Intensity correlations in decoy-state BB84 quantum key distribution systems

TL;DR

This study demonstrates significant intersymbol intensity correlations in two industrial decoy-state BB84 QKD implementations, revealing that higher-order correlations can be as impactful as, or more than, nearest-neighbor ones and thus threaten the decoy-state security assumption. By combining direct pattern-based energy measurements with Savitzky–Golay and SVD denoising, the authors quantify correlation strengths up to $\\xi$ values as high as 6 and apply a Cauchy–Schwarz-based security analysis to bound Eve's information. Using a truncated Gaussian correlation model for $g_{a_k,a_{k-1}}$, they compute the asymptotic secret-key rate $K_{\\infty}$ and show substantial degradation from even low-order correlations, emphasizing the need for improved security proofs and practical mitigations such as higher-bandwidth modulators and optimized intensity settings. The findings underscore the importance of hardware stability and rigorous source characterization for reliable QKD deployment and motivate ongoing theoretical work to scale security proofs to higher-order correlations.

Abstract

The decoy-state method is a prominent approach to enhance the performance of quantum key distribution (QKD) systems that operate with weak coherent laser sources. Due to the limited transmissivity of single photons in optical fiber, current experimental decoy-state QKD setups increase their secret key rate by raising the repetition rate of the transmitter. However, this usually leads to correlations between subsequent optical pulses. This phenomenon leaks information about the encoding settings, including the intensities of the generated signals, which invalidates a basic premise of decoy-state QKD. Here we characterize intensity correlations between the emitted optical pulses in two industrial prototypes of decoy-state BB84 QKD systems and show that they significantly reduce the asymptotic key rate. In contrast to what has been conjectured, we experimentally confirm that the impact of higher-order correlations on the intensity of the generated signals can be much higher than that of nearest-neighbour correlations.

Figures (9)

• Figure 1: Correlation measurement. (a) Simplified scheme of the measurement setup for intensity correlations. Alice generates optical pulses with a laser diode (Laser). These pulses propagate through an intensity modulator (IM) and an encoding modulator (EM) and obtain a random intensity and encoding state according to the prescriptions of a decoy-state BB84 protocol. Both modulators are controlled with a field-programmable gate array (FPGA) that sets different operating voltage levels according to the signal received from a (quantum) random number generator (Q)RNG (one of the Alices tested is equipped with a classical pseudo-RNG). To ensure a classical energy level of the optical pulses at Alice's output, we remove a variable optical attenuator (VOA) from the optical scheme. The measurement setup consists of a fast photodetector (PD) and a digital high-bandwidth oscilloscope (OSC). (b) Conceptual view of five consecutive optical pulses emitted by Alice. The intensity of the latest-emitted correlated pulse D depends on the previously emitted pulses, whose number determines the correlation length $\xi$. In this work, we shall consider correlations up to $\xi=6$. (c) A short fragment of the recorded data oscillogram measured on system B with five registered consecutive optical pulses. The intensity settings and the chronological order of the pulses are the same as in (b). Dashed lines and arrows highlight the deviations in the maximum amplitude values of the D intensity setting pulses. (d) Distributions of calculated energies for the studied intensity settings (gray) and $\xi=1$ in system B. Each intensity setting, denoted by X, is represented by four distributions. We normalize the energy by the mean value of the S-state distribution. (e) Zoomed-in fragment of (d) with decoy-state distributions. We show both unnormalized (top) and normalized (bottom) groups of distributions. The latter group is normalized such that each distribution area's integral equals 1. The mean value for each distribution is marked by a vertical line of the same color. From these distributions, it follows that, in general, SD pulses have less energy than DD or VD pulses. The same trend can also be seen in (c).
• Figure 2:
• Figure 3: Comparison of averaged oscillogram waveforms of the two most-intensity-separated patterns, (a) measured in system B---SSSSSDS (red solid line) and VVVVVSS (black solid line). Individual measured pulses are plotted in pink (SSSSSDS, 250 pulses) and gray (VVVVVSS, 250 pulses). A noticeable deviation in the waveforms of the S state when considering $\xi=6$ is additionally shown in the zoomed-in sector (b). A similar effect of the long correlations on the pulses' waveforms is observed in system A as well (c). In the latter, we compare the waveforms of the most separated decoy-state third-order patterns---DSSD (black and gray, 50 pulses) and VVDD (red and pink, 50 pulses).
• Figure 4: Asymptotic secret key rate $K_{\infty}$ for the case of first-order pulse correlations ($\xi=1$, solid lines) and for the ideal scenario without correlations ($\xi=0$, dashed lines). For system B, we consider different attenuations to study the dependence of the secret key rate on the decoy state intensities. It is apparent from the figure that lowering the intensities in the presence of correlations is beneficial for long-distance transmissions. For completeness, we have also included the attainable secret key rate for system B if the attenuation is optimized for each distance. For the simulations, we use the channel model described in \ref{['app:channel']}.
• Figure 5: Asymptotic secret key rate $K_{\infty}$ for the case of second-order pulse correlations ($\xi=2$, solid lines) and for the ideal scenario without correlations ($\xi=0$, dashed lines). Similarly to \ref{['fig:1_vecinos']}, we consider different values of the attenuation for system B and we use the channel model presented in \ref{['app:channel']}.