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Experimental Twin-Field Quantum Key Distribution Through Sending-or-Not-Sending
Yang Liu, Zong-Wen Yu, Weijun Zhang, Jian-Yu Guan, Jiu-Peng Chen, Chi Zhang, Xiao-Long Hu, Hao Li, Cong Jiang, Jin Lin, Teng-Yun Chen, Lixing You, Zhen Wang, Xiang-Bin Wang, Qiang Zhang, Jian-Wei Pan
TL;DR
Controlling the relative phase between Alice’s and Bob’s independent lasers is identified as a major SNS-TF-QKD challenge. The paper uses phase-reference pulses and post-processing, with an X-basis baseline error rate around 2% in the reported results.
Problem
Controlling the relative phase between Alice’s and Bob’s systems is the biggest challenge in the SNS-TF-QKD experiment.
Method
The experiment uses phase-reference pulses to estimate phase differences and post-processing to compensate fiber-induced relative phase.
Results
Around 2%: the X-basis baseline error rate is reported at approximately this level for the tested setting.
Takeaways & Limitations
Phase-reference-based estimation and compensation provide the experimental route used to address fiber-induced relative phase variation.
Abstract
from arXiv · showhide
Channel loss seems to be the most severe limitation on the practical application of long distance quantum key distribution. The idea of twin-field quantum key distribution can improve the key rate from the linear scale of channel loss in the traditional decoy-state method to the square root scale of the channel transmittance. However, the technical demanding is rather tough because it requests single photon level interference of two remote independent lasers. Here, we adopt the technology developed in the frequency and time transfer to lock two independent lasers' wavelengths and utilize additional phase reference light to estimate and compensate the fiber fluctuation. Further with a single photon detector with high detection rate, we demonstrate twin field quantum key distribution through the sending-or-not-sending protocol with realistic phase drift over 300 km optical fiber spools. We calculate the secure key rates with finite size effect. The secure key rate at 300 km ($1.96\times10^{-6}$) is higher than that of the repeaterless secret key capacity ($8.64\times10^{-7}$).
I. THEORY OF SNS-TF-QKD PROTOCOL
The SNS-TF-QKD protocol uses decoy-state sending-or-not-sending decisions, basis and intensity announcements, and detector outcomes to estimate key-relevant yields and phase-flip errors. Finite-size statistical bounds then determine a secure, correct final key rate.
- Protocol preparation: Alice and Bob randomly choose signal or decoy sources and, in the Z-basis, encode bits by sending or not sending phase-randomized coherent pulses.The signal source has intensity µz, while decoy sources use vacuum and two coherent intensities.
- Event selection: Mismatched bases or X-basis intensities are discarded, while Charlie’s announcement of exactly one detector click defines an effective event.Alice and Bob retain effective events for subsequent parameter estimation and key processing.
- Parameter estimation: Z-basis errors are estimated from a random subset, while X-basis errors estimate the phase-flip error rate of single-photon Z-basis events.The sending and not-sending choices cannot all be publicly announced because they encode the secret key.
- Phase slicing: X-basis phase-slice conditions select events whose phase differences fall near 0 or π after accounting for the phase offset ΔϕT.The slice width Ds is determined by the threshold Λ, and the security proof allows a general ΔϕT.
A. Relative Phase Drift between Alice and Bob
Relative phase fluctuations arise from both frequency differences between Alice’s and Bob’s lasers and changes in the fiber path. The experiment therefore separates source-related and fiber-related contributions to phase drift.
- Experimental challenge: The SNS-TF-QKD experiment identifies control of the relative phase between Alice and Bob as its biggest challenge.This challenge motivates phase-locking and phase-fluctuation compensation procedures.
- Drift sources: The relative phase fluctuation contains a source-frequency term and a fiber-path-change term between Alice and Bob.The optical frequency ν, fiber length L, and light speed s enter the phase-drift relation.
- Prior measurements: Earlier reports measured phase-drift rates of 2.4 rad · ms^-1 at 100 km and 6.0 rad · ms^-1 at 550 km between fiber spools.These values provide prior distance-dependent phase-drift context for the experiment.
- Source stabilization: Frequency differences between independent lasers create beats, so their relative frequency must be measured rapidly before the phase drifts substantially.For δba = 0.01 and L = 2 km, the required frequency difference is Δν < 159 Hz.
B. Measuring the Phase Drift due to Frequency Differences in the Sources
The experiment measures phase drift under increasingly realistic source and fiber configurations, including independent phase-locked lasers separated by 150 km of fiber. The measured drift rates remain within the compensation target used for the protocol.
- 150 km fiber: 7.1 rad · ms^-1 is the measured standard deviation of the phase-drift rate with one laser source and 150 km fiber.The measured phase-drift rate follows a Gaussian distribution in this configuration.
- Phase-locked sources: 5.8 rad · ms^-1 is the measured standard deviation with two phase-locked independent lasers and 0 km fiber.This scenario directly tests the phase-locked laser sources.
- Compensation timescale: 7.4 rad · ms^-1 bounds the drift-rate standard deviation across tested scenarios, including independent sources and 150 km fiber spools.The maximum drift rate is 31 rad · ms^-1, corresponding to about 0.31 rad in 10 µs; the related error is below 3% for equal phases.
C. Detailed Encoding Method in Experiment
The experimental encoding uses continuous-wave lasers, phase and intensity modulation, and timed phase-reference pulses. These components generate signal, decoy, vacuum, and not-sending states while supporting relative-phase estimation.
- Signal level: Independent continuous-wave signals are attenuated to the single-photon level before leaving Alice’s or Bob’s secure zone.This attenuation is part of the practical optical implementation.
- Optical encoding: Continuous-wave lasers are modulated into 16 phases and then encoded with three intensity modulators.The modulators control signal, decoy, vacuum, reference-pulse intensity, and pulse width.
- Not-sending state: Vacuum and not-sending states are produced by blocking the light with the modulators.This implements the not-sending choice used for Z-basis encoding.
- Control electronics: The arbitrary-waveform generator uses pre-generated quantum random numbers to control pulse modulation in Alice’s and Bob’s laboratories.The AWG operates at 2 GHz with 14-bit depth.
- Pulse timing: Each 5 µs period contains 100 signal pulses in the first 3 µs, four phase-reference pulses in the next 1.2 µs, and a final 0.8 µs vacuum interval.The reference pulses estimate the relative phase, while the vacuum interval provides SNSPD recovery time.
D. Estimating the Relative Phase Drifts in Alice’s and Bob’s Fibers
The experiment estimates relative phase drift between Alice’s and Bob’s fibers using phase-reference pulses and interference measurements at Charlie. It then minimizes an error model to infer the most likely phase difference and filters detections using the estimation quality.
- Pulse pattern: Four phase-reference pulses probe relative fiber phase during each 5 µs period.Alice cycles through phases 0, π/2, π, and 3π/2 while Bob holds phase 0.
- Phase measurement: The reference detections at Charlie estimate the total relative phase between Alice and Bob’s links.The measured interference intensity represents the probability of detection at Charlie’s beam-splitter output.
- Phase estimation: The estimated phase difference is obtained by scanning ∆ϕT from 0° to 360° and selecting the value minimizing Err(∆ϕT).The model compares measured and theoretical detection probabilities for the four phase settings.
- Phase-estimation quality: rc measures agreement between the measured phase-count distribution and the theoretical distribution, approaching zero for a perfect match and one for equal counts.The parameter is defined from the ratio of minimum to maximum errors.
- Experimental optimization: For 100 km, rc = 0.01 retained about 14% of the data and was selected as the best choice, although the optimum varies with channel conditions.The acceptance threshold was optimized to maximize the final key rate.
E. Compensating for the fiber-induced relative phase via post-processing
The experiment compensates fiber-induced relative phase through post-processing rather than active real-time correction. Charlie post-selects detections using phase estimates and a digital time window, while weak reference pulses reduce noise and avoid fast feedback circuitry.
- Post-processing compensation: Post-processing replaces active phase compensation at Charlie by using the estimated relative phase ∆ϕT.The revised criterion selects X-basis events whose phase difference falls within the accepted range.
- Phase-slice selection: Detections are retained when the compensated phase lies within Ds or Ds + π, reducing interference-related error.The accepted conditions are expressed as bounds on |θA − θB + ∆ϕT|.
- Reference-based estimation: Phase-reference pulses estimate the relative phase between the two fibers using fewer than 50 total detections over 10 µs.The required reference intensity remains practical over several hundred kilometers.
- System implications: Weak reference pulses reduce noise, and the approach does not require a fast feedback circuit.These features simplify the compensation system for long-distance communication.
III. DETAILED EXPERIMENTAL PARAMETERS
The experiment uses fixed timing and basis/intensity ratios to accommodate amplifier and modulator response limits. Detector dead time and modulation stability constrain the realized detection counts and decoy-state settings.
- Decoy-state settings: The weak decoy intensity µ1 was fixed at a mean photon number of 0.05 because stable modulation was required.The chosen value was the lowest intensity that maintained stable decoy intensities.
- Electronic constraints: Fixed X-basis fractions avoid nonlinear amplifier and modulator responses caused by long signal-free intervals.The relevant component cutoff frequencies range from 75 kHz to 10 GHz.
- Optimization scope: Better electronic components could enable parameter optimization that further enhances the final key rate.The current parameter choices were constrained by amplifier and modulator response limitations.
- Detection effects: Detector dead time reduced actual detection counts below theoretical estimates at high count rates.The peak phase-reference intensity was approximately 2–5 photons per 100 ns, without affecting signal-pulse detection efficiency.
IV. DETAILED EXPERIMENTAL RESULTS
The study characterizes the optical and detector system and tests SNS-TF-QKD over multiple fiber distances, optimizing phase-selection parameters for secure key-rate calculations. An improved detector and phase-estimation system lowers the X-basis baseline error rate and permits less restrictive post-selection.
- Improved system: The improved system used an SNSPD with dark count probability approximately 10^-7 per pulse and reduced the X-basis baseline error rate to approximately 2%.A linear voltage amplifier and modified phase-calculation algorithm improved phase estimation.
- Fiber-distance tests: SNS-TF-QKD was tested at 100 km, 200 km, and 300 km under realistic phase-drift conditions.The second test used the same fiber conditions and therefore the same phase-drift rate as the first test.
- Reported metrics: The experimental-results table reports single-photon yield s1, phase error rate eph1, and final key rate R for optimized parameters.It also includes accepted phase range Ds, estimation success probability rc, and the retained fraction rrc.
- Key-rate optimization: The optimized settings were obtained by searching parameter ranges, including accepted phase differences and rc, to maximize the final key rate.For the 150 km data set, Ds/2 = 10° and rc = 0.04 gave the optimized key rate.