# MQC-MB: Multiphoton Quantum Communication Using Multiple-Beam Concept in Free Space Optical Channel
## 3 Formulation of FSO Multiphoton Quantum Cryptography  
[PERSON] as a sender has ownership of a list of symbols or raw keys. She needs to share it with [PERSON] over a quantum channel. To extract a short secret key from the list of raw keys, one-way post-processing is used. There are two steps in one-way post-processing [24]. The first step is error correction (EC) or information reconciliation and the second step is privacy amplification (PA). After EC is implemented, the raw key becomes shorter and the symbols or keys are completely correlated, while [PERSON]'s understanding of the raw keys would be diminished in PA. Therefore, the final secret key length relies on [PERSON]'s information about the raw keys. Secret key rates (SKRs) and quantum bit error rates (QBERs), to be referred to in [11; 13; 20], are used to evaluate QKD's output over FSO's. SKR can be defined as the probability of obtaining a bit of secret key per transmitted quantum signal pulse [4], while QBER is determined by the probability ratio of error bits and the overall probability of [PERSON] detection. SKR, \(K\) consists of two parts, which are \(R\) and \(r\)[25; 26], and can be calculated by:  
\[K=Rr \tag{1}\]  
where \(R\) is the raw key rate, which is the rate that depends on the implementation setup and protocol used, while \(r\) is a secret fraction that is known from the parameter estimation. The first term, raw key \(R\), becomes shorter and the key is correlated after the error correction procedure. The second term \(r\) is intended to amplify privacy and is intended to remove [PERSON]'s knowledge of the raw keys. \(R\) can be evaluated by:  
\[R=vs\ P_{Bob}(N_{max})=vs\sum_{m=1}^{Nmax}p_{A}(n)[1-(1-\eta FSO.\eta.\eta det)^ {n}] \tag{2}\]  
where \(vs\) is the repetition rate and \(P_{Bob}(N_{max})\) is [PERSON]'s detection probability. The standard values of detector efficiency \(\eta det\) is 10% at telecom and \(\eta\) is the fluctuating transmission of the turbulence atmosphere [27]. The distribution number of photon \(n\) until the maximum number of photons \(Nmax\) was determined according to the Poissonian statistic of \(\mu=<n>\), and \(p_{A}(n)\) is the distribution of the photon number, given by:  
\[p_{A}(n)=\frac{\mu^{n}}{n!}e^{-\mu} \tag{3}\]  
Referring to Equation (2), the \(n_{FSO}\) is the attenuation due to channel loss [14; 28], and is calculated with geometric and atmospheric loss, shown by the following equation:  
\[n_{FSO}=geometrical\ loss\ \times atmospheric\ loss \tag{4}\]  
The equation of geometrical loss as suggested in [29] has been used in standard FSO using a single beam specifically in quantum communication [30; 31], respectively, and can be calculated using:  
\[geometricalLoss_{single-beam}=10 log_{10}\left[\frac{\mu\left(\frac{dr}{2} \right)^{2}}{\mu\left(\frac{D}{2}L\right)^{2}}\right] \tag{5}\]  
Therefore, the total attenuation of channel loss in dB using the single-beam model can be derived as:  
\[n_{FSOsingle-beam}=10 log_{10}\left(\left[\frac{\mu\left(\frac{dr}{2}\right)^ {2}}{\mu\left(\frac{D}{2}L\right)^{2}}\right]\times exp(-\beta L)\right) \tag{6}\]  
The problem of the single beam is high geometrical losses, which are also referred to as optical beam attenuation. Some of the light beams would not hit the lenses of the receiver. As a consequence, the diverging light beam causes a loss of power signal. The solution to this issue is to use multiple apertures of the receiver (multiple beam), equivalent to a greater diameter of one aperture of the receiver. By increasing the area of apertures or using multiple apertures, the telescope will be able to collect more light or photons as shown in Figure 2. Thus, the geometrical loss can be reduced.