## Abstract

In this paper we investigate the chromatic dispersion impact on the quantum key distribution system based on multi-mode weak coherent phase-coded states. We provide an asymptotic secure key rate estimation, taking into account error detection probability due to chromatic dispersion. We demonstrate numerically and experimentally that the effect of chromatic dispersion in an optical fiber without any compensation hinders the secret key distribution at a distance more than 53 km. Finally, we propose a modification to the considered quantum communication system in order to mitigate the influence of chromatic dispersion on its performance.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Fiber-based quantum key distribution (QKD) systems have been under an intense study over the past two decades, with multiple in-lab and field implementations reported [1–11]. One of the main challenges for experimental QKD systems is to maximize secure key rate and propagation distances over the optical fiber. Record transmission distances over 300 km [2] and over 400 km [3] have been achieved recently. Secure key rates over 1 Mbps were reported in [5,7]. Such impressive results can be obtained as a result of analyzing different parameters of the real QKD system and minimizing the influence of various distortions on the quantum channel. These signal impairments can be caused by such factors as spontaneous Raman scattering, channel crosstalk, fiber losses, and chromatic dispersion [12]. The latter is a particularly acute problem for the protocol described in this paper, since it leads to intersymbol interference and a dramatic increase in quantum bit error rate (QBER).

Digital dispersion compensation, which is commonly applied in classical communication [11,12], cannot be used in QKD systems due to the low value of mean photon number per pulse, therefore using fully analog approaches is required. Known analog methods for chromatic dispersion compensation in fiber-based telecommunication systems are described in [12,13]. These include:

- • Using dispersion compensating fiber (DCF) as a complement to a regular fiber. This method leads to a substantial attenuation and cumbersomeness of the overall system. Nevertheless, this approach is involved, for example, in [3] with reported DCF attenuation of 0.5 dB/km;
- • Compensation by applying regular or chirped fiber Bragg gratings (FBG), that results in much lower attenuation compared to the previous method. For instance, FBGs for chromatic dispersion compensation are used in [8];
- • Dispersion shifted fiber (DSF) with zero dispersion at the signal wavelength is used to mitigate the effect of chromatic dispersion [5];
- • Reduction of the chromatic dispersion effect by calculating the optimal initial pulse duration at which its broadening caused by chromatic dispersion is minimal [12].

In this paper we consider the subcarrier wave (SCW) QKD system presented in [14–16] which, due to its peculiarity, was found to be especially susceptible to chromatic dispersion. Generally, in SCW QKD signal photons are not emitted directly by the laser source but are generated at the subcarrier frequencies of the classical electromagnetic field modulated by means of electro-optic modulation of light on the central frequency (carrier wave) [14–18]. This approach has many advantages, such as simplification of phase matching between Alice and Bob, using the carrier wave as a reference signal, and the ability to generate several secret keys on different sidebands at once [18,19], hence demonstrating unsurpassed spectral efficiency. Another benefit of the SCW method is its versatility. For instance, a free space quantum communication scheme invariant to telescopic rotation [20], a continuous variable quantum key distribution with carrier wave as local oscillator [21], and a twin-field setup [22] have been presented using the SCW method. Altogether, the protocol is reasonably well-resistant to external conditions affecting the channel and is prospective for integrating into existing telecommunication infrastructure.

The effect of chromatic dispersion was previously studied for Coherent One-Way QKD protocol [11] and for the SCW QKD scheme with a pair of phase modulators (PM-PM) [23]. However, a similar study for the SCW setup covered in this paper has not been performed yet. Our analysis includes analytical and numerical studies of the classical field propagation in the fiber as well as the security estimation based on a quantum model. Current security models of SCW QKD [24] are relatively simple and do not take into account various real-life effects, such as chromatic dispersion. This problem is quite nontrivial due to the peculiarities of the mathematical description of the protocol which uses multi-mode weak coherent states and a complex algebraic apparatus for describing the signal evolution in the electro-optical modulator taking into account different group velocities for various spectral components. Therefore, the scope of this work is to develop and substantiate an approach combining the quantum-mechanical description of SCW quantum states generation, transmission and detection with the classical description of chromatic dispersion in optical fiber.

This paper is organized as follows. Section 2. describes general principles of the SCW QKD protocol and contains the parameters of the experimental SCW QKD device that are used in subsequent calculations. In Section 3. we describe the classical chromatic dispersion model and its effects on the SCW QKD system. We demonstrate two models: one to illustrate the principles in simplified but clear form, and another, suitable for precise numerical calculations. We also compare the signal visibility for different durations of the pulse and find an optimal one for the described protocol. In Section 4. the mathematical models of the proposed system and the dispersion model are combined to calculate the quantum bit error rate and the secret key generation rate. In Section 5. we demonstrate and discuss the feasibility of the method of mitigating chromatic dispersion without using dispersion compensation devices described above. The performed analysis is experimentally proven. Section 6. concludes the paper.

## 2. SCW QKD principles and setup

SCW QKD experimental setup [14,24] is shown in Fig. 1. A coherent monochromatic light beam at optical frequency $\omega$ is modulated in an electro-optical phase modulator by a running wave with frequency $\Omega$ and phase $\varphi _{A}$. The output signal is obtained in course of energy transfer from the carrier wave to the $2S$ sidebands at frequencies $\omega _k=\omega +k\Omega$, where integer $k$ is limited by $\pm S$. The modulation index and beam intensity are chosen to provide an optimal mean photon number. The randomly chosen phase shift $\varphi _{A}$ encodes Alice’s bit at a transmission window $T$. In the receiver module, Bob performs similar operations. The carrier wave is cut by an optical filter, which lets the sidebands pass. The resulting quantum signal amplitude depends on the phase difference $\varphi _{A}-\varphi _{B}$. A detailed mathematical model is described in Appendix A.

In this work we consider a phase-coded version of BB84 protocol which allows the receiver to decode only a half of the states in each basis. Implementation of this protocol via SCW method had been presented in [14], and security analysis against collective attacks was provided later in [24].

Table 1 summarizes the SCW QKD system parameters.

The fiber parameters used in calculations are shown in Table 2.

## 3. Chromatic dispersion in the SCW QKD system

Initial study of the effect of chromatic dispersion on SCW QKD was performed in [23]. Here we perform a similar analysis applying it to the developed SCW QKD system and also taking into consideration the effect of temporal deviation of sidebands due to second order dispersion. It is known that wavelength dependence on propagation constant of the fundamental mode in single-mode optical fibers can be represented as Tailor series within the vicinity of some wavelength [25].

We consider the case when maximum visibility at a given fiber length $L$ can be achieved. One can see that the maximum and minimum intensities, respectively, correspond to following phase matching conditions:

Thus, the interference visibility can be derived as

One can notice from Eq. (4) that when $L\beta _{2}\Omega ^{2}/2 = -\pi /2$, intensity will not depend on the induced phases, and visibility will always be zero. This means that at certain fiber lengths secure key generation is fundamentally impossible without any compensation for this dispersion effect. It can be seen that this analytical derivation considers only the two first-order sidebands. We use this simple model to show principles behind this effect and to describe phase-matching conditions required to achieve constructive and destructive interference in the second modulator.

However, due to the fact that full spectrum of sideband frequencies is taken into account in subsequent security analysis in Section 4, the analytical model cannot be used any further. Hence we shall utilize the numerical approach and use the same modulator model without any simplifications in the protocol model. This approach is based on the split-step Fourier transform method of solving nonlinear Schrödinger equation, where phase modulation is represented in its general form and therefore full spectrum of sideband frequencies within the computational window is considered. This form can be written as follows

After we apply this to a Gaussian pulse, we use nonlinear Schrödinger equation of the following form to calculate propagation of this field through an optical fiber with chromatic dispersion which we solve, as was mentioned above, in a Fourier domain using split-step methodHaving obtained the intensities, we can now calculate the mean photon number using the following relation

where $E_{pulse}$ is the pulse energy, $\lambda$ is the carrier wavelength and $c$ is the speed of light in vacuum. This value will be used in the next section to calculate the performance parameters of the QKD link.An experiment was conducted to assess the effect of chromatic dispersion on the visibility. High power continuous wave signal was launched into the SCW QKD scheme connected by Corning SMF-28e+ single-mode fiber (with known attenuation). The output power was measured at different distances while the modulator at Bob’s side was tuned to zero phase shift in order to observe the interference visibility. We compared the experimental results with the ones predicted by the analytical and numerical models (Fig. 2). The main difference in the results is that in the experimentally achieved visibility does not reach zero. This could be caused by multiple factors: nonlinear response of the detectors to low power signals (optical power attenuates to the level of nanowatts at 150 km and longer fiber lengths), fiber attenuation can be slightly different for the right and left sidebands, or the signal can exhibit nonlinear noise from the phase modulator. Still, we observed the expected trend of the visibility reaching its minimum at the distance between 150 and 200 km, and then rising again as phase difference between the sidebands becomes closer to $2\pi$.

Another effect to consider is temporal pulse broadening caused by the second order dispersion. When the pulse temporal width becomes much lower than 5 ns the effect is significant enough to impact the interference. Since different sidebands propagate at different group velocities, they temporally deviate from the carrier pulse. After the second modulation at Bob’s side, the new sideband pulses do not completely overlap with the ones that originated from Alice, thus disrupting the interference and reducing the visibility. In Fig. 3 we can see visibility curves for three different FWHM of the incoming pulse. To determine the cause of this effect, we consider the propagation of the modulated 0.5 ns pulse through 340 km fiber by separating the carrier and the sideband components and plotting them together on a time line (Fig. 4). It can be seen that the left and right sideband pulses deviate from the central one by the value of $\Delta t = \beta _{2}\Omega L \approx 0.2$ ns.

## 4. Quantum bit error rate and secure key rate

To estimate the effect of fiber chromatic dispersion on SCW QKD protocol security we combine the quantum model of the discussed system (described in Appendix A) along with the dispersion model. We introduce corrections to Eq. (31) determining the change in photon number due to dispersion impact and to obtain its dependence on the relative phase shift for the cases of constructive and destructive interference. Thus, we define a negative photon fraction decrement $\Delta _1$ for the case of constructive interference and a positive increment $\Delta _2$ for the case of destructive interference. These parameters essentially change the interference visibility in accordance with the model described in the previous section.

Then, depending on the selected phase $\varphi _B$ with a correctly chosen basis, we observe

For a well-founded combination of the two models and for subsequent calculations, we use the remarkable asymptotic property of a Wigner d-function [28]

In this asymptotic form, the number of sideband frequencies is infinite, which brings the used model closer to the conventional approaches [26,29]. As noted in Ref. [26], within the framework of the quantum model, the result of such transition remains physically correct.The dispersion parameters obtained from the numerical model are calculated as follows

The probability for a single photon detector to produce a click beyond a time window $T$ is

Detection probability $(1-G)$ and error probability $E$ are expressed through Eq. (17) as follows

To evaluate the chromatic dispersion effect on the protocol security we consider a collective attack in the asymptotic limit on infinitely long keys and compute the corresponding asymptotic key rate using the Devetak-Winter approach [30]. We also estimate an upper bound for Eve’s knowledge about the data using Holevo bound [31] for weak coherent states considering collective attacks.

The dependencies of average secret key rate $K$ on losses in the channel is

where $\nu =T^{-1}$ is the repetition rate, $P_B=(1-G)/N$ is the probability of successful decoding and accepting of a bit in a single transmission window including only the data with guessed basis out of $N$ possible, $\operatorname {leak}_{E C}(Q)\geq h(Q)$ is the amount of information revealed by Alice during the error correction limited by the Shannon bound ($h(x)$ is binary Shannon entropy function), the last term is the Holevo bound.Since Eve is not subject to additional effects due to the dispersion, the Holevo bound in the case of the CBS attack remains the same as in Ref. [24]. So the key rate can be rewritten as follows

Figure 6 shows asympotic secure key rate of SCW QKD system with the effect of chromatic dispersion taken into account. Without dispersion compensation the secure key generation rate drops dramatically to zero after 53 km, and there is no way to even observe its increase due to visibility rising at longer distances.

## 5. Method to reduce the impact of chromatic dispersion

We have shown how the quantum signal encoded into two sidebands degrades due to different phase rotation rates caused by chromatic dispersion. A common way to mitigate this sort of effects is to use dispersion compensation devices such as dispersion-shifted fiber or chirped Bragg gratings. Here we propose a new method for the SCW QKD systems which consists in using optical filters that cut off one side of sideband spectrum. By doing so we mitigate the problem related to different phase rotation speeds: the only negative effect that is left is pulse broadening which was found to be negligible for nanosecond pulses.

We conducted an experiment (see Fig. 7) using off-the-shelf spectral filters for the updated scheme. Similarly to the original setup, we launched continuous wave signal with high power and observed the visibility at different fiber distances. The scheme contained a passive cascade filtering system. On the Alice’s side we set the spectral filter SF1 to prevent the propagation of one of the sidebands into the quantum channel. At Bob’s side, the two filters, SF2 and SF3, reflected the idle sideband (appeared after Bob’s modulation at the "empty" side of the spectrum) and the central mode, respectively.

Figure 8 shows the comparison of numerically calculated and experimentally measured sideband interference visibility versus fiber length in case one of the sidebands is filtered out. It is clear that both the calculated and the measured visibilities remain close to unity.

In order to calculate the secure key rate for the case of only one sideband we use addition formulas for Wigner d-function [33]. Appendix C contains the detailed mathematical description. As described above, we introduce corrections determining the lost and gained fractions of photons due to the influence of dispersion on destructive and constructive interference. Then, depending on the relative phase with a correctly chosen basis, and taking attenuation factors as negligible, we obtain

For this case, a recalculation of the Holevo information is also required, and the result is as follows

Then the secret key generation rate is as follows

As can be seen from Fig. 9, in case of only right sideband remaining the secure key rate dependence is almost equivalent to the case when chromatic dispersion is not taken into account. Hence our method has allowed to compensate for the chromatic dispersion effect was sufficiently enough to increased the secure key distribution distance up to 196 km (i.e. roughly four times).

It should be noted that a work analyzing the similar protocol shows that the probability of the USD attack increases when the same information is kept in two halves of the spectrum simultaneously (i.e. both in the lower and higher sidebands) [34]. Thus, filtering out one half of the spectrum also helps to strengthen the security against such attacks.

## 6. Conclusion

In this paper we analyzed the performance of SCW QKD link with chromatic dispersion taken into account. We developed an analytical model that shows how chromatic dispersion results in different phase rotations of the sidebands with respect to the carrier wave. We also used a numerical model based on split step Fourier method solving the classical pulse propagation by a nonlinear Schrödinger equation. Our experimental results coincide with the trends predicted by the classical model. We integrated the results of more complete numerical simulations into the quantum model and calculated secure key rate in SCW QKD with dispersion against the CBS attack. Finally, we proposed a method to mitigate the impact of chromatic dispersion by filtering out one of the sidebands and developed an adjusted quantum model which has proven the validity of such approach in terms of security and reach. Our approach has allowed to increase SCW QKD distance in realistic fiber channels with dispersion almost four times.

## Appendix A: SCW QKD mathematical model

The initial state of the field produced by a laser source is

where $|\mathrm {vac}\rangle _{S B}$ is the sidebands vacuum state and $|{\sqrt {\mu _{0}}}\rangle _0$ is the carrier wave coherent state with a given average number of photons $\mu _{0}$. The phase of the carrier wave is taken as zero. The state after the Alice’s modulation is with coherent amplitudesThe average photon number arriving at the detector on Bob’s side in the transmission window $T$ is given by

## Appendix B: Analytical dispersion model

A wave formed by the modulator at the Alice’s side can be written as:

We consider that the maximum and minimum intensities correspond to a phase matching condition of

Respectively,

Thus, the interference visibility can be derived as

## Appendix C: SCW QKD with one sideband: mathematical model

In case of the one-side sideband SCW QKD, the "right" sideband and the carrier are filtered by Alice and sent through the quantum channel, while the "left" sideband remains attenuated by the factor of $\vartheta ^{\prime }$. Then the input state at the Bob’s modulator is

Two-modulator case was considered in Ref. [33], where a general definition of the temporal evolution operator of the photon annihilation operator $a_n(\varphi _{A},\varphi ,t)$ is given as

Hence, taking all the attenuation factors as negligible, the average number of photons arriving at the detector on Bob’s side in the transmission window is

In the generalized case with all the attenuation factors, the average number of photons is

Figure 10 shows the dependencies obtained from expressions (49) and (50). The error in approximate calculation is extremely low, so we can use the simplified mathematical model to further calculate the chromatic dispersion influence.

## Appendix D: Holevo bound

Holevo information is given by the von Neumann entropy of a density operator

The von Neumann entropy of a density operator is the Shannon entropy of its eigenvalues, so the eigenvalues of $\rho$ are as follows

where $I(\varphi _1, \varphi _2)$ is a state overlap and## Funding

Ministry of Science and Higher Education of the Russian Federation (Passport No. 2019-0903).

## Disclosures

The authors declare no conflicts of interest.

## References

**1. **H. Takesue, S. W. Nam, Q. Zhang, R. H. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, “Quantum key distribution over a 40-db channel loss using superconducting single-photon detectors,” Nat. Photonics **1**(6), 343–348 (2007). [CrossRef]

**2. **B. Korzh, C. C. W. Lim, R. Houlmann, N. Gisin, M. J. Li, D. Nolan, B. Sanguinetti, R. Thew, and H. Zbinden, “Provably secure and practical quantum key distribution over 307 km of optical fibre,” Nat. Photonics **9**(3), 163–168 (2015). [CrossRef]

**3. **A. Boaron, G. Boso, D. Rusca, C. Vulliez, C. Autebert, M. Caloz, M. Perrenoud, G. Gras, F. Bussières, M.-J. Li, D. Nolan, A. Martin, and H. Zbinden, “Secure quantum key distribution over 421 km of optical fiber,” Phys. Rev. Lett. **121**(19), 190502 (2018). [CrossRef]

**4. **H.-L. Yin and Z.-B. Chen, “Coherent-state-based twin-field quantum key distribution,” Sci. Rep. **9**(1), 14918 (2019). [CrossRef]

**5. **Q. Zhang, H. Takesue, T. Honjo, K. Wen, T. Hirohata, M. Suyama, Y. Takiguchi, H. Kamada, Y. Tokura, O. Tadanaga, Y. Nishida, M. Asobe, and Y. Yamamoto, “Megabits secure key rate quantum key distribution,” New J. Phys. **11**(4), 045010 (2009). [CrossRef]

**6. **P. Eraerds, N. Walenta, M. Legré, N. Gisin, and H. Zbinden, “Quantum key distribution and 1 gbps data encryption over a single fibre,” New J. Phys. **12**(6), 063027 (2010). [CrossRef]

**7. **A. R. Dixon, Z. Yuan, J. Dynes, A. Sharpe, and A. Shields, “Continuous operation of high bit rate quantum key distribution,” Appl. Phys. Lett. **96**(16), 161102 (2010). [CrossRef]

**8. **J. Dynes, I. Choi, A. Sharpe, A. Dixon, Z. Yuan, M. Fujiwara, M. Sasaki, and A. Shields, “Stability of high bit rate quantum key distribution on installed fiber,” Opt. Express **20**(15), 16339–16347 (2012). [CrossRef]

**9. **K. Patel, J. Dynes, I. Choi, A. Sharpe, A. Dixon, Z. Yuan, R. Penty, and A. Shields, “Coexistence of high-bit-rate quantum key distribution and data on optical fiber,” Phys. Rev. X **2**(4), 041010 (2012). [CrossRef]

**10. **I. Choi, Y. R. Zhou, J. F. Dynes, Z. Yuan, A. Klar, A. Sharpe, A. Plews, M. Lucamarini, C. Radig, J. Neubert, H. Griesser, M. Eiselt, C. Chunnilall, G. Lepert, A. Sinclair, J.-P. Elbers, A. Lord, and A. Shields, “Field trial of a quantum secured 10 gb/s dwdm transmission system over a single installed fiber,” Opt. Express **22**(19), 23121–23128 (2014). [CrossRef]

**11. **M. Mlejnek, N. A. Kaliteevskiy, and D. A. Nolan, “Modeling high quantum bit rate qkd systems over optical fiber,” in * Quantum Technologies 2018*, vol. 10674 (International Society for Optics and Photonics, 2018), p. 1067416.

**12. **M. Mlejnek, N. A. Kaliteevskiy, and D. A. Nolan, “Reducing spontaneous raman scattering noise in high quantum bit rate qkd systems over optical fiber,” arXiv preprint arXiv:1712.05891 (2017).

**13. **S. K. Juma, “Chirped bragg gratings compensate for dispersion,” Laser Focus World **33**, 125–130 (1997).

**14. **A. V. Gleim, V. I. Egorov, Y. V. Nazarov, S. V. Smirnov, V. V. Chistyakov, O. I. Bannik, A. A. Anisimov, S. M. Kynev, A. E. Ivanova, R. J. Collins, S. A. Kozlov, and G. S. Buller, “Secure polarization-independent subcarrier quantum key distribution in optical fiber channel using bb84 protocol with a strong reference,” Opt. Express **24**(3), 2619–2633 (2016). [CrossRef]

**15. **A. V. Glejm, A. A. Anisimov, L. N. Asnis, Y. B. Vakhtomin, A. V. Divochiy, V. I. Egorov, V. V. Kovalyuk, A. A. Korneev, S. M. Kynev, Y. V. Nazarov, R. V. Ozhegov, A. V. Rupasov, K. V. Smirnov, M. A. Smirnov, G. N. Goltsman, and S. A. Kozlov, “Quantum key distribution in an optical fiber at distances of up to 200 km and a bit rate of 180 bit/s,” Bull. Russ. Acad. Sci.: Phys. **78**(3), 171–175 (2014). [CrossRef]

**16. **A. V. Gleim, V. V. Chistyakov, O. I. Bannik, V. I. Egorov, N. V. Buldakov, A. B. Vasilev, A. A. Gaidash, A. V. Kozubov, S. V. Smirnov, S. M. Kynev, S. E. Khoruzhnikov, S. A. Kozlov, and V. N. Vasil’ev, “Sideband quantum communication at 1 mbit/s on a metropolitan area network,” J. Opt. Technol. **84**(6), 362–367 (2017). [CrossRef]

**17. **J.-M. Merolla, Y. Mazurenko, J.-P. Goedgebuer, H. Porte, and W. T. Rhodes, “Phase-modulation transmission system for quantum cryptography,” Opt. Lett. **24**(2), 104–106 (1999). [CrossRef]

**18. **J. Mora, A. Ruiz-Alba, W. Amaya, A. Martínez, V. García-Mu noz, D. Calvo, and J. Capmany, “Experimental demonstration of subcarrier multiplexed quantum key distribution system,” Opt. Lett. **37**(11), 2031–2033 (2012). [CrossRef]

**19. **A. Ortigosa-Blanch and J. Capmany, “Subcarrier multiplexing optical quantum key distribution,” Phys. Rev. A **73**(2), 024305 (2006). [CrossRef]

**20. **S. Kynev, V. Chistyakov, S. Smirnov, K. Volkova, V. Egorov, and A. Gleim, “Free-space subcarrier wave quantum communication,” in Journal of Physics: Conference Series, (2017), 5.

**21. **E. Samsonov, R. Goncharov, A. Gaidash, A. Kozubov, V. Egorov, and A. Gleim, “Subcarrier wave continuous variable quantum key distribution with discrete modulation: mathematical model and finite-key analysis,” Sci. Rep. **10**(1), 10034 (2020). [CrossRef]

**22. **V. Chistiakov, A. Kozubov, A. Gaidash, A. Gleim, and G. Miroshnichenko, “Feasibility of twin-field quantum key distribution based on multi-mode coherent phase-coded states,” Opt. Express **27**(25), 36551–36561 (2019). [CrossRef]

**23. **J. Mora, A. Ruiz-Alba, W. Amaya, and J. Capmany, “Dispersion supported bb84 quantum key distribution using phase modulated light,” IEEE Photonics J. **3**(3), 433–440 (2011). [CrossRef]

**24. **G. Miroshnichenko, A. Kozubov, A. Gaidash, A. Gleim, and D. Horoshko, “Security of subcarrier wave quantum key distribution against the collective beam-splitting attack,” Opt. Express **26**(9), 11292–11308 (2018). [CrossRef]

**25. **G. P. Agrawal, * Fiber-optic communication systems*, vol. 222 (John Wiley & Sons, 2012).

**26. **J. Capmany and C. R. Fernández-Pousa, “Quantum model for electro-optical phase modulation,” J. Opt. Soc. Am. B **27**(6), A119–A129 (2010). [CrossRef]

**27. **P. Y. Amnon Yariv, * Optical Waves in Crystals* (John Wiley & Sons, 2002).

**28. **D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, * Quantum theory of angular momentum* (World Scientific, 1988).

**29. **D. Horoshko, M. Eskandary, and S. Y. Kilin, “Quantum model for traveling-wave electro-optical phase modulator,” J. Opt. Soc. Am. B **35**(11), 2744–2753 (2018). [CrossRef]

**30. **I. Devetak and A. Winter, “Distillation of secret key and entanglement from quantum states,” Proc. R. Soc. A **461**(2053), 207–235 (2005). [CrossRef]

**31. **A. S. Holevo, “Bounds for the quantity of information transmitted by a quantum communication channel,” Problemy Peredachi Informatsii **9**, 3–11 (1973).

**32. **A. Kozubov, A. Gaidash, and G. Miroshnichenko, “Finite-key security for quantum key distribution systems utilizing weak coherent states,” arXiv preprint arXiv:1903.04371 (2019).

**33. **G. P. Miroshnichenko, A. D. Kiselev, A. I. Trifanov, and A. V. Gleim, “Algebraic approach to electro-optic modulation of light: exactly solvable multimode quantum model,” J. Opt. Soc. Am. B **34**(6), 1177–1190 (2017). [CrossRef]

**34. **D. Horoshko, M. Eskandari, and S. Y. Kilin, “Equiprobable unambiguous discrimination of quantum states by symmetric orthogonalisation,” Phys. Lett. A **383**(15), 1728–1732 (2019). [CrossRef]

**35. **M. O. Scully and M. S. Zubairy, “Quantum optics,” (1999).