Robust diagnostics of dark counts for quantum networks N.S. Perminov1,2,3, M.A. Smirnov1, K.S. Melnik1,2, L.R. Gilyazov1,2, O.I. Bannik1,2, M.R. Amirhanov1,2, D.Y. Tarankova1, A.A. Litvinov1,2 qm.kzn@ya.ru | maxim@kazanqc.org | mkostyk93@mail.ru | lgilyazo@mail.ru | olegbannik@gmail.com | m.amirhanov85@gmail.com | tarankovadyu@ya.ru | litvinov85@gmail.com 1 Kazan National Research Technical University n.a. A.N. Tupolev, Kazan, Russia 2 «KAZAN QUANTUM COMMUNICATION» LLC, Kazan, Russia 3 Zavoisky Physical-Technical Institute, FRC Kazan Scientific Center of RAS, Kazan, Russia In this work, we study timestamps when registering counts of single-photon detectors in quantum communications. Post-pulse counts are analyzed based on several approaches. Explicit statistical accounting of the noise of quantum detectors allows you to most correctly select the mode of use of the detectors to realize the most efficient quantum communication with the highest signal to noise ratio. Direct statistical analysis and robust diagnostics of the noise of quantum detectors can be done by ranging the time's tags of quantum keys that are available for the online diagnostic system and analysis a significant amount of information about the quantum communication performance (the amount of dark noise and post-pulse counts, line interference, etc.). The conclusion is made about the proportion of dark noise and post-pulse counts in the total noise, and the limits of applicability of the theory are shown using a sequence of the ranged amplitudes. We offer non-parametric robust diagnostic of times tags in keys to increase the security of quantum networks, and also discuss the prospects of commercializing quantum-classical cloud-based security services. Keywords: quantum networks, quantum communications, single-photon detector, dark noise, sequence of the ranged amplitudes. In large-scale TQN, according to the theory of 1. Introduction reliability, control requirements should be higher than for Robust diagnostics of the working complexes of conventional QC due to the large number of elements and quantum communication (QC) [1] is crucial for the a high level of noise that cannot be eliminated for long- implementation of quantum networks capable of working range QC. A significant part of the noise in the TQN is due in both urban and trunk standard fiber-optic to various factors arising from the registration of single communication lines. At the same time, from the point of photons by highly sensitive single-photon detectors. view of communication systems, trunk quantum networks Therefore, explicit statistical accounting of the noise of (TQN) with lines of more than 100 km and losses between quantum detectors allows you to most correctly select the nodes of more than 25 dB, where the signal-to-noise ratio mode of use of the detectors to realize the most efficient cannot be considered large, are particularly difficult to quantum coupling with the highest signal to noise ratio. implement [2]. From the point of view of fundamental Direct statistical analysis and robust diagnostics of the statistics, the fundamental difficulty here lies in the fact noise of quantum detectors can be performed by that despite the relatively low average percentage of errors, distributing the time stamps of quantum keys, which are the magnitude of the error span and the variance of errors available for the online diagnostic system and, in our can be extremely large, which entails low reliability of opinion, carry a significant amount of information about diagnosing errors when performing continuous tests of the the QC operability and the level of quantum-classical TQN and especially large-scale TQN with a large number security of QC complexes as a whole. of nodes. In this work, we study timestamps when registering The necessary solution for a similar problem in counts of single-photon detectors in QC. Post-pulse counts communication theory is the use of prognostic monitoring are analyzed based on several approaches. The conclusion tools and error filtering, which will increase the reliability is made about the proportion of dark noise and post-pulse of security parameters for inter-city QC in continuous use. counts in the total noise, and the limits of applicability of We also note that due to large errors for the limit passport the theory are shown using a sequence of the ranged operating modes of QC complexes, the security of the amplitudes. We offer non-parametric robust diagnostic of communication complex should be determined by a times tags in keys to increase the security of quantum special diagnostic system that is different from the networks, and also discuss the prospects of diagnostic system for operating passport modes. This commercializing quantum-classical cloud-based security difference in diagnosis is also due to the fact that the services. reliability of error determination in the limiting mode 2. Modern diagnostic methods effectively depends on a significantly larger number of physical factors, which are not always easy to track within One of the modern statistical diagnostic methods is a the framework of a single structurally finished product. sequence of the ranged amplitudes (SRA), which is a That is, a more powerful specialized diagnostic system sequence of numbers obtained from the original sequence should be delivered with the QC complex as a separate by ordering numbers in descending (or ascending) order of product and be able to work in the background of the TQN their values. SRA are often used in histogram construction to track the entire history of changes in the line algorithms. The SRA sequences themselves are practically performance even with the QC complex in the node not analyzed, although they are very interesting, since disabled. In addition, such a system should be able to track when constructing them, there is no loss of information additional performance factors associated with the about specific values of the source data, as, for example, statistics of quantum and classical noise. using the histogram method. It also gives the advantage of Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0) using much less data for statistical analysis. Analytical The characteristic value of the parameter α for InGaAs/InP analysis of SRA is the essence of the so-called SRA detectors is 1.2 ± 0.2 [10]. method [3-5]. When using the SRA method, the Expression (4) describes with high accuracy the dependence of the values of the elements in the SRA statistics of post-pulse readings. The reasons for this are sequence on their serial number (index) in the sequence is that the charge traps in the avalanche photodiode have a analyzed. wide energy distribution. This circumstance does not To search for an analytical expression, it is convenient allow us to describe the statistics of readings by a single to use an approximate expression connecting the SRA with decaying exponent, which determines a single Arrhenia the empirical distribution function F(x) [6,7]: relaxation process. Of course, it is possible to improve the 𝑁𝑁 + 1 − 𝑛𝑛(𝑥𝑥𝑛𝑛 ) degree of fitting by choosing as the fitting function the sum 𝐹𝐹(𝑥𝑥, 𝑥𝑥𝑛𝑛 ) = , (1) 𝑁𝑁 of exponentials with different decay times, however, the number of exponents is not a fixed parameter and the where N is the total number of points in the sample. In physical meaning of the parameters obtained is not clear. some cases, knowing the analytical form for the It is important to note that each value of the probability distribution function F(x), one can find an analytical distribution function in (2) and (4) is obtained by expression for the SRA of the form that can be actively calculating a statistically suitable number of experimental used in the SRA method to quickly find the statistical points in a small time interval. Thus, to obtain the parameters of the initial sample. Next, we consider two dependence of the probability distribution function that is versions of expressions for SRA that describe random suitable for analysis on the inter-pulse intervals, a large readings of a single-photon detector. The first one number of experimental values are required (of the order considers samples caused only by random Poisson sources of 106 values [11]). (the probability of post-pulse counts is zero). The second For the data obtained in the experiment, we test one considers counts caused by post-pulse counts. statistical hypotheses of the form (2), (4) with an 3. Dark counts of detectors additional factor introduced for testing flexibility. In addition, we are testing a new hypothesis for a probability Light sources, as well as dark noise, corresponding to of the form P=A/(1-e-Bt), which is a solution of a nonlinear Poisson processes, correspond to the probability density equation of the form dP/dt=aP+bP2, potentially for the time intervals between samples, which is corresponding to possible strong nonlinear effects on approximately described by a decaying exponent [8,9]: microscopic level of description of quantum detectors. The 𝑑𝑑𝑑𝑑 results of quantitative parametrization of dark samples = 𝜌𝜌(𝑥𝑥𝑛𝑛 ) = 𝜆𝜆𝑒𝑒 −𝜆𝜆𝑥𝑥𝑛𝑛 , (2) 𝑑𝑑𝑑𝑑 with a small number of post-pulse counts for a working where λ is the average frequency of avalanche events, and QC system are shown in Fig. 1. xn are the time intervals between samples. From (1) and We see that for the new hypothesis and Poisson theory (2) can be obtained in form [6,9]: [7], the results of estimating noise in the region of small 1 𝑁𝑁 times t<3 us are more accurate than for Itzler's theory [10]. 𝑥𝑥𝑛𝑛 = 𝐿𝐿𝐿𝐿 . (3) However, the asymptotic behavior of the curves on a 𝜆𝜆 𝑛𝑛 − 1 logarithmic scale indicates the need for a more competent In gallium arsenide avalanche photodiodes used in QC account of the distribution tails in the region of large times quantum detectors, the exponential model for dark counts t>3 us, which is not so simple to do in the framework of is violated due to post-pulse counts. The effect of post- one consistent physical theory [10]. Here we see the pulse counts is the process of re-emission of charges imperfection of theoretical models, which is difficult to captured by the avalanche diode traps during the previous overcome within the framework of theoretical models of avalanche event. The traps inside the diode are not due to noise analysis with a small finite number of parameters, the ideality of the detector and its manufacturing but it is easy to overcome within the framework of technology. nonparametric statistical criteria for noise analysis, such as The first models describing post-pulse counts were SRA. constructed on the basis of a simple exponential dependence of the probability density on the inter-pulse interval [10,11]. This expression depends on the amplitude of the probability and the decay time. However, this expression describes well the processes in the "free-run" mode and does not accurately describe the readings of devices operating in the "gating" mode. In [10], it was shown that in this mode, post-pulse counts are well described by a power function of the form 𝑃𝑃 = 𝐶𝐶𝑡𝑡 𝛼𝛼 , (4) where C and α are positive parameters, t is time in counts. In this paper, it is assumed that a possible reason for this dependence is the wide distribution of trap energy in the Fig. 1. Noise parametrization according to Itzler's theory (4) used semiconductor avalanche diode. Moreover, the (red line), Poisson (green line) and the new theory (blue line). values of the parameters C and α depend on the shape of Here, the comparison is at the level of the empirical probability the density function of the energy distribution of the traps. density of dark counts for the working QC system 4. SRA analysis of dark noise times tags within in key for continuous diagnosis of changes in statistics. Expression (4) we will base the model of post-pulse counts based on SRA. In the case when the samples dominated by the counts due to the effect of post-pulses, we can substitute (4) in (1) and get: xn= xmin (N/(n-1))1/(α-1), (5) where xmin is the minimum time value equal to the dead time Tho, N is the number of points in the sample. Eq. (5) has only one adjustable parameter α. Note that Eq. (3) and (5) contain each experimental value. Thus, in this case, to obtain a dependence suitable for analysis, a much smaller number of experimental values (of the order of 103–104 are required in comparison with the probability distribution functions obtained from Fig. 3. Dependence of SRA[Constructive]-SRA[Dark] on the histograms. In Fig. 2 shows the parametrization of dark SRA[Dark] for precise distinguishing constructive and noise with a sufficient proportion of post-pulse samples destructive interference using times tags in keys for continuous based on formula (5) with an additional factor for testing diagnostics of changes in statistics flexibility, which corresponds to Itzler's theory [10]. Unlike parametric criteria with 2-3 evaluation parameters, the criterion described here is the entire resulting dependence, that is, 1000 fitting parameters. In this sense, the power of many advanced nonparametric criteria is almost incomparable with conventional parametrization methods. Therefore, despite the difficulty in comparing nonparametric criteria with real physical factors, they are hypersensitive even with small changes in the systems under study and are able to work in the absence of a priori information. Accordingly, such robust nonparametric methods as SRA can be used as the basis for quantum-classical diagnostics of QC. Fig. 2. Robust parametrization of dark photocounts using SRA xn. For t>24us, Itzler's theory does not work well. Axes are 6. Conclusion given on a logarithmic scale Precise error diagnostics in QC depends on a large In Fig. 2 on a logarithmic scale, we clearly see that number of physical factors that are difficult to track within starting from t> 24 us, Itzler's theory ceases to work. Thus, the framework of only one structurally finished product. In in addition to fast parametrization of noise by theoretical our opinion, needed an expanded robust diagnostic models, we can also quantify the applicability limits of system, which should be supplied with the QC complex models using non-parametric analysis of the tails of the and be able to conduct a joint statistical analysis of such distribution of SRA. seemingly different parameters as the time stamps of keys and the distribution of errors within the key. Such a 5. Relative SRA for different operating modes quantum-classical diagnostic subnet in the TQN, capable of QC of diagnosing even small noises in the network in the absence of a priori information about the type of Explicit statistical accounting of the noise of quantum interference, also opens up prospects for the detectors allows you to most correctly select the mode of commercialization of quantum-classical cloud services for use of the detectors to realize the most efficient quantum robust information protection. communication with the highest signal to noise ratio. Direct statistical analysis and robust diagnostics of the Acknowledgments noise of quantum detectors can be done by distributing the times tags of quantum keys that are available for the online The work was carried out with partial financial support diagnostic system and a significant amount of information in the framework of the topic of the laboratory "Integrated about the QC performance (the amount of dark noise and quantum optics" of the Kazan quantum center KNRTU- post-pulse counts, line interference, etc.). Moreover, a KAI (the basic idea of nonparametric analysis and analysis joint statistical analysis of the dynamics of bit errors in of results – NSP, AAL). Research in the field of statistics keys and key times tags can become the new standard of of quantum detectors and quantum sensors was carried out quantum-classical security of quantum communication with the financial support of RFBR grant No. 19-32-80029 complexes as a whole due to the high predictive power of (idea of a microscopic approach to the dynamics of post- nonparametric statistical criteria [3-7]. pulse counts and the basic statistical verification of this In Fig. 3 shows the dependence of SRA[Constructive]- hypothesis – MAS) and also grant of the Government of SRA[Dark] on SRA[Dark] for precise distinguishing the Russian Federation 14.Z50.31.0040, February 17, between constructive and destructive interference using 2017 (experiments on the generation of quantum keys – NSP, MAS, KSM, LRG, OIB, AAL). The work is also partially supported in the framework of the budget theme detectors,” Journal of Lightwave Technology, vol. 33, of the laboratory of Quantum Optics and Informatics of no. 14, pp. 3098–3107, 2015. Zavoisky Physical-Technical Institute FRC Kazan Scientific Center of RAS (numerical modeling in quantum About the authors informatics – NSP, DYT). Nikolay S. Perminov, junior researcher, Kazan Quantum Center, KNRTU-KAI. E-mail: qm.kzn@ya.ru References: Maxim A. Smirnov, junior researcher, Kazan Quantum [1] A.G. Vtyurina, V.L. Eliseev, A.E. Zhilyaev, A.S. Center, KNRTU-KAI, E-mail: maxim@kazanqc.org Konstantin S. Melnik, engineer, Kazan Quantum Center, Nikolaeva, V.N. Sergeev, and A.V. Urivskiy, “On the KNRTU-KAI. E-mail: mkostyk93@mail.ru principal decisions of the practical implementation of Lenar R. Gilyazov, junior researcher, Kazan Quantum the cryptographic devices with quantum key Center, KNRTU-KAI. E-mail: lgilyazo@mail.ru distribution,” Doklady TUSUR, vol. 21, no. 2, 2018. Oleg I. Bannik, junior researcher, Kazan Quantum Center, [2] O.I. Bannik, L.R. Gilyazov, A.V. Gleim, N.S. KNRTU-KAI. E-mail: olegbannik@gmail.com Perminov, K.S. Melnik, N.M. Arslanov, A.A. Marat R. Amirhanov, engineer, "KAZAN QUANTUM Litvinov, A.R. Yafarov, and S.A. Moiseev, “Noise- COMMUNICATION" LLC, E-mail: immunity kazan quantum line at 143 km regular fiber m.amirhanov85@gmail.com link,” arXiv preprint arXiv:1910.10011, 2019. Diana Y. Tarankova, student, Department of Radio- Electronics and Information-Measuring Technique, KNRTU- [3] R. Nigmatullin and G. Smith, “Fluctuation-noise KAI. E-mail: tarankovadyu@ya.ru spectroscopy and a “universal” fitting function of Aleksandr A. Litvinov, engineer, Kazan Quantum Center, amplitudes of random sequences,” Physica A: KNRTU-KAI. E-mail: litvinov85@gmail.com Statistical Mechanics and its Applications, vol. 320, pp. 291–317, 2003. [4] R.R. Nigmatullin, "Universal distribution function for the strongly-correlated fluctuations: General way for description of different random sequences." Communications in Nonlinear Science and Numerical Simulation vol. 15, no. 3, pp. 637-647, 2010. [5] R.R. Nigmatullin, “New noninvasive methods for ‘reading’ of random sequences and their applications in nanotechnology,” in New trends in nanotechnology and fractional calculus applications, pp. 43–56, Springer, 2010. [6] M.A. Smirnov, N.S. Perminov, R.R. Nigmatullin, A.A. Talipov, and S.A. Moiseev, “Sequences of the ranged amplitudes as a universal method for fast noninvasive characterization of SPAD dark counts,” Applied optics, vol. 57, no. 1, pp. 57–61, 2018. [7] N.S. Perminov, M.A. Smirnov, R.R. Nigmatullin, and S.A. Moiseev, “Comparison of the capabilities of histograms and a method of ranged amplitudes in noise analysis of single-photon detectors,” Computer Optics, vol. 42, no. 2, pp. 338–342, 2018. [8] C. Wiechers, R. Ram´ırez-Alarc´on, O.R. Mu˜niz- S´anchez, P.D. Y´epiz, A. Arredondo-Santos, J.G. Hirsch, and A.B. U’Ren, “Systematic afterpulsing- estimation algorithms for gated avalanche photodiodes,” Applied optics, vol. 55, no. 26, pp. 7252–7264, 2016. [9] D. Horoshko, V. Chizhevsky, and S.Y. Kilin, “Afterpulsing model based on the quasi-continuous distribution of deep levels in single-photon avalanche diodes,” Journal of Modern Optics, vol. 64, no. 2, pp. 191–195, 2017. [10] M.A. Itzler, X. Jiang, and M. Entwistle, “Power law temporal dependence of InGaAs/InP SPAD afterpulsing,” Journal of Modern Optics, vol. 59, no. 17, pp. 1472–1480, 2012. [11] G. Humer, M. Peev, C. Schaeff, S. Ramelow, M. Stipˇcevi´c, and R. Ursin, “A simple and robust method for estimating afterpulsing in single photon