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Abstract
Long-term quantum key distribution (QKD) using polarization encoding requires a random drift compensation method. We propose a method to compensate any state of polarization based on the quantum bit error rate (QBER) of two states from two non-orthogonal mutually unbiased bases. The proposed method does not require dedicated equipment, and through a simple but highly efficient feedback loop it compensates the polarization random drift suffered by photons while transmitted over the optical fiber quantum channel. A QBER lower than 2% was observed even considering imperfect single photon detectors. Besides, we verify a 82% secret key rate generation improvement in a finite-key size BB84 implementation for a 40 km fiber-optics quantum channel.
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1. Introduction
Quantum key distribution (QKD) assures unconditional secure key distribution even in the presence of an eavesdropper with unlimited quantum resources [1,2]. Different photon degrees of freedom can be used to implement QKD: polarization, time, frequency, phase, and orbital angular momentum. The usage of polarization encoded qubits to transmit information is the earliest and may be the simplest form to implement QKD in fiber-optic systems [2]. Nevertheless, polarization fluctuations in the quantum channel can lead to a considerable increase on quantum bit error rate (QBER) [3]. Therefore, the polarization random drift suffered by any state of polarization (SOP) must be compensated ensuring a QBER suitable with standard boundaries of QKD to be implemented.
Photonic qubits are generally encoded using polarization [4]. Nevertheless, the state of polarization that carries quantum information can be affected when goes through a transmission channel that does not preserves the polarization state, such as the standard optical fibers [1]. Furthermore, the information flow may even be interrupted when the error rate among the legitimate users of quantum channel rises to a value that makes it unsuitable for a reliable information transmission [5,6]. Most of the proposed methods to compensate polarization random drift in real-time require additional signals, such as using wavelength or time division multiplexing techniques [7,8]. Both require additional classical hardware. In the case of time division multiplexing the transmission rate of secure information is reduced, while the wavelength division multiplexing (WDM) technique require additional spectral bands [9]. By using WDM techniques, the reaching distance between parties is also limited since the information decorrelation between the two wavelengths increases with distance [10]. An alternative approach is the use of feedback algorithms free of additional classical reference signals [11]. Moreover, in [12] the suitability of the use of quantum frames in QKD systems with polarization encoding was demonstrated. In [13], a protocol-dependent method to compensate polarization drift using the unveiled bits was proposed. Even though this method does not require additional reference signals neither additional dedicated bits, it depends on the implemented protocol which limits its large deployment. Despite being a blind searching algorithm, in [14] the authors proposed an upper-layer agnostic polarization compensation method. In [15] a feedback loop is used to also implement a blind stochastic searching algorithm, which finds the set of voltage values to apply on the electronic polarization controller to minimize the QBER. This method achieves a QBER as lower as $3 \%$. However, the use of blind searching methods increase the required bandwidth consumption, which issue can be overcome using heuristic searching methods. In [16] a heuristic search method is presented, where the reversal operator to compensate polarization random drift in standard fiber-optics channels is found based on the QBER value. This method is capable of finding the reversal operator for a specific SOP that has suffered polarization random drift, and compensate it in less than tens of microseconds with a very low overhead. For instance, in the case of BB84 protocol it is required the use of two non-orthogonal mutually unbiased bases to generate 4 states of polarization [17,18]. Also, a complete and general polarization compensation method for any arbitrary input SOP can be based on the compensation of two states obtained from two different bases within the three non-orthogonal mutually unbiased bases [19]. As far as we know there is still no heuristic searching method to compensate any arbitrary SOP proposed until now.
We propose a novel heuristic searching method to compensate polarization random drift of any state based on the QBER of only two states of polarization belonging to two different non-orthogonal mutually unbiased states of polarization. By compensating these two states of belonging to two non-orthogonal mutually unbiased bases, we are able to compensate the remaining states of polarization. Through the use of quantum frames, the proposed method continuously monitors the QBER of two states of polarization being able to maintain a real-time drift tracking. Therefore, in a prepare and measure finite-key QKD implementation that includes the proposed compensation method, we show an improvement of $82 \%$ in secret key rate generation, and a long-term QBER bellow $2 \%$.
This paper contains 4 sections. In section 2, the method to compensate any polarization random drift through fiber-optics quantum channel is proposed. In section 3, we detail the implemented system model, and we present the numerical validation for the method to compensate polarization random drift of the SOPs to be used in BB84 QKD. In section 4, we assess the performance of BB84 protocol in a finite-key size implementation using the proposed algorithm in a discrete variable polarization encoding quantum communication system. Finally, in section 5 the main conclusions are summarized.
2. Heuristic polarization compensation method
In this section, we propose a method to compensate arbitrary SOP drifts induced by birefringence through a fiber-optics based quantum channel. The method is based on the estimation of the QBER computed from a set of quantum frames encoded in non-orthogonal mutually unbiased bases. [8]. The method starts by compensating the SOP whose its QBER first rises above a user defined QBER threshold. In this context, it is not relevant which state is first compensated. The polarization random drift compensation is performed applying reversal rotations on the SOP that arrives at receiver with a misaligned polarization compared with the one it had when left the transmitter. Nevertheless, it is crucial that the rotation applied to compensate the non-orthogonal SOP does not impact the one already compensated. In this way, the more important consideration is to guarantee that the rotation applied to compensate the non-orthogonal SOP is performed around the axis on which the already compensated one is placed on.
Lets detail the operation mode of the polarization drift compensation method. The communication between parties is performed using quantum frames [12]. The transmitted frame is divided in two quantum frames: control frame and data frame. The proposed method continuously keep track on the QBER of the control qubits individually, which are known in advance by both parties, and these values are continuously compared with a previously defined threshold above which starts the compensation method.
Without loss of generality, we choose for the control qubits the $|{H}\rangle$ and the $|{45}\rangle$ polarization states. Lets assume a threshold value above which the method must be applied, $\textrm {QBER}_{th}$. By continuously monitoring both QBER values, $\textrm {QBER}_{|{H}\rangle }$ and $\textrm {QBER}_{|{45}\rangle }$, the algorithm starts the operation mode as soon as one of them crosses the threshold. The first SOP whose QBER rises above the threshold is the first to be compensated. We assume a six-waveplate electronic polarization controller (EPC), where four of the six wave-plates are used to induce the required rotations, one is used for basis selection and the last one is connected to zero birefringence and has no impact in the system. Regardless of which one is the first SOP to be compensated, the first wave-plate of the EPC is responsible for a first small arbitrary rotation as suggested in the algorithm presented in [16]. With that rotation we aim to change the QBER value on the physical system, and simultaneously we apply the same rotation in software on the first circle of a sphere drawn due the first estimated QBER. Next, a new QBER is estimated, and the two possible locations of the SOP result from the intersection between those two circles. In order to fulfill this step of the algorithm, two angles must be chosen to induce a small rotation on the first wave-plate of the EPC, $\alpha$ and $\delta$. The transformation matrix of a linear retarder with fast angle, $\alpha$, and retardation, $\delta$, is given by $R{(\alpha,\delta )}$ [20],
Note that the angles used in this rotation do not depend of which state is being first compensated,
where $\alpha _{max}$ and $\delta _{max}$ are the rotation and ellipticity angles, respectively, defined by the circle of a sphere corresponding to the estimated QBER with relation to the reference state of polarization, see Fig. 1 [16]. The second and third wave-plates of the EPC are used to compensate the control qubit of which its QBER first crossed the threshold with the transformation matrix $R(\alpha _1,\delta _1)$ and $R{(\alpha _2,\delta _2)}$, respectively. In this way, the transformation matrix used to compensate the first SOP is the concatenation of three matrices,For instance, lets assume the first state to be compensated is the $|{45}\rangle$, and the following must be guaranteed,
where $\textrm {R}_{\textrm {F}}$ is a Jones matrix that represents the random rotation induced by the standard fiber-optics channel. Fig. 2(a) shows the Stokes representation of the rotations induced by $R{(\alpha _1,\delta _1)}$ and $R{(\alpha _2,\delta _2)}$. According to Fig. 2(a), the first rotation must be performed such that, where $(S_1,S_2,S_3)^{T}$ are the stokes parameters corresponding to the intersection points found by the polarization compensation method. Thereafter, the second state to be compensated is $|{H}\rangle$, and the fourth wave-plate is responsible for performing the polarization drift compensation of this state with no influence on the first which is already compensated. In this way, the main rotation axis of the fourth wave-plate must be aligned with the SOP already compensated. Before inducing any rotation, the QBER of each control qubit is again estimated, see Fig. 3 step (iii). The $\textrm {QBER}_{|{45}\rangle }$ should approaches the zero bound. At this stage, the transformation matrix to be applied to decrease this QBER to a value close to the zero bound is which must guarantee that,
Fig. 1. Circle of a sphere defined by all possible states of polarization corresponding with the QBER of the control qubits that first rises above the threshold. $\alpha _{max}$ and $\delta _{max}$ correspond to the rotation and ellipticity angles defined by the circle of a sphere, respectively.
Fig. 2. Stokes space representation of the rotations induced by $R_1$ and $R_2$. (a) Rotations induced at wave-plate 2 and 3 to compensate the first SOP. At this point both angles $\alpha _i$ and $\delta _i$ can vary. (b) Rotation induced by wave-plate 4 to compensate the second SOP without influence the first SOP already compensated. At this point $\alpha$ is maintained constant and aligned with the axis on which the first SOP is on. Only $\delta _3$ varies.
Since we assumed the first state to be compensated is the $|{45}\rangle$, the non-orthogonal state of polarization to be compensated now is the $|{H}\rangle$. Fig. 2(b) shows the rotation performed using the fourth wave-plate, and at this point the $\alpha _3$ must be maintained constant and coincident with the axis $S_2$ such that the rotation induced does not have influence in the first compensated SOP. Note that, in any case the matrix $R{(\alpha _3,\delta _3)}$ can only vary one of the angles and maintain the other. In the case we are currently analyzing, the $\alpha _3$ angle must be constant and coincident with the axis $S_2$ therefore the induced rotation has no influence on the first SOP already compensated $|{45}\rangle$, since the performed rotation is around the axis which it is on. In this way,
and $\delta _3$ can assume two different values, $\pi - \delta _k$ or $\pi + \delta _k$, where $\delta _k$ is defined as following,The polarization compensation method chooses one of the two possible $\delta _3$ and estimates the QBER. If the QBER does not approaches zero another rotation is induced using the other angle. Otherwise, the compensation procedure stops and runs as initially, i.e. monitoring the QBER using quantum frames. The concatenation of both transformation matrices $R_1$ and $R_2$ must compensate the polarization drift induced in $|{H}\rangle$ by $\textrm {R}_{\textrm {F}}$,
Once assuring (4), (7) and (10) the polarization random drift compensation method compensates two states of polarization from two different non-orthogonal mutually unbiased bases, which implies the compensation of any state of polarization on the Poincaré sphere.
3. Method validation
A conventional physical system to implement QKD is shown in Fig. 4. To support any QKD upper-layer protocol assuring unconditional security, the physical layer system must allow to encode the photons in four states of polarization using two non-orthogonal bases [21]. In this work, we use the two non-orthogonal linear bases: rectilinear basis, $|{H}\rangle$ and $|{V}\rangle$, and diagonal basis, $|{45}\rangle$ and $|{-45}\rangle$.
Fig. 4. Schematic of the quantum key distribution (QKD) system configuration using polarization encoded single-photons in two non-orthogonal basis, with active polarization drift compensation. EPC: Electronic polarization controller. D1 and D2: single-photon detectors.
3.1 Transmission system modelling
3.1.1 Transmitter
In the transmitter side, it is assumed a well defined horizontal polarized state at the input of the electronic polarization controller (EPC1), which is obtained from a strongly attenuated laser source whose optical pulses have an average number of photon per pulse of $n_{\textrm {H}_0} (t_k)$. In this way, the number of photons per pulse is randomly generated from a Poisson distribution,
where $\mu$ is the average number of photons per pulse. The EPC1 is responsible for encoding any state of polarization applying a rotation on the input state, for instance the four required for QKD implementation. For that purpose only one wave-plate of the EPC1 is required, and its transformation matrix is assumed to be described as a rotation imposed by a wave-plate [20],3.1.2 Fiber-optic quantum channel
The fiber-optic quantum channel is modeled based on the work presented in [23], and we represent it as a random matrix parameterized by the random parameters $\gamma _k = (\gamma _1,\gamma _2,\gamma _3)$,
3.1.3 Receiver
In the receiver side, the first optical component the polarization encoded single-photons face is EPC2. The EPC2 is the head device of our system, since it is the one responsible for actuating in the modelled system as a compensation of the polarization random drifts occurred throughout the quantum channel. This component is modelled as a concatenation of five wave-plates similar to (12), and can be represented as
In [16], an algorithm to find the reversal operator for one particular SOP was proposed. Here, we present a more comprehensive method for compensating the drift of any arbitrary SOP during continuous transmission of polarization states while a QKD protocol is running. The proposed method assures that polarization rotations induce by matrix $R_{\textrm {F}}$ are reverted after the EPC2, see Fig. 4. Every wave-plate has a particular role in the measurement scheme, assuring the polarization drift compensation by the first four wave-plates as described in Fig. 3, and the basis choice by the fifth wave-plate.
In the next sub-section, we present numerical results to demonstrate the effectiveness of the proposed method to compensate the polarization random drift suffered by an arbitrary SOP along the fiber-optics quantum channel.
3.2 Assuming a perfect receiver
The QBER has two main contributions, the QBER resulted from channel errors induced by polarization random drift ($\textrm {QBER}_{pol}$) [9,16], and the QBER resulted from single-photon detectors imperfections such as dark-counts, detectors efficiency or after-pulses ($\textrm {QBER}_{det}$) [1].
Lets start taking only into account the QBER resulted by polarization random drift, in order to demonstrate that with the proposed method we can compensate the polarization drif of any SOP looking only into the QBER calculated for two of the four required SOPs, since they belong to mutually unbiased bases [8]. Control frame is composed by the two mutually unbiased states of polarization $|{H}\rangle$ and $|{45}\rangle$, and the data frame is composed by the four random states of polarization belonging to two non orthogonal bases considering in the simulation. Considering the study case presented in Fig. 4, we present numerical results for polarization drift compensation method validation. Note that the control qubits are chosen by the user with a determined position on the frame and previously known by both parties. The data and control qubits are both implemented using a $0.2$ average number of photons per pulse. In this subsection, we use a $\Delta _p = 4 \times 10^{-8}$, a symbol repetition rate of $100$ MHz, and a $\textrm {QBER}_{th}= 3.0\%$.
Figure 5(a) shows the individual QBER evolution for control qubits, $|{H}\rangle$ and $|{45}\rangle$. The first SOP to be compensated was $|{45}\rangle$ since, as one can see in Fig. 5, the QBER associate with this SOP is the first to reach the predefined $\textrm {QBER}_{th}$. The polarization random drift compensation occurred at $175$ ms, and one can see that both $\textrm {QBER}_{|{H}\rangle }$ and $\textrm {QBER}_{|{45}\rangle }$ decrease approaching the zero bound. Also, looking into Fig. 5(b), where the QBER of the four BB84 SOP are presented, they also decrease approaching zero at the time of polarization compensation.
Fig. 5. (a) QBER for two non-orthogonal states with the random drift compensation method as a function of time. (b) QBER evolution with time for the data qubits for each of the four SOPs.
3.3 Assuming a realistic receiver
Once the effectiveness of the proposed polarization drift compensation method has been demonstrated considering a perfect receiver, we will now consider the single-photon detectors dark-counts, and detection efficiency. Besides polarization random drift, we also consider the fiber-optic channel attenuation of $0.2$ dB/km. In this subsection we consider an average number of photons per pulse of $0.2$ at transmitter output for data qubits, and at receiver input for control qubits, a channel length of 40 km, a polarization linewidth of $\Delta _p = 2 \times 10^{-8}$, a symbol repetition rate of $100$ MHz, and a $\textrm {QBER}_{th}= 1.0\%$ on control qubits. Note that the length of control and data frames had to be increased since with the addition of the dead time later, the number of photons that are effectively measured decreased. In this way, a lower threshold had to be also defined to guarantee a QBER on data qubits is maintained bellow $2.1 \%$. We divided the transmitted frame in 50000 control qubits and 50000 data qubits. Again, the control qubits frame is a predefined sequence of $|{H}\rangle$ and $|{45}\rangle$, and the data frame is a random sequence that includes the four SOP. Figure 6(a) shows the data qubits QBER of each one of the four SOP used in this work, considering single-photon detectors efficiency of $25 \%$ and a dark-count probability of $5 \times 10^{-4}$. The polarization compensation method actuated three times in this session. The QBER does not returned to zero since there are other error sources apart from the polarization random drift. Figure 6(b) shows the QBER of each one of the four SOP considering a $0.1 \mu$s dead-time of single-photon detectors. In this case, the average QBER is higher than the previous case as expected, since one more error source was added because the correlation time between samples was decreased. Nevertheless, in both cases presented in Fig. 6 we observe the robustness of our algorithm, since in both cases we observe a QBER for the data qubits lower than $2.1 \%$. Furthermore, a system parameter that has impact on the polarization random drift compensation method performance is the single-photon detectors dead-time. We consider the single-photon detectors are operating in gated mode, and the dead-time holds-off consecutive gates maintaining the bias voltage of the avalanche photo-diode well bellow the breakdown voltage [24]. The dead-time increases the time interval between consecutive samples, which can lead to uncorrelated consecutive samples [10].
Fig. 6. QBER time evolution for data qubits, considering a dark count probability of $5 \times 10^{-4}$ , a detection efficiency of $25\%$, and a dead-time of: a) - dead time null; b) - dead time equal to $0.1 \mu s$. An average QBER of $0.59\%$ and $0.65\%$ was calculated in a) and b), respectively.
4. Use case: BB84 protocol
In this section, we include the polarization random drift compensation method in a realistic QKD system, where the BB84 protocol is implemented [21].
4.1 Comparative analysis with a non-automatic compensation method
Lets assume a system that implements the BB84 protocol with polarization encoded single-photons using the linear and diagonal bases. In this case, we use a $5 \times 10^{4}$ qubits control frame with an alternated sequence of two mutually unbiased states of polarization, $|{H}\rangle$ and $|{45}\rangle$, and a $5 \times 10^{4}$ qubits data frame with the four randomly generated states of polarization from two non-orthogonal bases. For the method actuation boundary we chose a $\textrm {QBER}_{th} = 1 \%$. The sifted key is obtained from the events in which both Alice and Bob prepare and measure the qubits with the same basis, i.e. the sifted key is the key generated after basis reconciliation. The polarization control algorithm uses M bits of the sifted key (obtained after basis reconciliation) to estimate the sifted key QBER, and we call this procedure data check. After data check and discard the bits used for that step, a secret key is obtained. Figure 7 shows the QBER estimated using 1000 bits of the obtained sifted key when 50000 data qubits are transmitted. The qubits are prepared using an average of $\mu = 0.2$ photons per pulse and were exchanged throughout a quantum channel with $L = 40$ km length. The qubits measurement is performed using single-photon detectors with $25 \%$ efficiency, a dark-count probability of $P_{\textrm {dc}}=5 \times 10^{-4}$ and a dead-time of $0.1 \space \mu s$. As we can see in Fig. 7, the QBER without the proposed polarization compensation method increases with time, readily surpassing the security bound for the BB84 protocol which is close to $11\%$ [25]. We assume here that as soon as the QBER reaches the security boundary imposed by the protocol, an instantaneous compensation occurs. This is the best scenario we can consider for a compensation method, i.e. an instantaneous compensation time. On the contrary, the QBER using the proposed compensation method remains stable with a QBER lower than $2 \%$. In this way, even consuming $50 \%$ of extra bandwidth, the proposed method provides long-term key exchange between parties with an error bellow $2 \%$.
Fig. 7. QBER estimated using 1000 bits extracted from the sifted key with and without the actuation of the polarization random drift compensation method over time. The average number of photons per pulse at transmitter output is $\mu = 0.2$. The channel length between transmitter and receiver is $L = 40$ km with attenuation of $0.2$ dB/km. The dark-count probability of single-photon detectors is $P_{\textrm {dc}}=5 \times 10^{-4}$, and the dead-time of $0.1 \space \mu s$.
4.2 Impact of finite-key size effects on compensation method performance
We now analyze the length of the secret key as a function of the total number of transmitted qubits, including control and data qubits. For each session that produces non-zero secret key, we recorded the length of the sifted key, the number of control and data transmitted qubits, and the sifted key error rate. Finite-key unconditional security boundaries are used for the BB84 protocol for a practical prepare and measure implementation [26].
A set of sessions was recorded considering three different distances for the fiber-optics quantum channel, with and without the polarization random drift compensation method. We now analyze the numerical results taking into account the finite-key unconditional security boundaries defined for BB84 protocol prepare and measure implementations [27]. We adapt the eq. (5) presented in [28] to our case study, being the length of the final secret key defined as
Figure 8(a) shows the secret key length versus the total number of transmitted qubits for three different distances, with and without the actuation of the polarization random drift compensation method. As the fiber-optics quantum channel length increases the secret key generation is less efficient. However, for higher distances such as $40$ km and $60$ km the use of the polarization random drift compensation method considerably improves the efficiency of secret key generation since the average quantum bit error rate decreases. Figure 8(b) shows the secret key length versus the total number of transmitted qubits for different average number of photons per pulse in control qubits at the receiver input for a 40 km fiber-optic quantum channel. From the results in Fig. 8(b), we can see that for $\mu =0.2$ in the data qubits, the most efficient average number of photons in the control bits at receiver input is $\langle n_c \rangle = 0.2$. Note that all curves in Fig. 8(a) and (b) were obtained over the same time window acquisition. In Fig. 8(a), the curves that represent the "with compensation" correspond to scenarios where the QBER never reaches the limit defined by BB84 to produce a secret key, which means that even increasing the time window under analysis a secret key is continuously produced over long time. On the contrary, the curves that represents the "without compensation" correspond to scenarios where the QBER eventually reaches the limit imposed by BB84 to produce a secret key, and with an increasing of the time window the system eventually stops to produces a secret key even the qubits are transmitted over long time. In this way, a secret key improvement can be calculated taking into account a certain time window.
Fig. 8. Secret key length versus total number of transmitted qubits. (a) Numerical results for different fiber-optic channel lengths with and without polarization compensation method. (b) Numerical results for different average number of photons per pulse in control frame at the receiver input for a 40 km fiber-optics quantum channel.
4.3 Bandwidth consumption analysis
One important assessment measurement is the bandwidth consumption of the proposed polarization random drift compensation method. Here, we also present the final secret key length versus the total transmitted qubits. Figure 9(a) shows the secret key length as a function of the total transmitted qubits for different polarization random drift compensation method bandwidth consumption. The bandwidth consumption of $50 \%$ is the most advantageous since the secret key generation is more efficient requiring less transmitted qubits to generate a final secret key with the same size. Figure 9(b) shows the secret key ratio, calculated by dividing the secret key length by the total transmitted qubits required to generate it, versus the bandwidth consumption. Similarly, the most advantageous is the $50 \%$ bandwidth consumption since provides the highest secret key ratio. The reason for this is related to the polarization compensation method capacity of maintaining the lowest QBER when comparing with other bandwidth consumption. Even consuming $50 \%$ bandwidth less qubits needed to be transmitted to generate a secret key because the error correction codes are more efficient requiring less qubits to correct the errors.
Fig. 9. (a) Secret key length versus total transmitted qubits for three different polarization random drift compensation method bandwidth consumption. A fiber-optics quantum channel with 40 km is considered. (b) Secret key ratio versus bandwidth. The secret key ratio is calculated in relation to the total transmitted qubits.
5. Conclusion
We propose a polarization random drift compensation method able to compensate arbitrary SOP time drifts induced by the propagation over standard fiber-optic channel. We demonstrated that only monitoring the QBER induced by two states of polarization prepared from two different non-orthogonal mutually unbiased bases, the proposed method is able to compensate any SOP.
Moreover, we have shown that the implementation of the proposed method in discrete-variables polarization encoded based systems assures a long-term key exchanging between parties with a QBER lower than $2 \%$. Furthermore, the compensation method provides an average QBER bellow $2\%$ for a realistic system with a 40 km fiber-optics quantum channel, and average number of photons per pulse of $0.2$ at the transmitter output in the data frame, and $0.2$ at receiver’s input in the control frame. This QBER value is lower than a half of a system with no compensation method, which strongly impacts the secret key generation efficiency in BB84 protocol.
We have demonstrated that by employing the proposed method in a finite-key implementation, the secret key rate generation is improved in $82 \%$, even consuming part of the transmission bandwidth for polarization random drift compensation. To obtain this result, we divided the average secret key length, resulted from the case where no polarization compensation method was applied, by the average secret key length, resulted from the case where the proposed method was applied. Note that to obtain the secret key length in all situations we always used the same time transmission window. Regarding the bandwidth consumption, we analysed the secret key ratio for different bandwidths, and this analysis allowed us to conclude that the optimum value for bandwidth consumption is $50\%$, being the efficiency of BB84 higher for this value.
Funding
Fundação para a Ciência e a Tecnologia (Ph.D. Grant SFRH/BD/145670/2019, UIDB/50008/2020-UIDP/50008/2020 (action QUESTS)); European Regional Development Fund (Q.DOT Ref.POCI- 01-0247-FEDER-039728).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002). [CrossRef]
2. S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. L. Pereira, M. Razavi, J. S. Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden, “Advances in quantum cryptography,” Adv. Opt. Photonics 12(4), 1012–1236 (2020). [CrossRef]
3. R. Liu, H. Yu, J. Zan, S. Gao, L. Wang, M. Xu, J. Tao, J. Liu, Q. Chen, and Y. Zhao, “Analysis of polarization fluctuation in long-distance aerial fiber for QKD system design,” Opt. Fiber Technol. 48, 28–33 (2019). [CrossRef]
4. E. Diamanti, H.-K. Lo, B. Qi, and Z. Yuan, “Practical challenges in quantum key distribution,” npj Quantum Inf. 2(1), 16025 (2016). [CrossRef]
5. J. Dynes, A. Wonfor, W.-S. Tam, A. Sharpe, R. Takahashi, M. Lucamarini, A. Plews, Z. Yuan, A. Dixon, J. Cho, Y. Tanizawa, J. Elbers, H. Greisser, I. White, R. Penty, and A. Shields, “Cambridge quantum network,” npj Quantum Inf. 5(1), 101 (2019). [CrossRef]
6. S. K. Joshi, D. Aktas, S. Wengerowsky, M. Lončarić, S. P. Neumann, B. Liu, T. Scheidl, G. C. Lorenzo, Ž. Samec, L. Kling, A. Qiu, M. Razavi, M. Stipcevic, J. G. Rarity, and R. Ursin, “A trusted node–free eight-user metropolitan quantum communication network,” Sci. Adv. 6(36), eaba0959 (2020). [CrossRef]
7. Y.-Y. Ding, H. Chen, S. Wang, D.-Y. He, Z.-Q. Yin, W. Chen, Z. Zhou, G.-C. Guo, and Z.-F. Han, “Polarization variations in installed fibers and their influence on quantum key distribution systems,” Opt. Express 25(22), 27923–27936 (2017). [CrossRef]
8. G. Xavier, G. V. de Faria, G. Temporão, and J. Von der Weid, “Full polarization control for fiber optical quantum communication systems using polarization encoding,” Opt. Express 16(3), 1867–1873 (2008). [CrossRef]
9. D.-D. Li, S. Gao, G.-C. Li, L. Xue, L.-W. Wang, C.-B. Lu, Y. Xiang, Z.-Y. Zhao, L.-C. Yan, Z.-Y. Chen, G. Yu, and J.-H. Liu, “Field implementation of long-distance quantum key distribution over aerial fiber with fast polarization feedback,” Opt. Express 26(18), 22793–22800 (2018). [CrossRef]
10. N. J. Muga, M. F. Ferreira, and A. N. Pinto, “QBER estimation in QKD systems with polarization encoding,” J. Lightwave Technol. 29(3), 355–361 (2011). [CrossRef]
11. T. Pengyi, L. Guochun, G. Song, Y. Gang, D. Yunqi, X. Yao, L. Dongdong, Z. Yinghua, W. Bing, Z. Ziyan, G. Dequan, L. Jianhong, and W. Jian, “Fast polarization feedback algorithm for quantum key distribution with aerial fiber for power grid,” Acta Opt. Sin. 38(1), 0106005 (2018). [CrossRef]
12. I. Lucio-Martinez, P. Chan, X. Mo, S. Hosier, and W. Tittel, “Proof-of-concept of real-world quantum key distribution with quantum frames,” New J. Phys. 11(9), 095001 (2009). [CrossRef]
13. Y.-Y. Ding, W. Chen, H. Chen, C. Wang, S. Wang, Z.-Q. Yin, G.-C. Guo, and Z.-F. Han, “Polarization-basis tracking scheme for quantum key distribution using revealed sifted key bits,” Opt. Lett. 42(6), 1023–1026 (2017). [CrossRef]
14. Á J. Almeida, N. J. Muga, N. A. Silva, J. M. Prata, P. S. André, and A. N. Pinto, “Continuous control of random polarization rotations for quantum communications,” J. Lightwave Technol. 34(16), 3914–3922 (2016). [CrossRef]
15. Y. Shi, H. S. Poh, A. Ling, and C. Kurtsiefer, “Fibre polarization state compensation in entanglement-based quantum key distribution,” arXiv preprint arXiv:2107.07654 (2021).
16. M. F. Ramos, N. A. Silva, N. J. Muga, and A. N. Pinto, “Reversal operator to compensate polarization random drifts in quantum communications,” Opt. Express 28(4), 5035–5049 (2020). [CrossRef]
17. A. V. Sergienko, “Experimental cryptography using continuous polarization states,” in Quantum Communications and Cryptography, (CRC Press, 2018), pp. 111–133.
18. I. B. Djordjevic, Quantum-Key Distribution (QKD) Fundamentals (Springer International Publishing, Cham, 2019), pp. 211–265.
19. G. Xavier, N. Walenta, G. V. De Faria, G. Temporão, N. Gisin, H. Zbinden, and J. Von der Weid, “Experimental polarization encoded quantum key distribution over optical fibres with real-time continuous birefringence compensation,” New J. Phys. 11(4), 045015 (2009). [CrossRef]
20. D. H. Goldstein, Polarized Light, revised and expanded (CRC, 2003).
21. C. Bennett, “G. brassard in proceedings of ieee international conference on computers, systems, and signal processing, bangalore, india, new york,” (IEEE, 1984).
22. L. Xi, X. Zhang, F. Tian, X. Tang, X. Weng, G. Zhang, X. Li, and Q. Xiong, “Optimizing the operation of LiNbO3 -based multistage polarization controllers through an adaptive algorithm,” IEEE Photonics J. 2(2), 195–202 (2010). [CrossRef]
23. C. B. Czegledi, M. Karlsson, E. Agrell, and P. Johannisson, “Polarization drift channel model for coherent fibre-optic systems,” Sci. Rep. 6(1), 21217 (2016). [CrossRef]
24. G. Ribordy, N. Gisin, O. Guinnard, D. Stuck, M. Wegmuller, and H. Zbinden, “Photon counting at telecom wavelengths with commercial ingaas/inp avalanche photodiodes: current performance,” J. Mod. Opt. 51(9-10), 1381–1398 (2004). [CrossRef]
25. N. Lütkenhaus, “Security against individual attacks for realistic quantum key distribution,” Phys. Rev. A 61(5), 052304 (2000). [CrossRef]
26. R. Y. Cai and V. Scarani, “Finite-key analysis for practical implementations of quantum key distribution,” New J. Phys. 11(4), 045024 (2009). [CrossRef]
27. D. Bacco, M. Canale, N. Laurenti, G. Vallone, and P. Villoresi, “Experimental quantum key distribution with finite-key security analysis for noisy channels,” Nat. Commun. 4(1), 2363 (2013). [CrossRef]
28. P. Chaiwongkhot, S. Sajeed, L. Lydersen, and V. Makarov, “Finite-key-size effect in a commercial plug-and-play qkd system,” Quantum Sci. Technol. 2(4), 044003 (2017). [CrossRef]
