2}}+e^{-2|\alpha|^{2} }+e^{-|\alpha|^{2}-i|\alpha|^{2}}},\\ \left\|\left|\phi_{1}\right\rangle\right.\right\|=\sqrt{\left(\langle\alpha|-i \langle i\alpha|-\langle-\alpha|+i\langle-i\alpha|\rangle(|\alpha\rangle+|i \alpha\rangle-|-\alpha\rangle-i|-i\alpha\rangle\right)}=\\ \hskip 14.226378 pt2\sqrt{1+ie^{-|\alpha|^{2}+i|\alpha|^{2}}-e^{-2|\alpha|^{2} }-ie^{-|\alpha|^{2}-i|\alpha|^{2}}},\\ \left\|\left|\phi_{2}\right\rangle\right.\right\|=\sqrt{\left(\langle\alpha| -\langle i\alpha|+\langle-i\alpha|-\langle-i\alpha|\rangle(|\alpha\rangle-|i \alpha\rangle+|-\alpha\rangle)-|-i\alpha\rangle\right)}=\\ \hskip 14.226378 pt2\sqrt{1-e^{-|\alpha|^{2}+i|\alpha|^{2}}+e^{-2|\alpha|^{2} }-e^{-|\alpha|^{2}-i|\alpha|^{2}}},\\ \left\|\left|\phi_{3}\right\rangle\right.\left\|=\sqrt{\left(\langle\alpha|+i \langle i\alpha|-\langle-\alpha|-i\langle-i\alpha|\rangle(|\alpha\rangle-|i \alpha\rangle-|-\alpha\rangle+i|-i\alpha\rangle\right)}=\\ \hskip 14.226378 pt2\sqrt{1-ie^{-|\alpha|^{2}+i|\alpha|^{2}}-e^{-2|\alpha|^{2} }+ie^{-|\alpha|^{2}-i|\alpha|^{2}}}.\end{array} \]  
Secondly, it is assumed that \(X_{0}\) represents the input signal source of user \(1\), and \(Y\) represents the output signal source of the receiver. \(X_{0}\) takes the value of \(\left\{x_{l}^{0}|k\in\left\{0,1\right\}\right\}\), and \(X_{1}\) takes the value of \(\left\{x_{l}^{1}|l\in\left\{0,1\right\}\right\}\). \(Y\) takes the value of \(\left\{y_{ij}|i,j\in\left\{0,1,2,3,4\right\}\right\}\) according to different measurement results. The transition probability of a coherent-state two-user channel measured by USD is  
\[P\left(y_{01}\middle|x_{0}^{0}x_{0}^{1}\right)=P\left(y_{03}\middle|x_{0}^{0}x _{1}^{1}\right)=P\left(y_{21}\middle|x_{1}^{0}x_{0}^{1}\right)=P\left(y_{23} \middle|x_{1}^{0}x_{1}^{1}\right)=P_{D}^{2}, \tag{26a}\] \[P\left(y_{41}\middle|x_{k}^{0}x_{0}^{1}\right)=P\left(y_{43}\middle|x_{k}^{0}x _{1}^{1}\right)=P\left(y_{04}\middle|x_{0}^{0}x_{I}^{1}\right)=P\left(y_{24} \middle|x_{1}^{0}x_{I}^{1}\right)=P_{D}(1-P_{D}),\] (26b) \[P\left(y_{44}\middle|x_{k}^{0}x_{I}^{1}\right)=(1-P_{D})^{2}. \tag{26c}\]  
For the remaining cases, the transition probability is always \(0\). Therefore, according to Equation (26), it can be deduced that the maximum channel capacity of the coherent-state two-user channel measured by USD is  
\[ \begin{array}{l}\text{C}^{\text{USD}\_2}=I(X_{0}X_{1};Y)\\ =H(Y)-H(Y|X_{0}X_{1})\\ =-\left(P_{D}^{2}log\frac{P_{D}^{2}}{4}+2P_{D}(1-P_{D})log\frac{P_{D}(1-P_{D}) }{2}+(1-P_{D})^{2}log(1-P_{D})^{2}\right)\\ \hskip 14.226378 pt+P_{D}^{2}logP_{D}^{2}+2P_{D}(1-P_{D})logP_{D}(1-P_{D})\\ \hskip 14.226378 pt+(1-P_{D})^{2}log(1-P_{D})^{2}\\ =2P_{D}\\ =2\times min\left(1+ie^{-|\alpha|^{2}+i|\alpha|^{2}}-e^{-2|\alpha|^{2}}-ie^{-| \alpha|^{2}-i|\alpha|^{2}},1-e^{-|\alpha|^{2}+i|\alpha|^{2}}+e^{-2|\alpha|^{2}}\right. \\ \hskip 14.226378 pt\left.-e^{-|\alpha|^{2}-i|\alpha|^{2}},1-ie^{-|\alpha|^{2} +i|\alpha|^{2}}-e^{-2|\alpha|^{2}}+ie^{-|\alpha|^{2}-i|\alpha|^{2}}\right). \end{array}  \tag{27}\]