# Multi-Access Channel Based on Quantum Detection in Wireless Optical Communication
## 2 Multi-Access Channel and Quantum Detection
### Multi-Access Channel Based on Symmetrical Coherent States  
Theoretically, the complex envelope \(\alpha\) of the coherent-state signal can take on any value from the complex plane. Therefore, the following situation is considered here: assuming that \(\alpha\) is taken as \(\{0,1,\ldots,2K-1\}\), there are \(2K\) symmetric complex numbers with different arguments and the same modulus. The specific description is shown in Figure 1.  
In Figure 1, \(\alpha_{j}=|\alpha|e^{\frac{\pi i}{N}}\); all plural numbers \(\{\alpha_{j}\}\) have a modulus equal to \(|\alpha|\). Here,  
\[|\alpha_{j}\rangle=e^{-\frac{1}{2}|\alpha|^{2}}\sum_{n=0}^{\infty}\frac{ \alpha_{j}n}{\sqrt{n!}}|n\rangle,(j\in\{0,1,\cdots 2K-1\}). \tag{1}\]  
It can be seen from Figure 1 that \(\alpha\), which is symmetrical with respect to the origin of the complex plane, has an opposite \(-\alpha\). It appears \(K\) pairs of \(\alpha\) and \(-\alpha\) in total. According to the above analysis, we propose a multi-access channel model based on symmetric coherent-state signals. In order to facilitate the unification of the formula indicators, we assume that the number of users ranges from \(0\) to \(K-1\), with a total of \(K\) users. If \(2K\) coherent states are assigned to \(K\) users, the jth user is required to enjoy a \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation or Symbol** & **Definition** \\ \hline BPSK & Binary phase shift keying \\ M-PSK & M-ary phase shift keying \\ QAM & Quadrature amplitude modulation \\ POVM & Operators of positive-operator-valued measure \\ K & Number of users in the multi-access channel \\ \(|\alpha_{j}\rangle\) & Coherent state \\ \(E_{j}\) & POVM operators \\ \(|\mu|^{2}\) & Signal average photon number, characterize the power of the signal. \\ \(N\) & Thermal noise average photon number, characterize the power of the noise. \\ Density operator for signal superimposed thermal noise, usually \\ \(\rho\) & \(\rho=(\pi N)^{-1}\int exp\left(\frac{-|\
u-\alpha|^{2}}{N}\right)|\mu\rangle \langle\mu|d^{2}\mu\). \\ BER & Bit error rate \\ \(SNR\) & Signal-to-noise rate; in this paper, it is denoted as \(|\mu|^{2}/N\). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Definition of some useful notations and symbols in the text. Figure 1: \(2K\) complex numbers symmetric to the origin of the complex plane. pair of coherent states \(\left\{\left|\alpha_{j}\right\rangle,\left|\alpha_{j+K}\right\rangle\right\}\), as shown in Figure 1, where \(j\in\left\{0,1,\cdots K-1\right\}\) and satisfy \(\left|\alpha_{j+K}\right\rangle=\left|-\alpha_{j}\right\rangle\). Each user adopts the BPSK method to modulate the transmitted information to the coherent state \(\left\{\left|\alpha_{j}\right\rangle,\left|\alpha_{j+K}\right\rangle\right\}\), as shown in Figure 2.  
The signal of the user shown in Figure 2 has the following prior assumptions. Where  
\[\text{User}:\;j \begin{cases}H_{0}^{j}&\left|\alpha_{j}\right\rangle\\ H_{1}^{j}&\left|\alpha_{j+K}\right\rangle,\;(j=0,1,\cdots,K-1).\end{cases}  \tag{2}\]  
For an ideal quantum multi-access channel, when the signal states sent by different users are all orthogonal, there is no inter-user interference. For coherent-state multi-access channels, since the coherent states are always non-orthogonal, inter-user interference will always exist. Therefore, the average inter-user interference can be reflected by the degree of overlap of the coherent-state signals of different users, that is  
\[r=\frac{1}{4K(K-1)}\sum_{ \begin{subarray}{c}j,j^{\prime}=0,j^{\prime}\
eq j \\ j^{\prime}\
eq(j+K)mod2K\end{subarray} }^{2K-1}\left|\left\langle\alpha_{j^{ \prime}}\right|\alpha_{j}\right\rangle\right|^{2}=\frac{1}{2(K-1)}\sum_{ \begin{subarray}{c}j=1\\ j\
eq K\end{subarray} }^{2K-1}\left|\begin{subarray}{c}exp\left(-\left|\alpha \left(1-e^{\frac{\pi i}{K}}\right)\right|^{2}\right).\end{cases} \tag{3}\]  
The value of \(r\) is between 0 and 1. The closer to 0, the smaller the interference, and the closer to 1, the stronger the interference.