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In quantum key distribution (QKD), the times of arrival of single photons are important for the keys extraction and time synchronization. The time-of-arrival (TOA) accuracy can affect the quantum bit error rate (QBER) and the final key rate. To achieve a higher accuracy and a better QKD performance, different from designing more complicated hardware circuits, we present a scheme that uses the mean TOA of M frequency-entangled photons to replace the TOA of a single photon. Moreover, to address the problem that the entanglement property is usually sensitive to the photon loss in practice, we further propose two schemes, which adopt partially entangled photons and grouping-entangled photons, respectively. In addition, we compare the effects of these three alternative schemes on the QKD performance and discuss the selection strategy for the optimal scheme in detail. The simulation results show that the proposed schemes can improve the QKD performance compared to the conventional single-photon scheme obviously, which demonstrate the effectiveness of the proposed schemes.
In recent decades, the free space quantum key distribution (QKD)[1–3] has become a matured technic in the quantum communication field.[4–6] In the free space QKD, a crucial problem is to analyze the effects of practical noises on the QKD performance. The decrease in the number of signal photons caused by the transmission losses and the increase in the number of noise photons caused by background light commonly decide a lower signal-to-noise ratio (SNR).[7,8] A lower SNR will directly bring a higher quantum bit error ratio (QBER) and a lower key rate. It is not difficult to see that the effects of actual noises on the QKD performance essentially are its effects on the single photons. As the information carrier of QKD, the single photons play an important role in the keys extraction and time synchronization. Especially, the accuracy of the time of arrival (TOA) of a single photon can further affect the QBER and the final key rate. Specifically, the single photons sent by Alice are probed by single photon detectors (SPDs) of Bob. Alice and Bob have independent reference clocks and the difference cannot be completely eliminated. Then Bob records the TOAs of all the output signals of the SPDs using the time-to-digital convertor (TDC) and saves the TDC data to a local memory.[9,10] Therefore, the TOA accuracy reflects the system timing precision, which is affected by the time synchronization imperfection[11,12] and the system jitters caused by the single photons source,[13] detection device,[14,15] TDC,[10,16,17] etc. Moreover, these factors bring an inevitable result that the temporal distribution of the single-photon pulse is expanded. The width of the corresponding detection time window has to be set wider. For example, in Ref. [1], the cumulative effects of the timing jitter of the Alice electronics, reference clock noise, timing jitter of the photodetector, and noise in the timestamp unit led to a temporal distribution of signal events with a full width half maximum (FWHM) of 4 ns. Another work in Ref. [2], the cumulative effects of the timing jitter of the Alice electronics, GPS noise, timing jitter of the detector, timing jitter of TDC and other temporal noise led to a total temporal precision of signal events with a FWHM of 1.1 ns. At last, a lower SNR leads to an unsatisfactory QKD performance. Therefore, how to promote the TOA accuracy to improve the system timing precision, thereby optimizing the QKD performance is worth studying. This exactly is the original point of this paper.
To improve the TOA accuracy of a single photon, the conventional method is to design a kind of TDC with a higher precision, smaller dead time and lower jitter,[10,16] such as the TDC based on the field-programmable-gate-array (FPGA).[17–19] The FPGA-TDC with the resolution of 50 ps and the dead time of 30 ns, has been successfully used in the moving platform QKD experiment over 40 km, the floating platform QKD experiment over 20 km and the high-loss free space QKD experiment over 96 km.[2,10] However, the hardware complexity of designing the high-precision TDC makes this work difficult to be implemented easily. Moreover, the TOAs of the single photons obtained from this way would be inevitably limited by the hardware conditions. Therefore, the TOA accuracy could not be greatly improved through this conventional method.
In recent years, quantum entanglement has been maturely applied in quantum information processing, especially for the design of some quantum protocols, such as the quantum secret sharing protocol,[20,21] the quantum private comparison of equality (QPCE) protocol,[22,23] the quantum key management protocol,[24] and quantum signature protocol.[25] Therefore, how to use the entanglement nature to improve the TOA accuracy is an interesting problem. To avoid designing a high-precision TDC, in this paper, we use M entangled photons to achieve an improvement in the TOA accuracy and the performance optimization for the QKD system.
At first, a simple but efficient method is using M unentangled photons to replace a single photon, whose TOA accuracy is certainly better than that of a single photon. The entanglement can bring a better TOA accuracy in principle,[26,27] so employing M frequency-entangled photons has become an effective approach to improve the QKD performance. Nevertheless, the main drawback that entanglement is sensitive to the photon loss[28–30] is also inherited into this improved QKD scheme. If one or more of the photons fails to arrive at the detector because of the atmosphere attenuation and background noises, the M fully entangled photons will become useless for the QKD process. Addressing this problem, we further propose two schemes using partially entangled photons and grouping-entangled photons respectively.[31–33] These two entanglement schemes can enhance the TOA accuracy while resisting the photon loss. The TOA accuracy obtained from the partially entangled photons or the grouping-entangled photons is worse than that which stems from the fully entangled photons, the robustness of QKD would still be greatly improved. In this paper, we concretely introduce the partially entangled photons scheme and the grouping-entangled photons scheme. In the case of an ideal channel and of a lossy channel, the effects of these schemes on the QBER and the final key rate are further analyzed. Finally, the selection for the optimal scheme in the lossy channel, which depends on the comparisons of the QBER performance among these three schemes, is also presented.
Regarding a QKD system, it is important to record the arrival time of a single photon that is used for syncing Alice and Bob as well as performing keys extraction. Different from the conventional hardware approaches, we make use of the average TOA of M fully entangled photons to replace the TOA of a single photon. However, while improving the TOA accuracy of a single photon, this method also inherits the drawback of entanglement. In practice, once an entangled photon is lost, the TOA accuracy afforded by the entanglement scheme will become zero, which is fatal for the fully entangled photons scheme. For the sensitivity to loss, one may use the partially entangled photons to increase the resistance to loss, although at the cost of the TOA accuracy. The remaining TOA accuracy obtained from the partially entangled photons, however, is still better than that from the single photons.
Although the fully entangled photons scheme is easy to be affected by the photon loss, its accuracy is the highest of the four schemes. These schemes include the fully entangled photons scheme, the partially entangled photons scheme, the unentangled photons scheme, and the ordinary single photon scheme. Therefore, it is necessary to analyze the fully entangled photons scheme.
The TOA accuracy of M unentangled photons is
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![](cpb_26_11_110305/cpb_26_11_110305_ieqn1.gif)
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On the other hand, the TOA uncertainty of M frequency-entangled photons comes from[31]
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Therefore, the gain in the time-of-arrival accuracy can be denoted as
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![](cpb_26_11_110305/cpb_26_11_110305_ieqn13.gif)
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To protect against the photon loss, we use partially entangled photons to increase the robustness of QKD. By contrast with the fully frequency-entangled photons, the first Q of the M photons are maximally entangled and the other
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![](cpb_26_11_110305/cpb_26_11_110305_ieqn17.gif)
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![](cpb_26_11_110305/cpb_26_11_110305_ieqn19.gif)
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However, there is a problem that needs our attention. The above discussion for the accuracy is based on the unit quantum efficiency η = 1. Here, the channel quantum efficiency η expresses a possibility that one photon can be detected by Bob’s receiver. The probability that all M photons reach Bob is given by ηM. Since the atmosphere attenuation and background noise, not all the M photons can be detected, namely
For the fully entangled photons scheme, the uncertainty of the average TOA of M photons is given by
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Therefore, the accuracy in the single-photon scheme is represented as
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Besides, different from the principle of the partial entanglement scheme, quantum fault computation can be applied into calculating the TOA accuracy.[31] As shown in Fig.
The TOA accuracy in this grouping entanglement scheme is given by
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![](cpb_26_11_110305/cpb_26_11_110305_ieqn23.gif)
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![](cpb_26_11_110305/cpb_26_11_110305_ieqn25.gif)
![](cpb_26_11_110305/cpb_26_11_110305_ieqn26.gif)
![](cpb_26_11_110305/cpb_26_11_110305_ieqn27.gif)
![](cpb_26_11_110305/cpb_26_11_110305_ieqn28.gif)
![](cpb_26_11_110305/cpb_26_11_110305_ieqn29.gif)
![](cpb_26_11_110305/cpb_26_11_110305_ieqn30.gif)
![](cpb_26_11_110305/cpb_26_11_110305_ieqn31.gif)
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According to the above discussion, the accuracy gains of these three schemes for
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By virtue of the improvement on the time-of-arrival accuracy, the system timing precision of QKD can be increased effectively. Moreover, the increase of
Denoting μ,
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![](cpb_26_11_110305/cpb_26_11_110305_ieqn37.gif)
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In addition, we denote
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![](cpb_26_11_110305/cpb_26_11_110305_ieqn41.gif)
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![](cpb_26_11_110305/cpb_26_11_110305_ieqn42.gif)
![](cpb_26_11_110305/cpb_26_11_110305_ieqn43.gif)
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The improvement of QKD performance on the ideal channel (η = 1) is the theoretical basis of that on the lossy channel (
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![](cpb_26_11_110305/cpb_26_11_110305_ieqn47.gif)
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The analysis of the key rate is analogous to that of the QBER. Inserting Eq. (
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The improvement of the key rate obtained from the partially entangled photons is shown in Fig.
Due to the presence of the atmosphere attenuation and background noises, losing one or more entangled photons is easy to happen.[28,37] This phenomenon can make the fully entangled photons scheme completely useless for the improvement of QKD performance. Since the loss of a single photon from a maximally entangled state would destroy the entanglement property, the gain in time-of-arrival accuracy afforded by the fully entangled photons scheme would become zero. Therefore, in the case of a lossy channel, the partial entanglement scheme, particularly the grouping entanglement scheme has become an effective alternative. According to Fig.
In G groups, if the lost photon comes from the jth group, only the jth group’s photon arrival time needs to be discarded, the other TOA data may be retained and employed. On the contrary, all the TOA data need to be abandoned, if M photons are fully entangled. We use Eq. (
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To describe the effects of the grouping entanglement scheme on the QKD performance, the relationship among G, η, and
According to the above theoretical analysis and simulations, in reality, the partially entangled photons scheme, especially the grouping-entangled photons scheme can battle the effects of photon loss, enhance the robustness of QKD, and improve the QBER and key rate. It seems like an effective method to replace the conventional single-photon scheme, but it is not always the optimal choice under the practical circumstances. This optimal scheme stems from three different schemes: the grouping-entangled photons scheme, the fully entangled M-photon scheme, and the unentangled M-photon scheme. Moreover, it can bring the highest TOA accuracy and the greatest QKD performance compared to other unselected schemes. In this section, the selection strategy for the optimal scheme is presented.
In fact, the selection for the optimal scheme depends on the relationship between the channel quantum efficiency η and the number of photons M. For example,
The gain in TOA accuracy can directly influence the system temporal precision of QKD, so we need to determine an optimal scheme that can bring the best QKD performance compared to other schemes. To show the effects of the relationship between η and M on the QKD performance, we insert Eqs. (
In Fig.
In this paper, we have used M fully entangled photons to achieve an enhancement in the time-of-arrival accuracy, which can decrease the width of the detection time window, increase the system temporal precision of QKD, and thereby improve the QBER and the final key rate. Most importantly, considering the adverse effects of photon loss on the entanglement in reality, we have employed M grouping-entangled photons to increase the robustness of QKD. In the case of a lossy channel, the selection for the optimal scheme has also been investigated, which comes from three alternative schemes presented in this paper. However, no matter which scheme is selected from the grouping-entangled photons scheme and the unentangled photons scheme, its QKD performance is always better than the results obtained from the conventional single-photon scheme.
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