Continuous variables quantum key distribution (CV-QKD) allows for the generation and distribution of symmetric cryptographic keys without imposing any computational limitations on a supposed adversary, doing so while employing standard telecom equipment [1]. However, the security of CV-QKD is highly dependent on the excess noise level [1]. Different modulation parameters have different security limits, thus a careful choice of modulation scheme is of high importance.

Coherent-state CV-QKD typically encodes the information in the phase and amplitude of weak coherent states, thus allowing for implementation with current modulation methods [1]. The first implementations of CV-QKD protocols were carried out by using a transmitted local oscillator (LO) setup [2]. However, that was found to open security loopholes. As a result, local LO (LLO) techniques, are today the most common implementations of CV-QKD [2]. Lately, LLO CV-QKD implementations using single-sideband modulation with true heterodyne detection and using root-raised-cosine (RRC) signal modulation have been proposed, avoiding low-frequency and out-of-band noise [2]. Misalignments between the polarizations of the two laser fields interfering in the coherent detection scheme will severely reduce the efficiency of the detection scheme employed [3]. To tackle this, both active [3] and passive [4] solutions have been proposed. Coherent state CV-QKD can be grouped into Gaussian modulated (GM) or discrete modulated (DM) [1] methods. GM-CV-QKD based systems allow for maximizing the transmitted information, as a result exhibiting an optimal theoretical secure key rate and resistance to excess noise [1]. However, GM-CV-QKD protocols have historically exhibited low reconciliation efficiency [5,6], put an extreme burden on the transmitter’s random number source [7], and tend to be more susceptible to imperfect state preparation [8]. As a result, the majority of experimental work done in CV-QKD uses DM [5]. DM-CV-QKD started by using phase shift keying (PSK) constellations, 2-PSK and 4-PSK at first [1], followed quickly by the adoption of 8-PSK, since increasing the constellation size brings the performance of DM implementations closer to that of GM ones [9]. However, further increasing the PSK constellation does not produce an appreciable improvement [10]. Better results have been obtained by using M-symbol quadrature amplitude modulation (M-QAM) coupled with probabilistic constellation shaping (PCS) [11]. M-symbol amplitude- and phase-shift-keying (M-APSK) constellations can have the same number of symbols while using a smaller number of amplitude levels, thus exhibiting a peak-to-average power ratio 1.5 dB lower than that of an equivalent M-QAM constellation [12]. M-APSK constellations, also coupled with PCS, have been proposed for use in CV-QKD [6,13].

In this Letter we present, for the first time, as far as we know, an experimental implementation of a DM-CV-QKD scheme using two different PCS-128-APSK constellations, one regular and one irregular, coupled with a polarization diverse receiver setup employing true heterodyne detection. We present experimental results showing that, under the asymptotic regime and with an optical channel of 40 km, our system would be able to achieve 79% of the key rate performance of an equivalent GM system, performing secure transmissions using realistic, telecom-grade components. We further show that our system is capable of working over distances in excess of 185 km, in the asymptotic regime.

This work is organized as follows. We begin by describing our experimental system and the employed constellations. Secondly, we present and discuss experimental results extracted from the previously described system, comparing the two employed constellations with each other and to the 8-PSK and Gaussian ones. We finalize this work with a summary of the major conclusions.

A block diagram of our system is presented in Fig. 1. Alice starts by modulating the optical signal that she extracts from her local coherent source, which consists of a Yenista OSICS Band C/AG TLS laser, tuned to 1550.004 nm. We adopt RRC modulation because of the possibility of using matched filtering at the receiver without inter-symbol interference [14], thus allowing for optimum Gaussian white-noise minimization. The symbol rate was set at 153.6 MBd, with two PCS-128-APSK constellations chosen, one regular (all rings have the same number of states) and the other irregular (different rings have different number of states). The states for these constellations are chosen from the alphabet [6],

Within each ring, each state is equiprobable. The RRC signal is then up-converted in the transmitter to an intermediate frequency, $f_Q = 153.6$ MHz, and frequency multiplexed with a DC pilot tone, i.e., $f_P = 0$ Hz, which will be used for frequency and phase recovery at the receiver. This signal is fed into a Texas Instruments DAC39J84EVM digital to analog converter (DAC), which in turn drives a u2t Photonics 32 GHz IQ modulator coupled with a SHF807 RF amplifier. The modulated signal is then attenuated using a Thorlabs EVOA1550F variable optical attenuator until the signal has, on average, 1.29 photons per symbol for the irregular and 1.91 photons per symbol for the regular constellation. The signal is then sent through a single-mode fiber spool with length 40 km before arriving at the receiver. At the receiver side, the signal is fed into a polarization diverse receiver, where it is mixed with the LLO. The LLO consists of a Yenista OSICS Band C/AG TLS laser tuned to 1549.999 nm. In this situation, the signals have a frequency shift of $f_S\approx 800$ MHz. The mixed optical signals are evaluated by a pair of Thorlabs PDB480C-AC balanced optical receivers, connected to the inputs of a Texas Instruments ADC32RF45EVM analog to digital converter (ADC) board, which is running at a sample rate of 2.4576 GS/s. The digitized signals are then fed into the digital signal processing (DSP) stage, where they are subjected to frequency, phase and clock recovery, steps which are aided by the pilot tone inserted at $f_P$, and matched filtering. For a more detailed description of the polarization diverse receiver, see Ref. [4]. The state sequences present at the transmitter and receiver are synchronized through the use of a known header of 3000 states inserted at the start of the sequence by Alice.

 

Fig. 1. Block diagram of the experimental system, including a representation of the two employed PCS-128-APSK constellations.

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The security of DM-CV-QKD against collective attacks was first established in Ref. [1] and has since been updated in Ref. [10]. In particular, the security of CV-QKD against collective attacks using APSK modulation was explored in our previous work in Ref. [6], from which we take the methodology followed in this work. The achievable secure key rate is given by

The transmission and excess noise are then estimated through

For the results presented in this work, the finite size effect is not taken into consideration due to memory limitations. The experimental system was run continuously for roughly one and a half hours, with 7 ms snapshots taken every 30 s. The time between acquisitions was due to the limitations of the ADC available in our laboratory. A total of 200 snapshots were taken, each containing 1 048 576 symbols. In order to achieve security in the finite size scenario with our constellation formats, approximately eight times as many symbols would have had to have been captured in order to achieve security with a $10^{-10}$ confidence level [15], which is not possible due to our post-processing hardware capabilities. This, coupled with the fact that post-processing is done offline, means that our system remains a proof of concept one, albeit one with very promising results.

In Fig. 2 we present the evolution of the excess noise as a function of the acquisition time. The theoretical excess noise limits for a secure transmission using 8-PSK, irregular and regular PCS-128-APSK and Gaussian constellations, at a transmission distance of 40 km, are also included, as a dashed line, a dotted line, a dash–dot line, and a full line, respectively. For the theoretical limits of the Gaussian and 8-PSK constellations, 2.40 and 0.33 photons per symbol were assumed, respectively. When a point is located below a certain line, it would mean a transmission in those conditions would be secure. We can see that there is a considerable increase from the security limit of the 8-PSK to that of the irregular PCS-128-APSK, meaning that the system is able to transmit secure keys $1.9\times$ more often than if it were using an 8-PSK constellation, all while using the same, fully telecom-grade equipment. Meanwhile, switching to a GM system would only lead to a $1.1\times$ increase over the irregular PCS-128-APSK constellation in the number of secure keys, a less dramatic increase. Note that the system was not finely tuned beforehand, so we believe that this experiment decently approximates a real-world scenario. Crucially, our system was able to generate secure keys for the duration of the acquisition time. We also perform a comparative study between the irregular and regular PCS-128-APSK constellations, also shown in Fig. 2, where a graphic inset shows the region immediately around the security limit in greater detail. For the situation presented in this work, the use of a regular constellation would lead to only a 0.5% increase in the number of transmitted keys. However small this increase, it is achieved with only a small alteration in the pre- and post-processing algorithms and both retain the same advantage in ease of implementation when compared with the GM case. Again, in this scenario our system was able to generate secure keys for the duration of the experiment.

 

Fig. 2. Evolution of the estimated excess noise for the 200 snapshots taken of both the irregular and regular 128-APSK constellations. Theoretical limits for the 8-PSK, 128-APSK, irregular and regular, and Gaussian constellations are included. A zoomed inset, showing the region between the theoretical limits of the 128-APSK constellations and the Gaussian one, is also included.

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In Fig. 3, the secure key rate, obtained through Eq. (3), as a function of the observed excess noise is presented. Experimental results, using instantaneous measurements of the channel transmission, excess noise, and thermal noise, are presented as dots and squares, for the irregular and regular PCS-128-APSK constellations, respectively. The trend lines for the irregular PCS-128-APSK, regular PCS-128-APSK, and Gaussian constellations are presented as a dashed line, a dash–dot line, and a full line, respectively. In obtaining these theoretical trend lines, the average observed channel transmission, $\tilde {T}=0.142$, and the average observed thermal noise, $\epsilon _\text {thermal}=0.35$ SNU, are used. From Fig. 3, we can see that the regular constellation has a slightly higher resistance to excess noise when compared with the irregular one, and that in both cases the performance of our system as a function of excess noise is quite close to that of an equivalent GM system, when sufficiently far away from the noise limit value, with the advantage of a simpler key reconciliation stage.

 

Fig. 3. Secure key rate, given by Eq. (3) with $\beta =0.95$, as a function of excess noise. Experimental results identified as dots and squares; lines indicate the theoretical secure key rate for the PCS-128-APSK irregular, PCS-128-APSK regular, and Gaussian constellations. The trend lines for the 128-APSK and Gaussian constellations use the mean transmission and electrical noise observed experimentally.

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In Fig. 4 we present the secure key rate as a function of the transmission distance. The mean experimental result for the regular PCS-128-APSK constellation is presented as a star, with its theoretical line consisting of a dash–dot line, the mean experimental result for the irregular PCS-128-APSK constellation is presented as a square, with its theoretical line consisting of a dashed line, and the theoretical curve for a Gaussian constellation is presented as a full line. The average excess noise in each experiment, 0.035 SNU in the irregular and 0.026 SNU in the regular one, was used for their corresponding theoretical curve, while for the GM curve the average of the two excess noises was used. For all of these, the asymptotic regime was assumed. Here we see that, under the observed experimental parameters, the regular PCS-128-APSK constellation would theoretically be able to reach distances of upwards of 185 km, while the irregular would only reach 129 km, albeit requiring the use of roughly $2^{53}$ symbols during parameter estimation in order to achieve security under a finite size scenario [15], this would necessitate data transmission for much longer than the channel parameters can be considered to remain stable. This indicates that, in a practical scenario, considering finite size effects and assuming a channel stability time of the order of seconds, achievable distances of around 60 km are expected. We see here that the experimental performance of the regular constellation approaches that of the GM one, achieving 79% of the latter’s performance, while the irregular constellation achieves only 58% of the GM system’s performance. These discrepancies are in part due to the different mean excess noise observed in each scenario. However, even when under the same channel parameters, regular constellations exhibit better performance than irregular ones with the same cardinality [13].

 

Fig. 4. Secure key rate, given by Eq. (3) with $\beta =0.95$, as a function of the transmission distance. Mean experimental result indicated as a star for the regular and as a square for the irregular 128-APSK scenario. The theoretical results for the 128-APSK constellations use the mean excess noise observed experimentally in each scenario, with the Gaussian results using the average of the noise observed in the other two.

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In summary, we present experimental results from our DM-CV-QKD communication system, using the PCS-128-APSK constellation format and employing only realistic, telecom-grade components. Our presented system greatly improves on the capability of M-PSK based systems, being able to work in worse conditions, in terms of excess noise, and our experimental results show that it is capable of reaching 79% of the performance of GM-CV-QKD at a distance of 40 km, thus being suitable for both metro network connections and for some short- to medium-range inter-city connections, provided that a substantial but ultimately manageable increase in the block size is performed. Furthermore, we show that, in the asymptotic regime, our system is capable of reaching distances in excess of 185 km, and, as a result, is compatible with medium- to long-range inter-city connections.