Robust continuous-variable quantum key distribution in the finite-size regime

Yuehan Xu; Tao Wang; Xiaojuan Liao; Yingming Zhou; Peng Huang; Guihua Zeng

doi:10.1364/PRJ.530729

1. INTRODUCTION

In order to achieve secure communication, current communication systems generally implement key distribution through public-key cryptography based on computational problems [1]. However, the rapid development of quantum computers renders these methods insecure in the future [2]. Quantum key distribution (QKD) utilizes the principles of quantum mechanics to provide theoretically secure keys that cannot be obtained by Eve [3]. However, due to the limitations of distance on key rate, the transmission distance of QKD falls far short of that of classical communication [4]. Therefore, the development of long-distance QKD systems has become a research focus [5 –13].

Many researchers have demonstrated excellent achievements in long-distance QKD. Currently, the distance of discrete-variable QKD (DV-QKD) can reach up to 830 km [12] and even 1002 km [13]. Although the system of continuous-variable QKD (CV-QKD) is simpler compared to DV-QKD, the challenging noise control makes it difficult to match the transmission distance of DV-QKD. The current CV-QKD can achieve distances of 150 km [6] and even 202 km [7]. The aforementioned long-distance CV-QKD systems are all constructed with a transmitted local oscillator (TLO). TLO systems generally have practical security issues, such as side-channel attacks [14,15]. These issues can be addressed by a local local oscillator (LLO) [16,17]. However, the LLO system will lead to difficulties in frequency and phase locking between different lasers. Although researchers have previously proposed some LLO systems to overcome these problems [8 –10], LLO systems continue to exhibit challenges in terms of high noise and low stability.

Under the effect of finite size, the problem of LLO will be further amplified. The finite-size effect stems from an assumption in the theoretical proof of QKD, which states that Alice and Bob can exchange an infinite number of signals in the asymptotic case of QKD. In practical implementations, this assumption is not available, as the block size cannot be infinite. Therefore, we need to consider the impact of finite size on the CV-QKD system [18]. In general, the transmission distance of CV-QKD under the finite-size effect is significantly shorter than that in the asymptotic case. In particular, one does not know in advance the characteristics of the quantum channel, which are known to lie inside some confidence regions. To make QKD more secure, parties must use lower transmittance and higher excess noise under a safer boundary. Therefore, the long-distance finite-size CV-QKD can be achieved only if the block size is high enough. When this issue is transferred to practical implementation, it will become a requirement for a longer continuous sampling time and larger bandwidth. Additionally, the system needs to maintain lower excess noise within the longer block. In the LLO system, this problem is particularly severe [19]. The sharp and fast frequency offsets and phase variations bring higher excess noise and more complex processing, leading to difficulties in generating secret keys with long blocks. Therefore, building a robust CV-QKD system to resist channel disturbances, especially in the case of large data blocks, is a challenge.

In order to address this issue, we propose a robust LLO CV-QKD system to realize secret key generation in the finite-size regime. Under time-variant parameter estimation, this system can achieve lower excess noise under arbitrary frequency shift and phase drift. First, the theory of this scheme is derived under the temporal modes of continuous-mode states. Then, we conduct experimental validation of this method. To simulate the frequency shift in field-test experiments, we use two lasers without locked frequencies. For the phase drift, we adopt a piezoelectric transducer (PZT) to simulate the severe disturbance experienced in the field-test optical fiber [20,21], which is often used for simulating vibrations [11,22,23]. The frequency shift is as rapid as 89.05 MHz/s, and the phase drift is as fast as 3.036 Mrad/s. Under severe environmental disturbances, the signal-to-interference ratio (SIR) is $- 51.67 dB$ . Through experimental verification, this method ensures that the quantum signal is not affected by these time-variant interferences. This solution achieves time-variant parameter evaluation by utilizing real-time calibration with a high-frequency pilot. In addition, the time complexity of this solution is independent of the block size, indicated as $O (1)$ . It takes less time and achieves lower excess noise. The achieved secret key rate is 0.29 Mbits/s under an attenuation of 16 dB or 80 km standard optical fiber. This achievement lays the foundation for subsequent field-test experiments in the finite-size regime.

In this paper, robust CV-QKD is introduced in detail. First, we describe the background of long-distance CV-QKD and our scheme. In addition, theoretical principles are derived under the temporal modes of continuous-mode states. Based on this optical structure and theoretical model, a proof-of-principle experiment is constructed to verify the feasibility of this method. Thus, we present the setup and results of this experiment. Finally, we draw a conclusion.

2. ROBUST CV-QKD IN THE FINITE-SIZE REGIME

In recent years, long-distance CV-QKD has developed rapidly. Researchers have verified many experiments, some of which are shown in Table 1. Jouguet et al. overcame many limitations and completed a CV-QKD experiment with a distance of 80.5 km and an attenuation of 16.1 dB [5]. Huang et al. achieved a CV-QKD experiment with a distance of 150 km and an attenuation of 30 dB by controlling system excess noise [6]. Zhang et al. accomplished work with a distance of 202.81 km and an attenuation of 32.45 dB using a fully automatic control system and a high-precision phase compensation method [7]. All the works mentioned above are based on TLO. Due to the significant practical security advantages of LLO over TLO, more and more LLO works are emerging. Hajomer et al. used machine learning for phase noise compensation and completed an experiment with a distance of 60 km and an attenuation of 8.76 dB [8]. Pi et al. used polarization-multiplexed pilot signals to compensate for the impact of phase noise, ultimately achieving a CV-QKD distance of 100 km and an attenuation of 18 dB [9]. Recently, Hajomer et al. achieved a finite-size experiment with a distance of 100 km and an attenuation of 15.4 dB [10]. The tasks mentioned above were all completed under the condition of stable channels. However, under the circumstances of unstable channels, it is essential to determine how to compensate for the signals and achieve coding with the finite-size effect.Table 1.

Long-Distance Continuous-Variable Quantum Key Distribution Experiments^a

Reference	Type	Attenuation	Distance	Secret Key Rate	Security	Interference
Jouguet et al. [5]	TLO	16.1 dB	80.5 km	200 bits/s	Finite-size	No
Huang et al. [6]	TLO	30 dB	150 km	40 bits/s	Finite-size	No
Zhang et al. [7]	TLO	32.45 dB	202.81 km	6.214 bits/s	Finite-size	No
Hajomer et al. [8]	LLO	8.76 dB	60 km	47.1 kbits/s	Asymptotic	No
Pi et al. [9]	LLO	18 dB	100 km	0.51 Mbits/s	Asymptotic	No
Hajomer et al. [10]	LLO	15.4 dB	100 km	0.03 Mbits/s	Finite-size	No
Current work	LLO	16 dB	80 km	0.29 Mbits/s	Finite-size	Yes

It can be mainly divided into transmitted local oscillator (TLO) and local local oscillator (LLO). Under the finite-size effect, the current work is the largest attenuation experiment in LLO CV-QKD. In order to demonstrate the system’s robustness, we also simulated various interferences that occur in field-test experiments.

The finite-size effect requires that the CV-QKD system should achieve stable low noise in the case of continuous sampling with a long block. In traditional CV-QKD channel estimation, it is generally assumed that the physical system within a frame is time-invariant. Based on this assumption, signal recovery can be performed easily. However, in the case of longer sampling time, if we use time-invariant parameter estimation, there will be a problem of higher noise or even inability to form a secret key. Therefore, the physical system cannot be considered time-invariant anymore. In this case, it is needed to implement an LLO CV-QKD system based on time-variant parameter estimation to achieve a longer transmission distance. By utilizing the method of real-time parameter calibration through high-speed pilot signals, we have achieved the highest attenuation of LLO CV-QKD. Moreover, the interference from field-test scenarios has been incorporated, such as rapid frequency shift and fast phase drift. Through theoretical analysis and experimental verification, we have accomplished accurate signal recovery in the presence of interference. Furthermore, the proposed method exhibits a time complexity of $O (1)$ , indicating that it is not affected by the block size.

The LLO CV-QKD system that we constructed is shown in Fig. 1. First, Alice’s laser 1 generates signal light that is transmitted to the in-phase and quadrature modulator (IQM). IQM performs base-band Gaussian modulation. Then, the modulated coherent state is split into two parts by the beam splitter (BS). A part of the light is transmitted to the automatic bias controller (ABC) for adjusting the bias of IQM, while another part is transmitted to the variable optical attenuator (VOA). After appropriate adjustment by VOA, the coherent state is sent into the optical fiber channel. In order to simulate the field-test environment, we have introduced PZT for vibration in the optical fiber channel. After passing through the optical fiber channel, the coherent state enters the polarization controller (PC) for polarization adjustment. Finally, the quantum signal outputs from the PC and local oscillator (LO) from Bob’s laser 2 are jointly input into the integrated coherent receiver (ICR) for coherent detection.

Figure 1.Finite-size and LLO CV-QKD optical layout. The signal light generated by Alice’s laser 1 is transmitted to the in-phase and quadrature modulator (IQM). Then, the modulated coherent state is divided into two parts by a beam splitter (BS). One part of the light is transmitted to the automatic bias controller (ABC), while the other part is transmitted to the variable optical attenuator (VOA). After intensity adjustment by VOA, the light is transmitted into the optical fiber channel. We introduced a piezoelectric transducer (PZT) in the optical fiber channel to simulate the field-test environment. Then, the coherent state enters the polarization controller (PC) for polarization adjustment. Finally, the quantum signal outputs from the PC and the local oscillator (LO) of Bob’s laser 2 are simultaneously input into the integrated coherent receiver (ICR) for coherent detection.

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3. TEMPORAL MODES OF CONTINUOUS-MODE STATES

A. Theoretical Principles

Compared to discrete mode, continuous mode better describes the non-uniform temporal waveforms induced by high-speed modulation in practical systems. Below, we will provide basic definitions for temporal modes of continuous-mode states. Referring to Refs. [24 –28], the annihilation operator $\hat{a}$ and creation operator ${\hat{a}}^{†}$ in the continuous-mode field are defined as ${\hat{a}}_{i} \to {(Δ ω)}^{\frac{1}{2}} \hat{a} (ω), {\hat{a}}_{i}^{†} \to {(Δ ω)}^{\frac{1}{2}} {\hat{a}}^{†} (ω),$ (1)where ${\hat{a}}_{i}$ and ${\hat{a}}_{i}^{†}$ denote the discrete-mode operators of the $i$ mode, and $Δ ω$ represents the mode spacing. They satisfy the commutation relation $[\hat{a} (ω), {\hat{a}}^{†} (ω^{'})] = δ (ω - ω^{'}),$ (2)in which $δ (\cdot)$ is the Dirac function. In the time domain, $\hat{a} (t)$ and ${\hat{a}}^{†} (t)$ can be obtained by the Fourier transforms $\hat{a} (t) = \frac{1}{\sqrt{2 π}} \int d ω \hat{a} (ω) \exp (- j ω t), {\hat{a}}^{⏧} (t) = \frac{1}{\sqrt{2 π}} \int d ω {\hat{a}}^{⏧} (ω) \exp (j ω t),$ (3)where $j$ is the unit imaginary number. For a photon-wavepacket, its annihilation operator ${\hat{A}}_{ξ_{i}}$ and creation operator ${\hat{A}}_{ξ_{i}}^{†}$ can be written as ${\hat{A}}_{ξ_{i}} = \int d t ξ_{i}^{*} (t) \hat{a} (t), {\hat{A}}_{ξ_{i}}^{⏧} = \int d t ξ_{i} (t) \hat{a} (t) .$ (4)In the above formula, $ξ_{i} (t)$ can be described as envelope $ξ_{i}^{0} (t) e^{- j ω (t) t}$ , in which $ω (t)$ is the optical carrier. If the temporal-mode field operator $ξ_{i} (t)$ meets the orthonormalization that $\int d t ξ_{i}^{*} (t) ξ_{i^{'}} (t) = δ_{i i^{'}}$ , for different modes $i, i^{'}$ , the temporal-mode operators also obey the commutation relation, which reads $[{\hat{A}}_{ξ_{i}}, {\hat{A}}_{ξ_{i^{'}}}^{†}] = δ_{i i^{'}} .$ (5)If we consider the coherent state as a unitary displacement operator, the photon-wavepacket coherent state ${| γ_{i} ⟩}_{ξ_{i}}$ on $ξ_{i}$ temporal mode can be represented as ${| γ_{i} ⟩}_{ξ_{i}} = {\hat{D}}_{ξ_{i}} (γ_{i}) | 0 ⟩ = \exp (γ_{i} {\hat{A}}_{ξ_{i}}^{†} - γ_{i}^{*} {\hat{A}}_{ξ_{i}}) | 0 ⟩,$ (6)in which $\hat{D} (\cdot)$ is the unitary displacement operator, $γ_{i}$ denotes the displacement parameter, and ${| γ_{i} |}^{2}$ represents the average photon number. The photon-wavepacket coherent state obeys the eigenvalue equation ${\hat{A}}_{ξ_{i}} {| γ_{i} ⟩}_{ξ_{i}} = γ_{i} {| γ_{i} ⟩}_{ξ_{i}}$ . Under these circumstances, the quadrature operators ${\hat{X}}_{ξ_{i}}$ and ${\hat{P}}_{ξ_{i}}$ can be defined as ${\hat{X}}_{ξ_{i}} = {\hat{A}}_{ξ_{i}}^{†} + {\hat{A}}_{ξ_{i}}, {\hat{P}}_{ξ_{i}} = j {\hat{A}}_{ξ_{i}}^{†} - j {\hat{A}}_{ξ_{i}} .$ (7)

B. Time-Variant Parameter Estimation

According to theoretical principles for temporal modes of continuous-mode states [28], we can provide the theoretical derivation of time-variant parameter estimation in LLO CV-QKD under the finite-size effect. The following derivation will be sequentially described from the aspects of modulation, transmission, and detection.

In the system shown in Fig. 1, the photon-wavepacket coherent state generated by laser 1 of Alice can be expressed as ${| γ_{0} ⟩}_{ξ_{s}}$ , where $ξ_{s} (t) = ξ_{s}^{0} (t) e^{- j ω_{s} (t) t}$ and $ω_{s} (t)$ is the optical carrier of signal. It is assumed that the IQM, after bias control through ABC, is an ideal balanced and linear quadrature modulator. We use frequency division multiplexing (FDM) to load the quantum signal and pilot signal on the same envelope $ξ_{s} (t)$ . This is equivalent to dividing the photons into two parts for the modulation of the quantum signal and the pilot signal separately. Thus, the photon-wavepacket coherent state used for quantum signal modulation on the $I$ or $Q$ component is ${| γ_{m} ⟩}_{ξ_{s}}$ , where ${| γ_{m} |}^{2} = {| γ_{0} |}^{2} / 4$ . It is assumed that the block size is $n$ , the start time of modulation is $t_{0}$ , and the base-band modulation rate is $F_{m}$ . At time $t \in (t_{0}, t_{0} + n / F_{m})$ , Alice encodes quantum signals ${x_{t}}$ and ${p_{t}}$ , respectively, on the $I$ and $Q$ components, in which ${x_{t}}$ and ${p_{t}}$ satisfy Gaussian distribution with variance $V_{A}$ and mean zero. It means that Alice performs a base-band Gaussian modulation of block size $n$ . The photon-wavepacket coherent state after quantum signal modulation can be represented as ${| x_{t} γ_{m} + j p_{t} γ_{m} ⟩}_{ξ_{s}} .$ (8)Meanwhile, Alice encodes pilot signals ${A_{p} \cos (ω_{p} t)}$ and ${A_{p} \sin (ω_{p} t)}$ with FDM, where $A_{p}$ denotes the amplitude and $ω_{p}$ is the carrier frequency of the pilot signal. The photon-wavepacket coherent state modulated by Alice can be expressed as [29] ${| x_{t} γ_{m} + j p_{t} γ_{m} ⟩}_{ξ_{s}} \otimes {| A_{p} \exp (j ω_{p} t) γ_{m} ⟩}_{ξ_{s}} .$ (9)Then, half of the coherent state is split by the beam splitter and enters ABC for bias control, while the other half enters VOA for power adjustment. After proper calibration, the photon-wavepacket coherent state output by Alice is ${| x_{t} + j p_{t} ⟩}_{ξ_{s}} \otimes {| A_{p} \exp (j ω_{p} t) ⟩}_{ξ_{s}} .$ (10)Thus, Alice’s modulation variance is $V_{A} = 2 {| x_{t} + j p_{t} |}^{2}$ .

Subsequently, the photon-wavepacket coherent state enters the channel for transmission as shown in Fig. 1, which consists of a PZT and two optical fiber cables. During this process, it will undergo phase rotation, power attenuation, and polarization changes. The phase angle can be expressed as $θ_{s} (t) = θ_{p} (t) + θ_{f} (t)$ , where $θ_{p} (t)$ denotes the phase rotation caused by the vibration of PZT and $θ_{f} (t)$ represents the phase rotation caused by the optical fiber cable. Considering linear and uniform attenuation, the attenuation factor can be described by the transmittance $T {= 10}^{- α_{L} L / 10}$ , where $α_{L}$ denotes the attenuation coefficient and $L$ represents the transmission distance. Polarization change is left for further consideration. Here we consider the phase rotation and transmittance attenuation acting on the envelope. Therefore, the photon-wavepacket coherent state reaching Bob can be regarded as ${| x_{t} + j p_{t} ⟩}_{ξ_{sc}} \otimes {| A_{p} \exp (j ω_{p} t) ⟩}_{ξ_{sc}},$ (11)where the envelope after passing through the channel is $ξ_{sc} (t) = \sqrt{T} ξ_{s}^{0} (t) \exp (j (θ_{s} (t) - ω_{s} (t) t)) .$ (12)

When the photon-wavepacket coherent state arrives at Bob, it first enters the polarization controller for polarization correction. We consider the PC to be ideal, so no consideration is given to polarization leakage. Bob’s laser 2 generates another photon-wavepacket coherent state ${| γ_{LO} ⟩}_{ξ_{LO}}$ , where $ξ_{LO} (t) = ξ_{LO}^{0} (t) e^{- j (ω_{LO} (t) t - θ_{LO} (t))}$ . $ω_{LO} (t)$ is the optical carrier of LO, and $θ_{LO} (t)$ is the phase difference between laser 1 and laser 2. Then, Bob inputs them into ICR for one mode measurement with the heterodyne detection. The corresponding output data ${\hat{I}}_{n}^{SNU}$ in shot noise unit (SNU) with a block size of $n$ can be obtained from the photocurrent flux operator [28], which can be given by ${\hat{I}}_{n}^{SNU} = {\hat{X}}_{Ξ_{DSP}} + {\hat{P}}_{Ξ_{DSP}},$ (13)in which ${\hat{X}}_{Ξ_{DSP}}$ and ${\hat{P}}_{Ξ_{DSP}}$ denote the quadrature operator of the temporal mode $Ξ_{DSP} (t)$ . Its photon-wavepacket expression is ${\bar{Ξ}}_{DSP} (t) = \frac{1}{σ_{cal}} ξ_{LO}^{0} (t) G_{dsp}^{n} (t) \exp (- j (ω_{LO} (t) t - θ_{LO} (t))),$ (14)where $σ_{cal} = \sqrt{\int d t {| ξ_{LO}^{0} (t) |}^{2} {(G_{dsp}^{n} (t))}^{2}}$ is the rescaled factor when calibrating output data by SNU. In the formula, $G_{dsp}^{n} (t)$ can be considered to be related to the weighted sum of the DSP algorithm and the impulse response function. Therefore, $G_{dsp}^{n} (t)$ for block size $n$ can be obtained by $G_{dsp}^{n} (t) = \sum_{m = 1}^{n} f_{dsp}^{m} \int_{t_{0} + (m - 1) / F_{m}}^{t_{0} + m / F_{m}} g (τ - t) d τ,$ (15)in which $f_{dsp}^{m}$ represents the digital signal processing function operated on the $m$ -th data, and $g (\cdot)$ denotes the impulse response function of the detector. Oversampling issues have not been considered here. Instead, it is defined based on block size $n$ and modulation rate $F_{m}$ . If there is an oversampling method (such as a root-raised cosine filter) present, downsampling (such as matched filtering) needs to be performed. Since the bandwidth of the detector is large enough, $g (\cdot)$ is considered to be the $δ$ -function.

After the photon-wavepacket coherent state ${| x_{t} + j p_{t} ⟩}_{ξ_{sc}} + {| A_{p} e^{(j ω_{p} t)} ⟩}_{ξ_{sc}}$ enters Bob’s ICR for heterodyne detection, the measurement result can be rewritten through the Gram–Schmidt process, which can be given by [28] ${\hat{I}}_{n}^{SNU} = \sqrt{η_{e} T} (\sqrt{η} {\hat{X}}_{ξ_{scn}} + \sqrt{1 - η} {\hat{P}}_{ξ_{scn}} + \sqrt{η} {\hat{P}}_{ξ_{scn}} + \sqrt{1 - η} {\hat{X}}_{ξ_{scn}}),$ (16)where $η_{e}$ is the quantum efficiency, and $ξ_{scn}$ means the normalization result of $ξ_{sc}$ . $\sqrt{η}$ can be considered as the mode-matching coefficient, and its expression is $\sqrt{η} = \int_{t_{0}}^{t_{0} + n / F_{m}} d t Ξ_{DSP}^{*} (t) ξ_{sc} (t) .$ (17)First, we make the DSP function work like a filter $f_{dsp}^{m} = H_{p}^{m}$ , where $H_{p}^{m}$ is the discrete impulse response function of the band-pass filter at the $m$ -th data. Therefore, we can separate the pilot signal ${| A_{p} e^{(j ω_{p} t)} ⟩}_{ξ_{sc}}$ , which matches the frequency band of the band-pass filter $H_{p}^{m}$ . The mode-matching coefficient $\sqrt{η_{p}}$ can be expressed as $\sqrt{η_{p}} = \int_{t_{0}}^{t_{0} + n / F_{m}} d t \frac{1}{σ_{cal}} ξ_{LO}^{0} (t) ξ_{s}^{0} (t) \exp (j (ω_{LO} (t) t - ω_{s} (t) t + θ_{LO} (t) + θ_{s} (t))) .$ (18)Thus, the quantum measurement result of the pilot signal’s photon-wavepacket coherent state in the temporal mode $Ξ_{DSP} (t)$ can be represented as $μ_{p} = {⟨ A_{p} \exp (j ω_{p} t) | {\hat{I}}_{n}^{SNU} | A_{p} \exp (j ω_{p} t) ⟩}_{ξ_{sc}} = \sqrt{η_{e} T η_{p}} A_{p} \exp (j ω_{p} t) \overset{N}{=} \exp (j (ω_{p} t + ω_{LO} (t) t - ω_{s} (t) t + θ_{LO} (t) + θ_{s} (t))), σ_{p}^{2} = {⟨ A_{p} \exp (j ω_{p} t) | {\hat{I}}_{n}^{SNU} {\hat{I}}_{n}^{SNU} | A_{p} \exp (j ω_{p} t) ⟩}_{ξ_{sc}} - {⟨ A_{p} \exp (j ω_{p} t) | {\hat{I}}_{n}^{SNU} | A_{p} \exp (j ω_{p} t) ⟩}_{ξ_{sc}}^{2} = η_{e} T (η_{p} A_{p}^{2} + A_{p}^{2} - η_{p} A_{p}^{2}) = η_{e} T A_{p}^{2},$ (19)in which $μ$ denotes the first-order moment, $σ^{2}$ represents the second-order moment, and $\overset{N}{=}$ means the normalization operation. The above derivation shows that the detection results of the pilot signal $μ_{p}$ can be used to calculate the $f_{dsp}^{m}$ algorithm function required for the quantum signal. At this point, a single detection is sufficient to accurately recover the signal, eliminating the need for traversal functions or machine learning. The recovery data of the quantum signal corresponds one to one with the detection data of the pilot signal; therefore the time complexity is $O (1)$ , which is independent of the block size $n$ . Recovery of arbitrary phase changes and frequency offsets under low algorithm complexity demonstrates the great robustness of the scheme. Moreover, due to the time-variant information carried by $μ_{p}$ , this algorithm remains highly effective in evaluating time-variant parameters. However, the DSP algorithm obtained by this method will still be affected by the variance of the pilot signal $σ_{p}^{2}$ . Since the photon wavepacket operator is normalized by SNU, it is affected by the shot noise limit. In the classical view, it is considered to be influenced by the signal-to-noise ratio (SNR) of the pilot signal ${SNR}_{p}$ . Through this method, the $f_{dsp}^{m}$ algorithm of the quantum signal can be expressed by $f_{dsp}^{m} = μ_{p}^{*} \exp (j ω_{p} t) H_{q}^{m},$ (20)where $H_{q}^{m}$ is the low-pass filter matching the frequency band of quantum signal ${| x_{t} + j p_{t} ⟩}_{ξ_{sc}}$ . When the SNR of the pilot signal ${SNR}_{p}$ is sufficiently high, the mode-matching coefficient of the quantum signal can be considered as $\sqrt{η_{q}} = 1$ . At this point, the quantum measurement of the detector achieves perfect mode-matching. Therefore, the quantum measurement result of the quantum signal’s photon-wavepacket coherent state can be derived as $μ_{q} = {⟨ x_{t} + j p_{t} | {\hat{I}}_{n}^{SNU} | x_{t} + j p_{t} ⟩}_{ξ_{sc}} = \sqrt{η_{e} T η_{q}} (x_{t} + j p_{t}) = \sqrt{η_{e} T} (x_{t} + j p_{t}), σ_{q}^{2} = {⟨ x_{t} + j p_{t} | {\hat{I}}_{n}^{SNU} {\hat{I}}_{n}^{SNU} | x_{t} + j p_{t} ⟩}_{ξ_{sc}} - {⟨ x_{t} + j p_{t} | {\hat{I}}_{n}^{SNU} | x_{t} + j p_{t} ⟩}_{ξ_{sc}}^{2} = η_{e} T (η_{q} V_{A} + V_{A} - η_{q} V_{A}) = η_{e} T V_{A} .$ (21)In this way, we have completed the quantum signal recovery under the time-variant model.

4. EXPERIMENTAL VERIFICATION OF ROBUST CV-QKD

A. Experimental Implementation

The experimental setup is shown in Fig. 2(a). The black line represents the optical fiber, and the blue line denotes the electrical cable. First, Alice’s narrow linewidth laser 1 generates the signal light. The center wavelength of laser 1 is set to 1550.1200 nm. The continuous wave is transmitted to IQM for modulation. The modulation signal is output through a 16-bit arbitrary waveform generator (AWG). The signal consists of pilot signals and quantum signals through FDM, with an amplitude ratio of $A_{p} = 20$ . The modulation format of the quantum signal is pulse-shaped base-band GMCS using a root-raised cosine filter, where its block size is $n = 10^{9}$ . The modulation variance is $V_{A} = 4.56 SNU$ , the modulation frequency is $F_{m} = 1 GHz$ , and the roll-off factor is 0.3. The modulation format of the pilot signal is carrier modulation, with a carrier frequency of $ω_{p} = - 1 GHz$ . Then, the modulated coherent states are divided into two halves by a 50:50 BS. One half of the light is output at ABC to adjust the bias of the IQM, and the other half of the light passes through a VOA. This part of light enters the fiber channel after proper adjustment. For different experiments, we use standard optical fiber cables of different lengths $L = 25 km$ , 60 km, and 80 km, with an attenuation coefficient of $α_{L} = 0.2 dB/ km$ . To simulate the field-test environment, we introduced a PZT for vibration in the fiber channel. PZT is wound around a 2.5 m optical fiber, with dimensions of $53.10 mm \times 55.00 mm \times 3.95 mm$ . It is controlled by the voltage waveform output from the PZT controller. After passing through the fiber channel, the coherent states enter the PC for polarization adjustment. Meanwhile, Bob’s laser 2 generates an LO, with a center wavelength of 1550.1515 nm. The signal and LO are input to ICR for heterodyne detection. Finally, the electrical signal is transmitted to the oscilloscope (OSC) for data acquisition and digital signal processing. OSC and AWG are clock-synchronized.

Figure 2.LLO CV-QKD system. (a) Experimental structure. (b) Complex frequency spectrum (CFS) of the signal modulated by Alice. (c) CFS of the signal detected by Bob. (d) CFS of the signal recovered by Bob. (e) Complex power spectrum (CPS) of the signal modulated by Alice. (f) CPS of the signal detected by Bob. (f) CFS of the signal recovered by Bob. (g) CPS of the signal recovered by Bob.

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The complex frequency spectrum (CFS) and complex power spectrum (CPS) of the signal modulated by Alice are shown in Fig. 2(b) and Fig. 2(e), respectively. According to it, the pilot tone is located at $- 1 GHz$ , while the quantum band is at 0 GHz. CFS and CPS of the signal detected by Bob are shown in Fig. 2(c) and Fig. 2(f), respectively. From left to right, there is $1 / f$ noise, pilot tone, quantum signal, and the beat signal at the center of the quantum signal. It can be observed that the pilot tone is no longer at its original position of $- 1 GHz$ and has moved to around 2.65 MHz. Furthermore, it is no longer a single pulse but has been irregularly expanded. This phenomenon is shown in the small figure of CFS [Fig. 2(c)]. However, as shown in CPS, Fig. 2(f), the power of the pilot tone has not been irregularly expanded. This phenomenon also occurs in the quantum signal and the beat signal at its center, which is caused by rapid frequency shift and fast phase drift. The frequency and phase between the two lasers change rapidly over time, as shown in the measured results, Fig. 3(a). PZT causes the change of optical fiber length, resulting in fast phase drift as shown in Fig. 3(b). Although the phase change after phase unwrapping in Fig. 3(b) looks like a sinusoidal signal with a fixed frequency, the open-loop PZT introduces an arbitrary phase change with directional and frequency variations in actual; see Ref. [23]. In addition, the spectrum broadening in Fig. 2(c) also proves this point. If the channel phase change were a sinusoidal change at a fixed frequency, there would be no spectrum broadening, but rather a spectral shifting. In the experiment, the mean frequency deviation is 89.05 MHz/s and the mean phase change is 3.036 Mrad/s. Although the vibration of the PZT increases the backward scattering of the signal, the rapid frequency shift and fast phase drift do not cause the power of the signal to change over time. Therefore, the phenomenon of spectrum expansion can only be observed in CFS, Fig. 2(c), and there is no power spectrum expansion in CPS, Fig. 2(f). Through the theoretical analysis in temporal modes of continuous-mode states, as long as we have the pilot signal, we can obtain a digital signal processing function $f_{dsp}^{m}$ that makes the matching factor of the quantum signal $η_{q} = 1$ . At this time, the optimal matching of the two modes is achieved, and the quantum signal is recovered using the time-variant parameter estimation method as mentioned in theoretical derivation. CFS and CPS of the signal recovered by Bob are shown in Fig. 2(d) and Fig. 2(g), respectively. From the graph, it can be seen that the signals in each component have been recovered to their original positions. Additionally, the pilot signals have also been recovered to their pulse form in CFS. Furthermore, the quantum signals modulated at the base-band mitigate the impact of $1 / f$ noise. The histogram in Fig. 4 presents the Gaussian distribution of the recovered signal, which validates the effectiveness of the recovery method on the other side. In the experiment, the SNR of the quantum signal can be denoted as ${SNR}_{q} = - 8.75 dB$ , and the SNR of the pilot signal can be represented as ${SNR}_{p} = 35.69 dB$ . Additionally, the interference-to-noise ratio (INR) is $INR = 42.92 dB$ , which is evaluated by the phase power spectrum in Fig. 5. The signal-to-interference ratio (SIR) can be calculated by $SIR = \frac{{SNR}_{q}}{INR} .$ (22)Therefore, the SIR of the experiment is $SIR = - 51.67 dB$ .

Figure 3.Time-variant parameters measured in the experiment. (a) Frequency deviation. The mean frequency deviation is 89.05 MHz/s. It is mainly caused by two lasers with the unlocked frequency, which is similar to the situation encountered during a field-test experiment where two lasers are difficult to synchronize. (b) Phase change. The mean phase change is 3.036 Mrad/s. It is mainly caused by the vibration of PZT, which is comparable to the scenario in the field-test experiment where vehicles pass through the optical fiber.

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Figure 4.Histogram of recovered signal. The signal processed by time-variant parameter digital signal processing demonstrates a Gaussian distribution.

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Figure 5.Phase power spectrum of the interference signal. The interference-to-noise ratio (INR) of the interference signal is 42.92 dB.

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B. Experimental Results

Based on the experimental setup mentioned above, we conduct two experiments: time-variant parameter estimation and time-invariant parameter estimation. Time-variant parameter estimation is performed using the method proposed in this paper, while time-invariant parameter estimation is based on the assumption that parameters do not change over time. Under the same interference environment, the results of the excess noise for 100 frames are shown in Fig. 6. The color of the scatter points represents the magnitude of the excess noise, and the values of the excess noise are the results back-calculated by the data of Bob. Since a high modulation frequency and long block size are used in the experiment, the excess noise results appear to vary continuously over time. In the time-variant parameter estimation experiment, the excess noise is indicated by the blue scale on the left side of the figure, with a mean value of $ε = 0.0275 SNU$ . In the time-invariant parameter estimation experiment, the excess noise is indicated by the red scale on the right side of the figure, with a mean value of 7.9428 SNU. As the estimation of time-invariant parameters cannot correctly demodulate signals, more noise will be introduced during the matched filtering process, leading to the phenomenon of excess noise exceeding the threshold $V_{A}$ . Therefore, in the case of rapid frequency shift and fast phase drift, only time-variant parameter estimation can maintain low noise, while time-invariant parameter estimation results in a sharp increase of noise. This result validates the effectiveness of the robust scheme.

Figure 6.Excess noise with time-variant estimation and time-invariant estimation. The color of the scatter points indicates the magnitude of the excess noise. In the time-variant estimation, the excess noise is represented by the blue scale on the left side of the figure, with an average value of $ε = 0.0275 SNU$ . Similarly, in the time-invariant estimation, the excess noise is depicted by the red scale on the right side of the figure, with an average value of 7.9428 SNU.

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Under the finite-size effect, the achievable SKR of this system can be calculated by $K = F_{m} (1 - a_{n}) (1 - e_{n}) (1 - v_{n}) (β I_{AB}^{het} - χ_{BE}^{het} - Δ (n)),$ (23)where $K$ denotes the achievable SKR, $F_{m}$ states the system repetition rate, $a_{n}$ represents the parameter estimation ratio, $e_{n}$ denotes the frame error rate, $v_{n}$ expresses the frame head ratio, $β$ states the reconciliation efficiency, $I_{AB}^{het}$ represents the classical mutual information between Alice and Bob, $χ_{BE}^{het}$ expresses the Holevo bound on the information leaked to Eve, and $Δ (n)$ is related to the security of the privacy amplification. The other formulas for calculating SKR are presented in Appendix A. The relative experimental parameters are shown in Table 2. After the practical measurement and calculation, the final SKR is shown in Fig. 7. The red line represents the theoretical SKR in the finite-size case. In addition, the three points on the red line correspond to the results achieved in three experiments: 32.37 Mbits/s at 25 km, 3.51 Mbits/s at 60 km, and 0.29 Mbits/s at 80 km. The blue line denotes the theoretical SKR in the asymptotic case. Additionally, the single point on the blue line corresponds to the result achieved in the 60 km experiment, which is 12.38 Mbits/s at 60 km. The experimental results demonstrate that the robust CV-QKD system can still achieve efficient signal recovery even in the presence of significant interference. Moreover, it achieves a low level of excess noise and successfully verifies long-distance experiments in finite size. This provides the foundation for field-test experiments in complex environments.Table 2.

Experimental Parameter in LLO CV-QKD System

No.	Parameter	Value
1	System repetition rate $F_{m}$	1 GHz
2	Modulation variance $V_{A}$	4.56 SNU
3	Electronic noise $v_{el}$	0.05 SNU
4	Block size $n$	$10^{9}$
5	Parameter estimation ratio $a_{n}$	50%
6	Frame error rate $e_{n}$	20%
7	Frame head ratio $v_{n}$	0.1%
8	Reconciliation efficiency $β$	98%
9	Quantum efficiency $η_{q}$	0.56

Figure 7.SKR of LLO CV-QKD system. The red line is the theoretical SKR in the finite-size case, where the three points correspond to the results achieved in three experiments. The blue line is the theoretical SKR in the asymptotic case, where the single point on the blue line corresponds to the result in one experiment.

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5. DISCUSSION

Through the theoretical and experimental evidence mentioned above, it is demonstrated that we can achieve effective digital signal processing in a long block size, thereby improving the transmission distance of LLO CV-QKD. This paper only considers frequency offset and phase variation. For practical field-test experiments, IQM imbalance, ICR imbalance, and polarization change will greatly impact the excess noise. Therefore, it is necessary to research a digital signal processing method that can maintain a mode-matching coefficient $η_{q} = 1$ in any environment. In future work, we will consider more complex scenarios and implement longer transmission distances in practical field-test experiments.

In order to remove the high device requirements brought by the pilot tone, the pilot-reference-free method has been proposed [30,31]. Subsequently, we will also consider implementing LLO CV-QKD using schemes that do not require pilot tones. The security analysis used here is only for the finite-size effect; composable security will be taken under advisement for the future.

6. CONCLUSION

This paper presents a robust CV-QKD system to overcome the problem from the finite-size effect and LLO structure. This system is capable of achieving lower excess noise during arbitrary frequency shift and phase drift through time-variant parameter estimation. First, we provide theoretical derivation based on the temporal modes of continuous-mode states. Then, we experimentally verify this robust CV-QKD system. Finally, we have achieved low noise that cannot be achieved by time-invariant parameter estimation, thus completing the experiment of long-distance CV-QKD under the finite-size effect. This work sets the foundation for future long-distance field-test experiments on finite-size effects.

Category: Quantum Optics

Received: May. 21, 2024

Accepted: Aug. 25, 2024

Published Online: Oct. 31, 2024

The Author Email: Tao Wang (tonystar@sjtu.edu.cn), Guihua Zeng (ghzeng@sjtu.edu.cn)

DOI:10.1364/PRJ.530729