Continuous-variable quantum key distribution with time-division dual-quadrature homodyne detection

Junsang Oh; Jeongsik Cho; June-Koo Kevin Rhee

doi:10.1364/OE.489253

1. Introduction

Quantum Key Distribution (QKD) is a method of cryptography that enables the sharing of key data between two authenticated parties through a quantum channel. These key data are used as one-time encryption keys, providing perfect secrecy and secure data communication as described in Shannon’s work on communication theory [1]. The development of quantum computers has threatened the security of traditional cryptography methods based on mathematical complexity. However, QKD, which utilizes irreversible physical properties to establish secret keys, is gaining significant attention as a next-generation secure communication technology.

One specific implementation of QKD is continuous-variable quantum key distribution (CVQKD), in which the sender, referred to as Alice, prepares displaced coherent states with quadrature components in optical phase space according to key information. Alice then transmits them to the receiver, referred to as Bob, through a quantum channel. Bob generates secret keys by performing coherent detection (homodyne or heterodyne) of the quantum states with a strong intensity local oscillator (LO) as a phase reference. The unconditional secrecy of the Gaussian-modulated coherent states (GMCS) protocol [2] against collective attacks has been demonstrated under both asymptotic and finite-key conditions [3–7], leading to rapid developments in CVQKD. Early CVQKD protocols require high signal-to-noise ratios (SNR) and are thus limited in transmission distance. However, advancements in reverse and multi-dimensional reconciliation techniques [8,9] have made long-distance transmission possible, even in low SNR conditions, thereby increasing the efficiency of the reconciliation process and improving the overall performance of the protocol. Furthermore, implementation of a stabilized system with a low level of excess noise allows CVQKD to generate secret keys even at a transmission distance of around 100km [10,11]. Recently, CVQKD has been demonstrated to distribute secret keys over 200 km using ultra-low-loss optical fiber [12].

A major advantage of CVQKD is its compatibility with standard telecom components and the use of high-efficiency detectors at room temperature. In particular, the LO plays a role in filtering quantum signals in an optical communication channel with multiple channels, making it possible to effectively generate secret keys in a wavelength-division multiplexing (WDM) environment [13–17].

However, as the transmission distance increases, insufficient LO power reduces the ratio of shot noise to electronic noise of the detector, making it difficult to generate secret keys. Nevertheless, transmitting a stronger LO also increases the excess noise due to the nonlinear effects in a fiber-optic channel, making it more difficult to achieve the desired key rate under certain conditions. A stronger LO also creates potential vulnerabilities for an eavesdropper (Eve) to exploit. [18–22]. To compensate for these disadvantages, methods of replacing the LO by using a separate light source on the receiver side [23,24] or measuring shot noise through the LO in real time [25,26] have recently been proposed. In the context of CVQKD, the feasibility of long-distance transmission remains a challenge. As a result, we have predominantly focused on achieving improved efficiency and enhanced performance within the short-reach access network domain in the development of a CVQKD system.

Coherent detection by a homodyne interferometer employs random modulation of the phase of the LO by 0 or $\frac {\pi }{2}$ to measure one of the two orthogonal quadrature components of a quantum state. Heterodyne detection, which is actually homodyne detection with two quadrature detections, is achieved by power splitting a quantum state into two homodyne interferometers in a way that both quadrature components are measured [27]. Heterodyne detection can achieve a higher secret key transmission rate compared to homodyne detection, particularly under high signal-to-noise ratio (SNR) conditions. This is because dual-quadrature detection increases the mutual information between Alice and Bob, which is effective for short distances and especially for channels with high excess noise. However, the high complexity and cost of implementing the system, which requires two interferometers even under high SNR conditions, have impeded progress in this scheme. In particular, heterodyne detection systems that require $\frac {\pi }{2}$-phase-difference optical hybrid and consist of two pairs of homodyne detectors have the drawback of requiring a separate channel to transmit the LO or a locally-local oscillator (LLO) at a receiver [23,28–31].

In this paper, we propose a new detection scheme with identical performance to heterodyne detection with only a single interferometer based on the GMCS protocol. Alice transmits quantum states and LOs alternatively in a time-division manner with an equal interval, and Bob performs homodyne detection twice for one quantum signal split by the balanced beam splitter (BS) of the asymmetric interferometer with unbalanced delays. The phase difference between the long and short arms of the asymmetric interferometer is kept constant through real-time control so that it allows the previous two consecutive measurements to achieve coherent detections in the two quadratures between a quantum state and two LOs in the front and back in time with a $\frac {\pi }{2}$ phase difference. After completing the transmission of all quantum states, Bob converts these measurements into Q and P quadrature values that have a linear correlation with Alice’s through proper post-processing.

In addition, we present a simple solution to the problem of receiving polarization alignment, which typically requires significant complexity in the implementation of a CVQKD system. The polarization state of the optical signals from Alice is transmitted to Bob while changing in real-time due to the birefringence of the fiber channel originating an irregular refractive index. Polarization alignment of the LO and quantum signals at the input to the coherence detector is essential. Most CVQKD experiments transmit quantum signals and LOs separated by orthogonal polarization using polarization-multiplexing techniques to prevent power loss of the received signals, and re-align polarization inside the interferometer to mix them. Here, polarization control at the receiver is critical to demultiplex the LO and the quantum state signal, while a fiber cable in the field rotates the polarization state randomly with a characteristic time of milliseconds. Uncontrolled polarization states can mix strong LOs into the quantum state signals. Almost all recent CVQKD systems have solved this problem by employing either a manual or a dynamic polarization controller [10,11,31,32]. However, our system constitutes a passive polarization homodyne receiver by adopting the Michelson interferometer with Faraday mirrors often used in plug-and-play protocol or DVQKD [33,34]. We also show the stability of the system and experimentally demonstrate the concept of dual quadrature homodyne detection with TDM signals based on the GMCS protocol. Here, we name this detection scheme time-division dual-quadrature homodyne detection (TDDQHD).

The paper is organized as follows. Section 2 introduces the experimental setup of the proposed system and describes the process of measuring quantum states with a Faraday-Michelson interferometer and obtaining two quadrature components through post-processing using the measured signal. Section 3 introduces a phase-locking method at the receiver and examines the phase stability of the signal detected from the self-created interferometer. Parameter estimation with the measurement results of our system, including real-time shot noise measurements (RTSNM), and the performance of the system with the proposed TDDQHD are described in Section 4. Finally, we conclude the paper in Section 5.

2. Scheme description

2.1 Experimental setup

Here, we describe our experiment based on the GMCS protocol directly transmitting quantum states and LOs from the sender, Alice. Figure 1 shows the layout of our experimental setup. Alice uses a low-noise CW laser (RIO lasers, ORION 1550 nm) with a narrow line width of 1.7 kHz at 1550.025 nm as the light source. A semiconductor optical amplifier (SOA) modulator (Thorlabs, BOA1004PXS) with a power extinction ratio of up to 75dB generates pulse amplitude modulation to produce strong and weak pulse sections with 10- and 90-ns widths, respectively, for LOs and quantum signals at a 10MHz repetition rate. Subsequently, an amplitude modulator (AM) and a phase modulator (PM) are employed to generate a quantum signal pulse in the Q and P quadratures with a zero-centered Gaussian distribution originating from the center of the weak pulse section, while the strong pulse section is kept to be used as the LO. The quantum signal pulses are situated at the center of the adjacent LOs in time, as shown in Fig. 1.

Fig. 1. Experimental setup for the CVQKD system with time-division dual-quadrature homodyne detection.CW laser, continuous wave laser; OS, optical switch based on semiconductor optical amplifier; AM, amplitude modulator; PM, phase modulator; eVOA, electronic variable optical attenuator; PD, photodetector; BS, beam splitter; SMF, single mode fiber; FOPM, Fiber-optic phase modulator; FRM, Faraday rotate mirror; VOA, variable optical attenuator; BPD, balanced photodetector.

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Then, an electronic variable optical actuator (eVOA) regulates and stabilizes Alice’s power from the feedback of a power monitor, and transmits LOs and quantum signals to the channel in a sequential manner. The coherence of the LOs and quantum signals remains due to the narrow line width laser source. A 20.06 km single-mode-fiber (SMF) spool with a loss of 0.2 dB/km is used as a channel in the experiment. On Bob’s side, the pulse sequences are transferred to an asymmetric Faraday-Michelson interferometer (aFMI) consisting of a three-port circulator which directs the signals from port 1 to port 2. The 50:50 beam splitter of the aFMI separates the optical pulses into two arms. The pulses in the long arm are delayed by the length difference between the two arms ($\sim 5$m), corresponding to one half of the interval between the quantum signal and the LO pulse ($50$-ns). Subsequently, the signal pulse and the LO pulse interfere when the pulses divided into both arms are reflected back to the beam splitter. In particular, the Michelson interferometer using Faraday mirrors allows LOs and quantum signals to interfere with the same polarization without using polarization-maintaining fibers and a dynamic polarization controller at the receiver. It achieves this by compensating for the states of the polarization of lights divided into short and long arms due to the time-reversal trip after reflection off the Faraday mirrors, as shown in Fig. 1. In this setup, a slow Fiber-optic phase modulator (FOPM) is installed to introduce phase modulation by stretching the optical fiber on the long arm of the interferometer by a piezo actuator controlled by an electrical signal. Phase control is applied in real time so that the phase differences between the optical signals arriving at the BS from the long and short arms are kept constant. Finally, two interference outputs from a quantum signal and an LO are detected by a shot-noise-limited balance detector (Insight, BPD-1) with a bandwidth of 400 MHz and a low noise equivalent power through a path of the same length. A VOA is used to balance the intensity of light and an isolator is used to remove back reflections.

2.2 Time-division Dual-quadrature homodyne detection

This section describes the proposed system’s behavior for the quantum states and LOs shown in Fig. 2 to introduce a time-division dual-quadrature detection scheme. Figure 3 (a) shows two LO pulses, $\text {LO}_i$ and $\text {LO}_{i+1}$, entering the aFMI on the time axis and one quantum signal $\text {S}_i$ interleaved between them. The quadrature value ($q_i$, $p_i$) of the quantum state $\text {S}_i$ is a realization of random variables $\textbf {Q}$ and $\textbf {P}\sim N(0,V)$ that follow a centered normal distribution and is defined based on the phase of the input LO. Figure 3 (b) shows signal pulses $\text {S}_i^l,\text {S}_i^s$ and LO pulses $\text {LO}_i^l,\text {LO}_{i+1}^s$ that are reflected back after being divided into a long arm and a short arm by a beam splitter. At this time, $\text {LO}_i^l$ and quantum states $\text {S}_i^l$ passing through the long arm rotate as much as the phase difference($\delta$) fixed by the FOPM and appear in the phase space. For one quantum state $\text {S}_i$ received from Alice, Bob obtains two homodyne detection results, $y_{r,i}^s = (y_i^s+n_0^s)$ and $y_{r,i}^l=(y_i^l+n_0^l)$, according to the phase relationship between $\text {LO}_i^l$ and $\text {S}_i^s$ and between $\text {LO}_{i+1}^s$ and $\text {S}_i^l$, where $y_i^s,y_i^l\in \textbf {Y}^s,\textbf {Y}^l \sim N(0,\frac {1}{2}V)$ and $n_0^s,n_0^l \in \textbf {N}_0^s,\textbf {N}_0^l \sim N(0,V_0)$, $V_0$ is the shot noise variance. Due to the phase relationship shown in Fig. 3 (b), each quadrature value of the input state and the corresponding detection result have the following relationship:

(1)$$\frac{1}{\sqrt{2}} \begin{bmatrix} \text{cos}\delta & \text{-sin}\delta \\ \text{cos}\delta & \text{sin}\delta \\ \end{bmatrix} \begin{bmatrix} q_i \\ p_i \\ \end{bmatrix} = \begin{bmatrix} y_i^s \\ y_i^l \\ \end{bmatrix}.$$

This means that each measurement result without considering shot noise is the same as the homodyne detection result for the rotation state of $\pm \delta$ for quantum states that have passed the channel due to arbitrary $\delta$ determined through a phase-locking circuit. Finally, Bob’s Q and P quadrature values for secret key generation can be simply obtained through reverse rotation transform from Eq. (2):

(2)$$\begin{bmatrix} \textbf{Q}_B \\ \textbf{P}_B \\ \end{bmatrix} =\frac{1}{\sqrt{2}} \begin{bmatrix} \textbf{Q} \\ \textbf{P} \\ \end{bmatrix} + \begin{bmatrix} \textbf{N}_0^s \\ \textbf{N}_0^l \\ \end{bmatrix} = \begin{bmatrix} \text{cos}\delta & \text{-sin}\delta \\ \text{cos}\delta & \text{sin}\delta \\ \end{bmatrix}^{{-}1} \begin{bmatrix} \textbf{Y}^s \\ \textbf{Y}^l \\ \end{bmatrix} +\begin{bmatrix} \textbf{N}_0^s \\ \textbf{N}_0^l \\ \end{bmatrix}.$$

After all quantum signals are transmitted, Bob performs post-processing to obtain the Q, P quadrature values to be used as actual secret key information from the measured raw data. In our experiment, we executed this process for a $n=10^6$ sized block of data. According to the GMCS protocol, data sets $\mathbf {Q}^A=\left \{q_1^A,q_2^A,\ldots,q_n^A\right \}$ and $\mathbf {P}^A=\left \{p_1^A,p_2^A,\ldots,p_n^A\right \}$ of each quadrature are generated by Alice, and Bob is post-processed according to the relationship of Eq. (2) using the measured raw data sets $\mathbf {Y}_s=\left \{y_1^s,y_2^s,\ldots,y_n^s\right \}$ and $\mathbf {Y}_l=\left \{y_1^l,y_2^l\cdots,y_n^l\right \}$ to obtain $\mathbf {Q}^B=\left \{q_1^B,q_2^B,\ldots,q_n^B\right \}$ and $\mathbf {P}^B=\left \{p_1^B,p_2^B,\ldots,p_n^B\right \}$. In the experiment, we employ the FOPM to fix the phase difference inside the interferometer to a specific value of $\delta =\frac {\pi }{4}$ based on the light path difference. This allows us to confirm the linear correlation between the data sent by Alice and the data received by Bob, as demonstrated in Fig. 4. To quantify this linear correlation, the Pearson correlation coefficient is used, defined as ${r_{XY}} = \frac{{{\Sigma }_i^n\left( {{X_i} - \bar{X}} \right)\left( {{Y_i} - \bar{Y}} \right)}}{{\sqrt {{{\left( {{X_i} - \bar{A}} \right)}^2}} \sqrt {{{\left( {{Y_i} - \bar{Y}} \right)}^2}} }}$, where $X$ and $Y$ represent the data sets of Alice and Bob, respectively. The Pearson correlation coefficient is a value between +1 and -1, indicating a strong positive linear relationship. As seen in Fig. 4(a), the Pearson correlation coefficient for both quadratures of $10^6$ samples is $r\sim \pm 0.63$ for the Q, P quadrature data of Alice and the raw data received by Bob. However, after post-processing, as shown in Fig. 4(b), the correlation coefficient has $r\sim 0.92$ for the same quadrature, and $r\sim 2\times 10^{-4}$ for different quadratures, indicating a strong positive linear relationship for the same quadrature and a lack of correlation for different quadratures. This allows us to assume the quantum channel between Alice and Bob as a linear model with Gaussian noise through a successful post-processing process.

Fig. 2. Data frame configuration. One data frame consists of 8192 pulses and is divided into 592 preambles and 7600 key signals. The preamble signals are made of timing trigger pulses for phase locking and phase control signals, and the key signals consist of quantum states and LO pulses including randomly inserted, null data for real-time shot noise measurement.

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Fig. 3. (a) Left - Transmitted signal and LO pulses from Alice before entering the interferometer at Bob. The time intervals between pulses are the same; Right - A quantum state $\text {S}_i$ and Local Oscillators $\text {LO}_i$ and $\text {LO}_{i+1}$ in phase space.(b) Left-Signal and LO pulses reflected back from the Faraday mirror within the interferometer after the 50:50 beam splitter; Right - Quantum states and LOs reflected from the short arm and long arm are indicated by the dotted line and solid line in the phase space, respectively.

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Fig. 4. Data correlation for Q and P quadrature values between Alice and Bob. (a) shows the correlation for Bob’s raw data, and (b) indicates the correlation after post-processing.

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3. Phase-locked circuit

In this section, we examine the outcomes of the TDDQHD for scenarios in which the phase difference $\delta$ of the optical signals reflected from the two paths of an aFMI, owing to the length difference between the long and short arms, has been fixed at a specific value. Here, we investigate the implementation of a phase-lock circuit that maintains a stable and constant value of $\delta$ through the use of preamble signals, which are subject to change due to ambient environmental factors such as temperature fluctuations. Specifically, we examine the performance of the produced phase noise control system. The phase control pulses that are part of the preamble signals, which are composed of strong pulses, are generated at a 50 ns interval preceding the key signals, as illustrated in Fig. 2. The E-field of any i-th phase control pulse is given as follows:

(3)$$E_i = A_{p}\text{exp}({-}i(\omega t+\phi_i)),$$

where $A_p,\omega$, and $\phi _i$ are the amplitude, angular frequency and modulated phase of the $i$-th control pulse, respectively. The voltage measured by the balance detector through the aFMI without phase control appears as a homodyne detection result between adjacent phase control pulses:

(4)$$V_{p,i} = k_{p}\text{cos}(\phi_i-\phi_{i+1} + \delta(t)) = k_{p}\text{cos}(\Delta\phi_i+ \delta(t)),$$

where $k_p$ is the proportional amplitude constant of the beat signal including the transimpedance gain of the balanced detector and $\Delta \phi _i$ represents the difference between the phase of the $i$-th and ($i+1$)-th phase control pulses. If $\Delta \phi _i$ is modulated to have the same value for all $i$ according to Eq. (4), we can confirm that the detection results of the phase control pulses depends solely on $\delta (t)$. The voltage readout of coherent detection is averaged from the control pulses through a low-pass filter and is sampled through a self-developed sample$\&$hold board to be used as the output of the PID control circuit. Then, $\delta (t)$ is fixed by driving the FOPM inside the interferometer to satisfy $\text {cos}(\Delta \phi _i+ \delta (t))=0$ with respect to $\Delta \phi _i$.

In our experiment, we modulate the phase difference between control pulses at the transmitter by $\Delta \phi$ to maintain a relative phase of $\delta =\frac {\pi }{4}$. Our FOPM uses a piezo actuator to precisely adjust the length of the optical fiber to the sub-micrometer level in response to the control signal, allowing the phase to be controlled within a two-cycle period. If it goes out of the desired range, a quick reset function can bring it back. Additionally, to counter the rapid phase fluctuations of the optical signal caused by temperature changes, a temperature controller (Stanford Research Systems, LDC501) is used to stabilize the internal temperature of the interferometer. To achieve high control speeds, an acquisition board (National Instruments, PCIe-6341) with above kHz-level control is used. The blue dots and red dots of the scatter plot in Fig. 5 show the Q-P phase distribution for $9.2 \times 10^4$ measurement samples for the same quantum state with- and without phase locking, respectively. The one-day measurements confirmed the error for the value $\delta =\frac {\pi }{4}$, the phase stability of the level $\Delta \delta =|\frac {\pi }{4}-\delta (t)|=0.1$, and the contribution of this error on the overall additional noise [11],

(5)$$\xi_{phase}\approx(1-\kappa)V_A,$$

where $\xi _{phase}$ is the phase noise, $\kappa =\text {cos}(\Delta \delta )$ represents the co-efficient of noise, and $V_A$ is the modulation variance. Accordingly, the term $(1-\kappa )$ is determined as $1.5\times 10^{-5}$. In this way, we can guarantee the security of the phase stabilization technique in a CVQKD experiment, and limit the phase error within tolerance.

Fig. 5. Phase-locked signal. The blue and red traces correspond to the measured quadrature values in phase space without phase locking and with phase locking. To show whether the phase lock works well, it represents $9.2 \times 10^4$ samples for a constant signal.

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4. Parameter estimation and system performance

The QKD protocol verifies the presence or absence of Eve by disclosing a portion of the raw key data exchanged between Alice and Bob after completing all quantum state transmissions, in order to generate the final secret key. This process, called parameter estimation, involves estimating the excess noise of the system in CVQKD. In this paper, we use the True Local Oscillator (TLO) method, which involves directly transmitting the LO from the transmitter. However, this method is vulnerable to a security loophole caused by the transmission of a strong LO. To address this vulnerability, the LLO method, which uses a separate LO in Bob, has been widely used. Nevertheless, it has been shown that even in the TLO transmission method of the CVQKD system, this problem can be solved by using a shot noise measurement method in real time. In this paper, we conducted a proof-of-concept experimental demonstration to verify the secret key rate of the TDDQHD system, including RTSNM. To achieve this, null data was deliberately transmitted from Alice, facilitating the measurement of shot noise. However, in a real system considering Eve’s attacks, it is necessary to accurately measure shot noise by temporally separating the quantum signal and the signal for shot noise measurement using an optical switch at the receiver [26].

4.1 Parameter estimation

After completion of transmission of all quantum states generated in the GMCS CVQKD protocol through TDDQH detection, the datasets shared by Alice and Bob for each quadrature are denoted as $Q = \{ (q^A_i,q^B_i)|i =1,2,\ldots,n \}$ and $P = \{(p^A_i,p^B_i)|i =1,2,\ldots,n\}$, respectively. Additionally, the dataset to measure shot noise is represented as a single vector $S^q_0 = \{s^q_{0, i}|i =1,2,\ldots,m\}$ and $S^p_0 = \{s^p_{0, i}|i =1,2,\ldots,m\}$. Here, $n$ and $m$ are the total number of data points for each quadrature and that for shot noise measurement, respectively. Since parameter estimation is performed in the same manner using data measured from each quadrature, we explain the process using $(x,y) \in \{(q^A,q^B)$ or $(p^A,p^B)\}$ and $s_0 \in \{s^q_0$ or $s^p_0\}$. Assuming the quantum channel follows a linear model with Gaussian noise, we satisfy the following relationship between Alice and Bob’s data:

(6) $$y = tx + z,$$

where $t = \sqrt {\frac {\eta T}{2}}$ and $z$ is the total noise that follows a centered normal distribution with variance $\sigma ^2=\frac {1}{2}\eta T \xi N_0 + N_0 + V_{el}$. The related parameters $\eta$ denotes the efficiency of the balanced detector, $T$ is the transmittance of the channel, $N_0$ is the variance of shot noise, $\xi$ is the excess noise in shot noise units, and $V_{el}$ represents the electronic thermal noise of the detector. In addition, when Alice does not transmit data for shot noise measurement, Bob’s detection result is $s_0=z_0$, where $z_0$ represents noise following a centered normal distribution with variance $\sigma _0^2=N_0+V_{el}$. Therefore, each random variable has the following relationship:

(7)$$\langle x^2 \rangle = V_A N_0, \quad \langle xy \rangle = \sqrt{\frac{\eta T}{2}} V_A N_0,$$

(8)$$\langle y^2 \rangle = \frac{1}{2} \eta T V_A N_0 + \frac{1}{2} \eta T \xi N_0 + N_0 + V_{el},$$

(9)$$\langle s^2 \rangle = N_0+V_{el},$$

where $V_A$ is modulation variance in shot noise units. In the experiment, Alice and Bob use $n$ samples in a block of size $N$ to generate secret keys, $k = N - n$ samples to estimate $T$ and $\xi$, and $m$ samples to measure shot noise in real time. The maximum likelihood estimators for the normal linear model, $\hat {t}$, $\hat {\sigma }^2$ and $\hat {\sigma _0}^2$ can be obtained as follows:

(10)$$\hat{t}=\frac{\Sigma_{i=1}^{k}x_i y_i}{\Sigma_{i=1}^{k}x_i^2},$$

(11)$$\hat{\sigma}^2 = \frac{1}{k}\Sigma_{i=1}^{k}(y_i-\hat{t} x_i)^2, \quad \hat{\sigma_0}^2 = \frac{1}{m}\Sigma_{i=1}^{m}(s_i)^2,$$

where the estimators $\hat {t}$, $\hat {\sigma }^2$ and $\hat {\sigma _0}^2$ are independent with the following distribution:

(12)$$\hat{t} \sim N \left( t,\frac{\sigma^2}{\Sigma_{i=1}^{k}x_i^2} \right),$$

(13)$$\frac{k\hat{\sigma}^2}{\sigma^2} \sim \chi^2 (k-1), \quad \frac{m\hat{\sigma}_0^2}{\sigma_0^2} \sim \chi^2 (m-1),$$

where $t$, $\sigma ^2$ and $\sigma _0^2$ represent the true values of each parameter. In the calculation of the secret key rate, we consider the scenario where Eve employs an optimized attack. As a result, the maximum amount of information that Eve can acquire on Bob’s key, denoted as $S_{EB}^{\epsilon _{PE}}$, can be determined through the lower bounds of $t$, as well as the upper bounds of $\sigma ^2$ and $\sigma _0^2$ corresponding to the error probability $\epsilon _{PE}$ that the predetermined estimated parameters will not be included in the confidence interval. In Eq. (13), the $\chi ^2$ distribution can be approximated by a normal distribution when the sample size is sufficiently large ($k,m \geq 10^6$). Therefore, the boundaries of each parameter can be calculated using this approximation through the measurement sample as follows:

(14)$$\begin{aligned} & t_{min} = \hat{t} - z_{\epsilon_{PE/2}} \sqrt{\frac{\hat{\sigma}^2}{k V_A}},\\ & \sigma^2_{max} = \hat{\sigma}^2 + z_{\epsilon_{PE/2}} \frac{\hat{\sigma}^2 \sqrt{2}}{\sqrt{k}},\\ & \sigma_{0,min}^2 = \hat{\sigma_0}^2 - z_{\epsilon_{PE/2}} \frac{\hat{\sigma_0}^2 \sqrt{2}}{\sqrt{m}}, \end{aligned}$$

where coefficient $z_{\epsilon _{PE/2}}$ satisfies $(1-erf(z_{\epsilon _{PE/2}}/\sqrt {2})/2=\epsilon _{PE}/2$ and the function erf(x) is the error function defined as $erf(x) = \frac {2}{\sqrt {\pi }} \int _0^x e^{-t^2}dt$. Finally, The minimum transmission rate $T_{min}$ and maximum excess noise $\xi _{max}$ of the quantum channel parameters are calculated using Eq. (14), which represents the bounds on the estimates $\hat {t}$, $\hat {\sigma }^2$ and $\hat {\sigma _0}^2$ taking into account the statistical additional noise caused by the finite-key effect. These values are used in the calculation of $S_{EB}^{\epsilon _{PE}}$ in the analysis of the final key:

(15)$$T_{min} = \frac{2t_{min}^2}{\eta}, \quad \xi_{max} = \frac{\sigma_{max}^2 - \sigma_{0,min}^2}{t_{min}^2 \left( \sigma_{0,min}^2 - V_{el} \right)}.$$

According to the aforementioned equation, we conducted a measurement of the total excess noise in our system, which includes the finite-key effect and real-time shot noise measurement, for a period of 24 hours. The data points in Fig. 6 represent the obtained values, which were calculated using a block size of $10^6$ and considers the worst case for excess noise.

Fig. 6. Excess noise measurement over 24 hours in the TDDQHD system. The red square points indicate the measured value of excess noise in shot noise units and the blue and black error bars represent the worst-case estimator with RTSNM and without RTSNM, respectively. The size of the data block and shot noise measurement block is equal to $10^6$. The green line shows the threshold of excess noise value which allows a positive secret key rate at 20.06km.

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4.2 System performance

We calculate the secret key rate and the maximum transmission distance using the correlated data measured from our system under the Trusted-device scenario, where eavesdropping attacks are only possible on the quantum channel. Finite-key effects are taken into account, and the final secret key rate $K$ using the reverse reconciliation is given by:

(16)$$K_{fin} = R \left( \frac{n}{n+k+m} \right) \left[ \beta I_{AB}-S_{EB}^{\epsilon_{PE}}-\Delta (n) \right],$$

where R is the system rate and $\beta$ is the reconciliation efficiency. $I_{AB}$ is the mutual information between Alice and Bob, which can be simply calculated using Bob’s measurement variance, $V_B$, and the conditional variance of Bob’s measurement considering the given state of Alice, $V_{B|A}$. For TDDQHD detection, both quadratures can be measured using a single quantum signal, similar to a standard heterodyne detection, resulting in the following calculation:

(17)$$I_{AB} = \text{log}_2 \frac{V_B}{V_{B|A}} = \text{log}_2\frac{\hat{t}^2 V_A+\hat{\sigma}^2}{\hat{\sigma}^2}.$$

In contrast, calculating $S_{EB}^{\epsilon _{PE}}$, which is Holevo’s bound between Eve and Bob, is more complex as it requires evaluation of the covariance matrix of Alice and Bob, including the value obtained from parameter estimation in the finite-key scenario through Eq. (15).

(18)$$\Gamma_{AB} = \begin{bmatrix} (V_A+1)\mathbb{I} & \sqrt{T_{min}(V_A^2+2V_A)}\sigma_z \\ \sqrt{T_{min}(V_A^2+2V_A)}\sigma_z & T_{min}V_A+1+\xi_{max} \end{bmatrix},$$

where the matrix $\mathbb {I}=\begin {bmatrix} 1 & 0\\0 & 1 \end {bmatrix}$ and $\sigma _z = \begin {bmatrix} 1 & 0\\0 & -1 \end {bmatrix}$. The process of obtaining $S_{EB}^{\epsilon _{PE}}$ from the covariance matrix $\Gamma _{AB}$ is described in detail in the Supplement 1. $\Delta (n)$ is a function of $n$ related to the security of privacy amplification in the finite-size regime, and has the following form:

(19)$$\Delta (n) \simeq 7 \sqrt{\frac{\text{log}_2(1/\bar{\epsilon}}{n}}+\frac{1}{n}\text{log}_2\frac{1}{\epsilon_{PA}},$$

where, $\bar {\epsilon }$ and $\epsilon _{PA}$ denote the smoothing parameter and the probability of failure for secrecy amplification, respectively. For the final key rate calculation, $\bar {\epsilon }$ and $\epsilon _{PA}$ were set to $10^{-10}$. Fig. 7 shows the final key rate calculated using the excess noise measured in our system. The blue and red dashed curves represent the secret key rates for homodyne detection and heterodyne detection, respectively, in asymptotic regimes. In this case, the analysis considers infinitely large data and calculates the secret key rate using the entire dataset, leading to a gap with finite-size data analysis. In asymptotic analysis, heterodyne detection can achieve a higher key rate, although the difference between heterodyne detection and homodyne detection becomes negligible as the transmission distance increases. The solid curves show the theoretical key rate for TDDQHD, which includes finite-size effects with a block size of $10^7$ for RTSNM. The curves from left to right correspond to the locking phase, with values of $\delta = 30^\circ, 35^\circ, 40^\circ, 45^\circ$. The detailed calculation is provided in the Supplement 1. The blue dots represent the calculated values of the final key rate, which is 0.187 Mbps at 20.06 km, using the parameters derived from the measurement data of our system. While homodyne detection achieves a secret key rate of 0.178 Mbps in real-time secret key extraction, it is noted that TDDQHD exhibits enhanced performance.

Fig. 7. Secret key rate of the 10MHz CV QKD system. The blue and red dashed curves show the ideal case of asymptotic secret key rate $(K = I_{AB}-S_{EB})$ of the CV QKD system with homodyne detection and heterodyne detection, respectively. The modulation variance $V_A$ is numerically optimized at each channel transmittance achieving the maximum secret key rate. The solid curves are the theoretical key rate of TDDQHD under general collective attacks in finite-size (N=$4.8\times 10^7$) key scenarios with real-time shot noise measurement. From left to right, the curves correspond to the locking phase in our interferometer of $\delta = 30^\circ, 35^\circ, 40^\circ, 45^\circ$. The blue dot is the expected final key rate through the experimentally measured excess noise. The modulation variance $V_A=4$, quantum efficiency is 0.65, reconciliation efficiency is 0.97, electronic noise is 0.09 and excess noise is 0.064 in shot noise units.

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5. Conclusions

Heterodyne detection systems using LO transmitted from Alice require sensitive phase stabilization in a couple of complex receiver interferometers; in general, they divide a quantum signal into orthogonal polarizations and measure two orthogonal components through respective asymmetric interferometers. In this paper, we propose a new detection method that can measure both orthogonal components using only a single homodyne interferometer by sequentially transmitting quantum signals and LOs at the same intervals and performing time-division measurements. In addition, the receiver, which consists of a Michelson-Faraday interferometer that compensates for irregular polarization drift in the channel, allows the system to operate independently of the input polarization. Through this, we measured a stable excess noise of 0.064 in shot noise units over 20.06 km fiber and estimated an expected secret key rate of 0.187 Mbps. Based on these two major characteristics, a simplified system can be utilized as a universal solution compatible with existing optical access networks by taking advantage of the CVQKD.

Acknowledgments

This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2021R1A2C2013790) and the MSIT(Ministry of Science and ICT), Korea, under the ITRC(Information Technology Research Center) support program(IITP-2023-2018-0-01402) supervised by the IITP(Institute for Information I & Communications Technology Planning & Evaluation)

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.

Supplemental document

See Supplement 1 for supporting content.

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Continuous-variable quantum key distribution with time-division dual-quadrature homodyne detection

Abstract

1. Introduction

2. Scheme description

2.1 Experimental setup

2.2 Time-division Dual-quadrature homodyne detection

3. Phase-locked circuit

4. Parameter estimation and system performance

4.1 Parameter estimation

4.2 System performance

5. Conclusions

Acknowledgments

Disclosures

Data availability

Supplemental document

References

Supplementary Material (1)

Data availability

Cited By

Figures (7)

Equations (19)

Optics Express