2. Theory and mathematical modeling

The complete mathematical model, as depicted in Figure 1, for generating quantum signals and the post processing algorithms are discussed in this section.

#### 2.1. CV-QKD signals

At the transmitter, Alice produces pseudo-random m-PSK symbols that can be controlled via pseudorandom binary sequence (PRBS), that is, I(t),Q(t) ∈f�1; þ1g. These randomly generted symbols are up-converted to radio-frequency (RF) signal levels with respective in phase and quadrature [25], that are denoted by SI (t) and SQ(t). These two parts can be mathematically expressed as in Eqs. (1) and (2).

$$S\_l(t) = I(t)\cos\left(\omega\_1 t\right) - Q(t)\sin\left(\omega\_1 t\right) \tag{1}$$

$$S\_{\mathcal{Q}}(t) = I(t)\sin\left(\omega\_1 t\right) - Q(t)\cos\left(\omega\_1 t\right) \tag{2}$$

Figure 1. Schematic of the m-PSK-based quantum transmitter (Alice) and quantum receiver (Bob) for QTTH applications.

where ω1 is the RF angular frequency of the signal. The output is used as the input of I/Q modulator, Mach-Zehnder modulator (MZM). The equivalent optical field can be expressed as in Eq. (3) and further be simplified as in Eq. (4).

m-PSK (where m = 2, 4, 8, 16 ….) to produce secure quantum keys and (2) limitations of using fast optical receivers in-terms of electronic and shot noise for commercially available coherent receiver to detect the CV-QKD signals. Furthermore, the transceivers, noise equivalent power (NEP) from ADCs and transimpedance amplifier (TIA) are emulated according to the physical parameters of the available off-the-shelf modules. Both single channel (suitable for high-speed point-to-point links) and especially wavelength division multiplexed (WDM) transmissions (suitable for multicasting) are investigated. We have also implemented: (1) local local oscillator (LLO) method to avoid possible eavesdropping or hacking on the reference laser signal and (2) a phase noise cancelation (PNC) algorithm for digital post-processing of the received signals. Moreover, we have depicted the trade-off between the secure key rates achieved and the splitratio of the access network considering the hybrid classical-quantum traffic. The proposed setup is further studied by incorporating different fiber types, for example, pure silica core fiber (PSCF) and low loss switch based on microelectromechanical systems (MEMS) for multiuser configurations. These detailed discussions will help the people from academics and industry to implement the QTTH concept in real-time networks. Furthermore, the designed system is

The complete mathematical model, as depicted in Figure 1, for generating quantum signals

At the transmitter, Alice produces pseudo-random m-PSK symbols that can be controlled via pseudorandom binary sequence (PRBS), that is, I(t),Q(t) ∈f�1; þ1g. These randomly generted symbols are up-converted to radio-frequency (RF) signal levels with respective in phase and quadrature [25], that are denoted by SI (t) and SQ(t). These two parts can be mathematically

Figure 1. Schematic of the m-PSK-based quantum transmitter (Alice) and quantum receiver (Bob) for QTTH applications.

SIðÞ¼ t I tð Þ cos ð Þ� ω1t Q tð Þ sin ð Þ ω1t (1)

SQðÞ¼ t I tð Þ sin ð Þ� ω1t Q tð Þ cos ð Þ ω1t (2)

energy efficient and cost-effective.

24 Telecommunication Networks - Trends and Developments

2.1. CV-QKD signals

expressed as in Eqs. (1) and (2).

2. Theory and mathematical modeling

and the post processing algorithms are discussed in this section.

$$E(t) = \left\{ \cos\left[AS\_1(t) + \frac{\pi}{2}\right] + j\cos\left[AS\_Q(t) + \frac{\pi}{2}\right] \right\} \sqrt{P\_s} \epsilon^{j\left[\left(\omega t + \varphi\_i t\right)\right]}\tag{3}$$

$$E(t) \approx \sqrt{2P\_s} \dot{e}^{\left[\left(\omega t + \frac{\dot{m}}{4}\right)\right]} - A[I(t) + Q(t)]\sqrt{2P\_s} \dot{e}^{\left[\left((\omega + \omega\_1)t + \varphi\_1 t d\right)\right]} \tag{4}$$

where A refers to the modulation index; ω, PS, and φ1t represent the angular frequency of the carrier, power and phase noise of the received signal. For investigating, the modulation variance VA of the optical received signal, evaluated as the shot-noise-units (SNUs), the parameter A and variable optical attenuator have been optimized at the input of the public communication. To further simplify the numerical model of the QTTH network, the quantum channel loss is expressed as the attenuation of the optical communication channel. Mathematically, noise variance produced by the communication channel is given as in Eq. (5).

$$
\chi\_{line} = \frac{1}{T} + \epsilon - 1 \tag{5}
$$

where T is the relationship between transmission distance and E is the excess noise. Realistically, excess noise measurements, expressed as SNUs [18, 32], may come from the laser phase noise, laser line width, imperfect modulation and coherent receiver imbalance [33]. In this chapter, we have implemented a local local oscillator (LLO) concept, which is considered as the vital configuration to keep the laser at the receiver side, that is, Bob, in order to stay away from any hacking attempt on the quantum channel to get the reference phase information of the incoming signal. The electric field of the LLO can be expressed as in Eq. (6).

$$E\_{LLO}t = \sqrt{P\_{LLO}} \mathcal{e}^{\left[\omega\_{LLO}t + \varphi\_2 t\right]} \tag{6}$$

where PLLO, ωLLO and φ2t represents the power, angular frequency and phase noise of the LLO, respectively. The structure of the Bob, that is, coherent receiver, consists of a 90� optical hybrid and two balanced photodetectors. The coherent receiver has an efficiency of η and electrical noise of Vel. Practically, Vel comprises electrical noise from transimpedance amplifiers (TIA) as well as the major contribution from the ADC. For this reason, the receiver added noise variance can be expressed as in Eq. (7).

$$\chi\_{det} = \frac{2 + 2V\_{vl} - \eta}{\eta} \tag{7}$$

Furthermore, the aggregate noise variance of the quantum network, including Alice and Bob, can be expressed as in Eq. (8).

$$
\chi\_{system} = \frac{\chi\_{line} + \chi\_{det}}{T} \tag{8}
$$

#### 2.2. Phase noise cancelation algorithm

Conventionally, in order to receive and process the weak quantum signals, a high-level local oscillator is required at the receiver. It is very vital to select the local oscillator with narrow line width so that the laser fluctuations cannot contribute to the overall system's excess noise that may damage the recovery of quantum signals. Furthermore, it will help the coherent receiver to have a low complex digital signal processing (DSP) module, that is, phase noise cancelation (PNC) algorithm, as explained in detail in Figure 2 [25]. For the efficient performance of the PNC module, it is essential that the photocurrents of the in phase and quadrature signals have to be measured with high precision.

rI <sup>¼</sup> LPF <sup>ι</sup>sð Þ<sup>t</sup> cos <sup>ω</sup>1<sup>t</sup> � <sup>π</sup>

rQ <sup>¼</sup> LPF <sup>ι</sup>sð Þ<sup>t</sup> sin <sup>ω</sup>1<sup>t</sup> � <sup>π</sup>

line width, local oscillator fluctuations and coherent detector imbalance [33].

(VOA) prior to the quantum channel, that is, based on the optical fiber.

fibers can be used to benchmark the hybrid quantum-classical optical networks.

3. Design of a hybrid quantum-classical network

any frequency and phase distortions.

3.1. Transmitter design

3.2. Quantum channel

3.3. Classical coherent receiver

4 h i � � ¼ � ffiffiffi

4 h i � � ¼ � ffiffiffi

where n'<sup>I</sup> and n'<sup>Q</sup> are the equivalent additive noise that is added during the quantum channel transmission and detection processes before the digital post-processing. By considering Eqs. (12) and (13), it is determined that the original m-PSK signals can be recovered without

In this section, we discuss the design challenges to optimize a hybrid quantum-classical network. More specifically, we discuss all the excess noise contributions, expressed as shotnoise-units (SNUs) [18, 32] may come from the imperfect modulation, laser phase noise, laser

The design of the simplified QTTH network with m-PSK (where m = 4, 8, 16, etc.) modulationbased quantum transmitter (Alice) and LLO-based coherent receiver (Bob) is as shown in Figure 1. At Alice, a narrow line width laser is modeled at the wavelength of 1550 nm having a line width of approximately <5 kHz allowing it to maintain low phase noise regime. A pseudorandom binary sequence (PRBS) of length 231–1 is programmed for single channel transmission and delay decorrelated duplicates copies are generated for the multichannel transmission. Furthermore, we execute pulse shaping at the transmitter according to the Nyquist criterion to generate intersymbol interference (ISI) free signals. Subsequently, 1 GBaud 4-PSK (four state phase-shift keying) signal is generated after the radio frequency (RF) signals are modulated with the help of an electro-optical I/Q modulator, where RF frequency is kept at 2 GHz. The modulation variance is applied with the help of a variable optical attenuator

For most of our numerical and experimental investigations, we have used two standard types of fibers, namely, standard single mode fiber (SSMF) and pure silica core fiber (PSCF). The physical parameters of the fibers are given in Table 2. These are the commercial fibers and deployed heavily around the globe for short and long range transmissions. Therefore, these

A standard commercially available coherent receiver has been modeled. The receiver module (Bob) consists of a 90� optical hybrid and 20 GHz balanced photodiodes. The gain of TIA,

2 <sup>p</sup> AI <sup>þ</sup> <sup>n</sup><sup>0</sup>

2 <sup>p</sup> AQ <sup>þ</sup> <sup>n</sup><sup>0</sup>

<sup>I</sup> (12)

27

Recent Progress in the Quantum-to-the-Home Networks http://dx.doi.org/10.5772/intechopen.80396

IQ (13)

Mathematically, they can be expressed as in Eqs. (9) and (10).

$$\begin{aligned} \mu\_l(t) &\approx \sqrt{2}\cos\left[\omega - \omega\_{\rm LO}\right]t + \varphi\_1 t - \varphi\_2 t + \frac{\pi}{4} - AI(t)\cos\left[(\omega + \omega\_1 - \omega\_{\rm LO})t + \varphi\_1 t - \varphi\_2 t\right] \\ &+ AI(\rm Q)\cos\left[(\omega + \omega\_1 - \omega\_{\rm LO})t + \varphi\_1 t - \varphi\_2 t\right] + \eta\_l \end{aligned} \tag{9}$$

$$\begin{aligned} \iota\_Q(t) & \propto \sqrt{2} \sin \left[ \omega - \omega\_{LO} \right] t + \varrho\_1 t - \varrho\_2 t + \frac{\pi}{4} - AI(t) \cos \left[ (\omega + \omega\_1 - \omega\_{LO})t + \varrho\_1 t - \varrho\_2 t \right] \\ & + AI(Q) \sin \left[ (\omega + \omega\_1 - \omega\_{LO})t + \varrho\_1 t - \varrho\_2 t \right] + \eta\_Q \end{aligned} \tag{10}$$

where nI and nQ describe the in phase and quadrature components of the additive phase noise that needs to be remunerated. We have implemented the phase noise cancelation (PNC) algorithm [25]. By combining the squares of the in phase and quadrature component of photocurrents, as in Eqs. (9) and (10), that is, i2 I(t) + i2 Q(t), and mitigating the DC component, the final result can be depicted as in Eq. (11).

$$\begin{split} \alpha\_{\rm S}(t) \propto 2\sqrt{2}Al(t)\cos\left(\omega\_{1}t - \frac{\pi}{4}\right) &+ 2\sqrt{2}AQ(t)\cos\left(\omega\_{1}t - \frac{\pi}{4}\right) \\ + 2\sqrt{2}\left\{n\_{\rm I}\cos\left[(\omega-\omega\_{\rm LO})t + \varphi\_{1}t - \varphi\_{2}t + \frac{\pi}{4}\right] + n\_{\rm Q}\sin\left[(\omega-\omega\_{\rm LO})t + \varphi\_{1}t - \varphi\_{2}t + \frac{\pi}{4}\right] \right\} \end{split} \tag{11}$$

The final step in the DSP module is to down-convert the RF signal. The resultant in phase and quadrature components can be expressed as in Eqs. (12) and (13).

Figure 2. Schematic of the digital signal processing (phase noise cancelation) module for quantum receiver.

$$r\_I = LPF\left[\iota\_s(t)\cos\left(\omega\_1 t - \frac{\pi}{4}\right)\right] = -\sqrt{2}AI + n\_I' \tag{12}$$

$$r\_Q = LPF\left[\iota\_s(t)\sin\left(\omega\_1 t - \frac{\pi}{4}\right)\right] = -\sqrt{2}AQ + \eta\_{\parallel Q}'\tag{13}$$

where n'<sup>I</sup> and n'<sup>Q</sup> are the equivalent additive noise that is added during the quantum channel transmission and detection processes before the digital post-processing. By considering Eqs. (12) and (13), it is determined that the original m-PSK signals can be recovered without any frequency and phase distortions.
