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Sending-or-Not-Sending with Independent Lasers: Secure Twin-Field Quantum Key Distribution Over 509 km
Jiu-Peng Chen, Chi Zhang, Yang Liu, Cong Jiang, Weijun Zhang, Xiao-Long Hu, Jian-Yu Guan, Zong-Wen Yu, Hai Xu, Jin Lin, Ming-Jun Li, Hao Chen, Hao Li, Lixing You, Zhen Wang, Xiang-Bin Wang, Qiang Zhang, Jian-Wei Pan
TL;DR
Long-distance SNS-TF-QKD requires precise relative-phase control and management of fiber-induced noise. The paper analyzes improved pairing and finite-key security methods, experimentally stabilizes independent sources, and reports improved key rates while identifying distance-dependent Re-Rayleigh scattering as a limitation.
Problem
Long-distance SNS-TF-QKD must control relative phase precisely while strong reference pulses introduce fiber-scattering noise.
Method
The paper combines AOPP pairing, finite-key security analysis, remotely locked independent lasers, phase estimation, narrow detection windows, and optimized parameter searches.
Results
AOPP can improve the final key rate by one or two times, while relative-phase drift over 509 km contributes less than 4% error under the stated estimation procedure.
Takeaways & Limitations
The reported techniques support high-performance SNS-TF-QKD over long fiber links, with Re-Rayleigh scattering acceptable at the 500 km scale.
Takeaways & Limitations
Re-Rayleigh scattering worsens with distance when reference detection rates must be maintained, degrading TF-QKD performance at longer distances.
Abstract
from arXiv · showhide
Twin field quantum key distribution promises high key rates at long distance to beat the rate distance limit. Here, applying the sending or not sending TF QKD protocol, we experimentally demonstrate a secure key distribution breaking the absolute key rate limit of repeaterless QKD over 509 km, 408 km ultra-low loss optical fibre and 350 km standard optical fibre. Two independent lasers are used as the source with remote frequency locking technique over 500 km fiber distance; Practical optical fibers are used as the optical path with appropriate noise filtering; And finite key effects are considered in the key rate analysis. The secure key rates obtained at different distances are more than 5 times higher than the conditional limit of repeaterless QKD, a bound value assuming the same detection loss in the comparison. The achieved secure key rate is also higher than that a traditional QKD protocol running with a perfect repeaterless QKD device and even if an infinite number of sent pulses. Our result shows that the protocol and technologies applied in this experiment enable TF QKD to achieve high secure key rate at long distribution distance, and hence practically useful for field implementation of intercity QKD.
THEORY OF IMPROVED SNS-TF-QKD PROTOCOL
The protocol combines four-intensity SNS-TF-QKD with AOPP post-processing and finite-key analysis. It estimates effective events and security parameters before extracting final keys.
- Protocol construction: Alice and Bob classify time windows as signal or decoy, then use one-detector heralded events in Z windows to form key strings.Signal windows randomly choose sending or vacuum pulses, while decoy windows support parameter estimation.
- Security estimation: The protocol estimates untagged-bit counts and phase-flip errors using decoy statistics and Chernoff bounds before privacy amplification.These estimates determine the secure final-key length through an entropy-based formula.
- AOPP post-processing: AOPP retains selected odd-parity pairs, reducing bit-flip errors while leaking one additional bit of information per pair.Only specified pair types survive, and the remaining bits form new strings for secure-key extraction.
- AOPP post-processing: AOPP can improve the final key rate by one or two times when it enables a larger sending probability and increases effective Z-window events.The improvement applies even though the procedure discards at least half of the original key bits.
- Security estimation: The protocol sets the total security coefficient to εtol = 22ξ = 2.2 × 10−9 under the stated failure-probability assignments.The assignments include εcor = ε̂ = εPA = ξ and a Chernoff-bound contribution of 6ξ.
THE CHERNOFF BOUND
The Chernoff-bound analysis converts observed counts into bounds on their expected values and supports finite-statistics estimation in the protocol.
- Finite-key estimation: Chernoff bounds relate observed values and expected values, including source-count quantities through lower and upper estimates.The relations are applied to NαβSαβ and corresponding expected counting rates.
CONTROLLING THE RELATIVE PHASE BETWEEN ALICE AND BOB
The experiment must precisely control the relative phase of the twin fields before they interfere at Charlie’s beam splitter. This phase control is central to the SNS-TF-QKD measurement setup.
- Phase control: The relative phase of the twin fields is controlled before interference at Charlie’s beam splitter.The requirement concerns the two fields sent by Alice and Bob in SNS-TF-QKD.
- Phase control: Accurate phase control is particularly significant for the SNS-TF-QKD experiment.
- Interference measurement: The beam splitter provides the interference stage where the controlled twin fields are measured.
independent sources
The experiment combines remotely locked independent lasers with reference-pulse phase estimation and post-selection across long fiber links. Relative phase drift remained bounded through 509 km, supporting the long-distance implementation.
- independent sources: Two independent lasers are remotely frequency-locked to compensate wavelength differences over a 500 km fiber distance.The setup uses time-frequency dissemination and sends reference light between Alice and Bob.
- Phase-drift measurements: 2.72rad · ms−1 is the relative phase-drift-rate standard deviation for 0 km total signal fiber links.
- Phase-drift measurements: 6.89rad · ms−1 is the relative phase-drift-rate standard deviation for 350 km total signal fiber links.
- Phase-drift measurements: 9.52rad·ms−1 and 9.58rad·ms−1 are the corresponding standard deviations for 408 km and 509 km links.
- Phase-drift measurements: At 509 km, 12 accumulated reference periods produce about 0.3 rad of phase drift and less than 4% error when both users send the same phase.The measured drift-rate standard deviation is below 9.6 rad · ms−1 across all tested scenarios.
- Phase estimation: Strong phase-reference pulses are periodically sent to Charlie, whose interference records estimate the twin-field relative phase.The estimation uses four phase differences and minimizes an error model between observed counts and theoretical detection probabilities.
- Phase compensation: Post-selection of X-basis events estimates the X-basis error rate instead of actively compensating the relative phase in real time.
ENCODING DETAILS OF THE EXPERIMENT
The experiment encodes pulses using phase slices and multiple intensity levels, while linear amplification controls modulation errors and phase-drift measurements characterize long links.
- Phase encoding: 16 phase slices encode the light, using an arbitrary-wave generator sampled at 2 GHz with 14-bit depth.The modulation signal is amplified by more than 25 dB before driving the modulators.
- Modulation control: Two linear amplifiers keep the X-basis error rate below 4% despite the approximately 9 V peak-to-peak phase-modulator drive.Normal RF amplifiers would introduce distortion after high-gain amplification of the waveform.
- Intensity encoding: Five intensity levels assign pulses to reference, signal, strong-decoy, weak-decoy, and vacuum states.Three intensity modulators set the required levels for phase-reference and signal pulses.
- Long-link characterization: The relative phase drift is measured for the 408 km total signal-fiber link.The figure reports the drift of the long-distance signal path used in the experiment.
DETAILED EXPERIMENTAL PARAMETERS
The experiment uses symmetric Alice–Charlie and Bob–Charlie links, distance-dependent sending parameters, and measured output intensities and detection counts; the 509 km link’s phase drift is characterized separately.
- Fiber and optical parameters: Alice–Charlie and Bob–Charlie fibers are set to equal lengths, with efficiencies, losses, state proportions, and intensities summarized by total fiber length.The signal pulse width is 1 ns, while each phase-reference pulse has an effective 100 ns width.
- Distance-dependent settings: Sending ratios and total signal-pulse counts vary with fiber distance, including basis, decoy-state, and sending/not-sending fractions.The parameters distinguish pX and pZ, p0, p1, p2, pz1, and pz0.
- Count estimation: The experiment calculates Alice’s and Bob’s output intensities and estimates detection counts after accounting for optical and detection efficiencies.These estimates use the selected experimental parameters.
- Phase-drift characterization: The 509 km total signal-fiber link has a measured relative phase-drift characterization.The corresponding phase-drift results are presented for the longest link.
THE RE-RAYLEIGH SCATTERING IN FIBER
Strong phase-reference pulses share the signal fiber and create Re-Rayleigh-scattering noise, whose modeled and measured levels remain acceptable at 500 km but worsen with distance.
- Noise origin: Strong reference pulses sharing the signal fiber inevitably generate scattering noise through elastic and inelastic fiber-scattering mechanisms.The reference pulses use the signal wavelength and are time-multiplexed through the same fiber.
- Re-Rayleigh mechanism: Re-Rayleigh scattering is defined as Rayleigh scattering of backward-scattered light that originated from the seed light’s initial Rayleigh scattering.The process involves a further scattering event at another point in the fiber.
- Noise model: The detection-noise model combines Re-Rayleigh-scattering noise with SNSPD dark counts and depends on fiber length, attenuation, input intensity, and photon energy.The model’s variables include d, Eν, l, α, P0, and Dc.
- Experimental validation: At 408 km and 509 km, measured detection noise with remotely locked lasers approximately agrees with theoretical estimates for a 2 MHz count rate.A single-source test was also performed over 250 km standard fiber.
- Distance limitation: Re-Rayleigh noise is acceptable at 500 km but increases with distance when reference detection rates must be maintained for phase estimation, degrading TF-QKD performance at longer distances.Forward spontaneous Raman noise can be blocked with a narrow-band filter, while forward spontaneous Brillouin noise is not considered relevant to overlapping signal pulses.
DETAILED EXPERIMENTAL RESULTS
The experiment characterizes optical components, filters detections by timing and phase-estimation criteria, and optimizes error rates and secure key rates across fiber distances, including 509 km.
- System characterization: The experimental system characterization includes SNSPD efficiencies, signal-fiber losses, optical-element transmissions, polarization components, circulators, DWDMs, PBSs, and beam splitters.These parameters are listed for the different fiber lengths.
- Detection filtering: Detection windows narrower than the signal pulses improve interference visibility but reduce accepted data as pulse jitter grows with fiber distance.The within-window fractions are about 51%, 48%, and 35% for 350 km, 408 km, and 509 km, respectively.
- Error analysis: The analysis reports basis-dependent error rates before and after AOPP, including Z-basis bit-flip and X-basis phase-flip or decoy-state error rates.The notation distinguishes QBER(Z-Before), QBER(Z-After), eph1 before and after AOPP, and QBER(X11).
- Key-rate optimization: The accepted phase-difference range Ds is optimized using phase-estimation success probabilities rc to maximize the final key rate R.The optimization is applied after filtering detections by the digital window and phase-estimation criterion.
- Count reporting: The results tabulate sent and detected pulse counts by basis, intensity, detector channel, accepted phase range, and correctness.These quantities support calculation of the X-basis error rate and identify optimized acceptance ranges.
- 509 km results: At 509 km, X-basis QBERs and detections are evaluated across phase-difference ranges and accepted phase-estimation probabilities before extracting optimized secure key rates.The corresponding analyses are reported in Tables IV–VIII.