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Generating the local oscillator "locally" in continuous-variable quantum key distribution based on coherent detection

Bing Qi, Pavel Lougovski, Raphael Pooser, Warren Grice, Miljko Bobrek

arXiv:1503.00662v3quant-ph

TL;DR

Existing coherent-detection CV-QKD sends the signal and LO from the same laser through the insecure channel, motivating a locally generated LO. The paper proposes pilot-aided feedforward recovery, demonstrates 0.04 rad^2 phase-noise variance with independent lasers over 25 km, and reports that the noise is tolerable for CV-QKD. A complete CV-QKD experiment is not presented.

  • Problem

    Existing CV-QKD implementations send both the quantum signal and LO from the same laser through the insecure channel, creating security and application limitations.

  • Method

    The paper proposes and demonstrates pilot-aided feedforward data recovery for coherent detection using an independently generated local LO.

  • Results

    0.04 rad^2 phase-noise variance was measured in a 25 km coherent communication system using two independent commercial lasers.

  • Takeaways & Limitations

    The scheme’s measured noise is reported as tolerable for CV-QKD and may support protocols such as MDI CV-QKD using independent light sources.

  • Takeaways & Limitations

    A complete CV-QKD experiment using the proposed scheme is not presented.

Abstract

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Continuous-variable quantum key distribution (CV-QKD) protocols based on coherent detection have been studied extensively in both theory and experiment. In all the existing implementations of CV-QKD, both the quantum signal and the local oscillator (LO) are generated from the same laser and propagate through the insecure quantum channel. This arrangement may open security loopholes and also limit the potential applications of CV-QKD. In this paper, we propose and demonstrate a pilot-aided feedforward data recovery scheme which enables reliable coherent detection using a "locally" generated LO. Using two independent commercial laser sources and a spool of 25 km optical fiber, we construct a coherent communication system. The variance of the phase noise introduced by the proposed scheme is measured to be 0.04 (rad^2), which is small enough to enable secure key distribution. This technology also opens the door for other quantum communication protocols, such as the recently proposed measurement-device-independent (MDI) CV-QKD where independent light sources are employed by different users.

I. INTRODUCTION

CV-QKD offers practical secure communication and strong noise filtering, but existing coherent-detection implementations send both signal and LO from the same laser through the insecure channel. The paper proposes pilot-aided feedforward recovery to support a locally generated LO.

  • I. INTRODUCTION: CV-QKD based on coherent detection is a viable practical approach with intrinsic filtering against background noise.The strong LO selectively suppresses noise photons, supporting operation over noisy fiber and free-space links.
  • I. INTRODUCTION: Existing coherent-detection CV-QKD implementations generate the quantum signal and LO from the same laser and transmit both through the insecure channel.
  • I. INTRODUCTION: This shared generation exposes the LO to Eve and can reduce QKD efficiency when a strong LO traverses a lossy channel.The required LO photon number can exceed 10^8 photons per pulse at the receiver.
  • I. INTRODUCTION: Generating the LO locally with an independent receiver laser requires a reliable phase reference, which existing classical recovery techniques do not directly provide for weak quantum signals.
  • I. INTRODUCTION: The proposed pilot-aided feedforward data recovery scheme enables coherent detection with a locally generated LO and achieves 0.04 rad^2 phase-noise variance in a 25 km demonstration.The experiment uses two independent commercial free-running lasers and opens a route toward MDI CV-QKD.

II. THEORETICAL ANALYSIS

The theoretical analysis frames locally generated-LO CV-QKD around phase-reference recovery after measurement. It describes GMCS encoding, Bob’s detection choices, and post-processing that restores correlated data.

  • II. THEORETICAL ANALYSIS: In GMCS QKD, Alice Gaussian-modulates coherent states using random amplitude and phase quadratures before transmission to Bob.The modulation variance is expressed relative to the shot-noise variance N0 = 1/4.
  • II. THEORETICAL ANALYSIS: Bob may use homodyne detection to measure a randomly selected quadrature or heterodyne detection to measure both quadratures after a 50:50 split.
  • II. THEORETICAL ANALYSIS: Alice and Bob estimate channel transmittance and excess noise from sampled raw-key data before determining whether a secure key can be established.
  • II. THEORETICAL ANALYSIS: With an independent LO, the central theoretical challenge is establishing a shared phase reference so Bob can perform the required quadrature measurement.
  • II. THEORETICAL ANALYSIS: Pilot-aided phase estimation lets Alice or Bob rotate measured data during post-processing, establishing correlation and applying in principle to homodyne and heterodyne CV-QKD.

A. CV-QKD using quadrature remapping scheme

Quadrature remapping allows Bob to measure with an arbitrarily rotated basis and correct the result after phase information becomes available. Precise phase estimation avoids adding noise through the rotation.

  • A. CV-QKD using quadrature remapping scheme: In heterodyne detection with free-running independent lasers, Bob’s LO phase is random relative to the signal laser.
  • A. CV-QKD using quadrature remapping scheme: Bob can use post-measurement phase information to rotate his two measured quadratures and recover data correlated with Alice’s encoding.
  • A. CV-QKD using quadrature remapping scheme: When the phase is determined precisely, quadrature rotation introduces no additional noise because the rotated noise terms remain independent Gaussian noises with unchanged variance.

B. Pilot-aided phase recovery scheme

The pilot-aided scheme uses strong reference pulses interleaved with quantum signals to estimate the relative phase of independent lasers and remap the quantum data. It must account for rapid phase fluctuations and stringent relative-noise requirements.

  • B. Pilot-aided phase recovery scheme: Averaging many calibration pulses reduces estimation noise, but quantum signals alone cannot provide a precise phase reference.
  • B. Pilot-aided phase recovery scheme: Independent lasers make quantum-signal phase estimation impractical because frequency instability and finite linewidth cause rapid phase fluctuations.
  • B. Pilot-aided phase recovery scheme: Alice sends an unmodulated strong reference pulse alongside each quantum signal, allowing Bob to estimate phase and perform quadrature remapping.
  • B. Pilot-aided phase recovery scheme: Quantum signals and reference pulses are sent alternately and periodically, while fewer reference pulses may improve efficiency when phase drift is slow.
  • B. Pilot-aided phase recovery scheme: A signal’s phase can be estimated from neighboring reference pulses, accounting for their measurement delay and laser frequency drift.
  • B. Pilot-aided phase recovery scheme: Unlike classical intradyne recovery, reference pulses must estimate phase during the quantum-signal arrival window, imposing stricter relative laser-noise requirements.

C. Security analysis

The proposed security analysis shows that phase-reference pulses do not compromise security and that quadrature remapping is equivalent to conventional CV-QKD. Phase-recovery uncertainty appears as excess noise, so minimizing it remains important for secure-key performance.

  • Security equivalence: Existing CV-QKD security proofs can be applied directly to the proposed scheme.The paper states that no new security proof is needed.
  • Security equivalence: Phase-reference pulses provide classical phase information and are not directly used for quantum-signal detection.Eve cannot access the LO itself, and the reference pulses give her no additional information under the standard phase-reference assumption.
  • Security equivalence: Quadrature remapping is security-equivalent to conventional CV-QKD because heterodyne detection commutes with unitary phase rotation.The phase-estimation channel can be treated as a classical channel controlled by Eve, yielding a virtual protocol equivalent to the proposed one.
  • Phase-recovery noise: Phase-recovery uncertainty is translated into excess noise εφ in the measured quadratures.Here VA is Alice’s modulation variance and σφ is the phase-determination noise variance.

A. Noise model

The noise model separates measurement noise from quantum laser phase noise in estimating the phase from reference pulses. Strong reference pulses suppress shot-noise contributions, while laser phase noise remains the main irreducible source.

  • Noise sources: The phase-estimation process has two major noise sources: reference-pulse measurement noise and quantum laser phase noise.The first depends on reference-pulse strength; the second arises from spontaneous emission between signal and reference pulses.
  • Measurement noise: A reference pulse with 1000 average photons and 50% detection efficiency gives approximately 0.001 phase-noise variance from shot noise.The paper treats this contribution as negligible in practice.
  • Laser phase noise: Quantum laser phase noise cannot be reduced simply by increasing reference-pulse amplitude and is therefore the scheme’s main noise source.It originates from amplified spontaneous emission and reflects the time separation between signal and reference pulses.
  • Laser phase model: The laser phase deviation Δθ(t) is modeled as a zero-mean Gaussian random variable whose variance depends on time.For a Lorentzian laser lineshape, the coherence time is related to the laser linewidth.
  • Laser phase model: The phase-estimation noise variance is determined by the signal- and LO-laser phase noises and the delay Td between signal and reference pulses.Td is defined as the time delay between those pulses.

B. Experimental setup

The experiment uses two independent commercial telecom-wavelength lasers to provide separate signal and LO sources. Signals and reference pulses travel through 25 km of single-mode fiber before balanced heterodyne measurement.

  • Laser sources: Two independent free-running, frequency-stabilized continuous-wave lasers serve as the signal and LO sources.They have no optical or electrical connections, and their frequency difference remains within 10 MHz without feedback.
  • Optical path: Both signal and reference pulses propagate through a 25 km single-mode fiber spool before measurement.The receiver uses a commercial 90° optical hybrid and balanced amplified photodetectors.
  • Detection: The receiver measures both X- and P-quadratures using a 90° optical hybrid and two 350 MHz balanced amplified photodetectors.The passive hybrid requires no temperature control to stabilize its internal interferometers.
  • Electronics: A 120 MHz arbitrary waveform generator drives the intensity and phase modulators and synchronizes the oscilloscope.This provides the modulation and timing signals for the experimental setup.

C. Experimental results

The experiments demonstrate phase recovery with an independently generated local oscillator: correction separates encoded phases and enables quadrature remapping for weak quantum signals. Measured residual phase noise is approximately 0.04 rad^2, consistent across classical and quantum-domain tests and compatible with simulated secure-key generation.

  • Before correction, random relative phase makes bit-0 and bit-1 measurements uniformly distributed, regardless of encoded phase.
  • After phase correction, the measurement results for bit 0 and bit 1 are clearly separated.
  • 0.040 ± 0.001 phase-noise variance is measured for bit 0, while bit 1 yields 0.039 ± 0.001.
  • 0.039 ± 0.001, 0.040 ± 0.001, and 0.054 ± 0.001 phase-noise variances are measured using reference pulses containing 10000, 1000, and 100 photons, respectively.The scheme works well even with reference pulses containing only a thousand photons.
  • 0.040 ± 0.001 expected phase-recovery noise matches the experimental result, with laser finite linewidth identified as the main noise source.Noise can be reduced using a smaller time delay or narrower-linewidth lasers.
  • 1.83 shot-noise units is the measured X-quadrature variance after remapping weak quantum signals, implying about 0.83 shot-noise units of detector excess noise.Residual phase noise makes the remapped P-quadrature variance larger than the X-quadrature variance.
  • 0.034 ± 0.01 phase noise is estimated in the weak-quantum-signal experiment, consistent with the strong-signal measurements.The result supports operation in both classical and quantum domains, although its uncertainty is higher.
  • 120 km secure-key generation is obtained in simulation under a realistic model where Eve cannot control Bob’s detector noise and loss.

IV. DISCUSSION

The discussion presents a pilot-aided feedforward scheme for coherent detection with a locally generated LO, addressing security and application limitations of transmitted LOs. Proof-of-principle results indicate tolerable noise, while a complete CV-QKD experiment remains future work.

  • IV. DISCUSSION: The conventional arrangement sends both the quantum signal and LO through the insecure channel, potentially exposing the LO to manipulation and limiting efficiency.The discussion also identifies transmitted strong-LO loss as a limitation in some applications.
  • IV. DISCUSSION: A pilot-aided feedforward data recovery scheme enables reliable coherent detection using a locally generated LO.The scheme uses post-measurement phase-reference information to recover the data phase.
  • IV. DISCUSSION: The proposed design removes cumbersome unbalanced fiber interferometers and their associated phase-stabilization system.This simplifies the CV-QKD system relative to the conventional scheme.
  • IV. DISCUSSION: Proof-of-principle experiments using commercial off-the-shelf components show that the scheme’s noise is tolerable for CV-QKD.The authors state that narrower-linewidth laser sources could further reduce the noise.
  • IV. DISCUSSION: A complete CV-QKD experiment using the proposed scheme is not presented, although the required components have been developed.The proposed system structure is described as simpler than the conventional scheme.
  • IV. DISCUSSION: The technology is also connected to other quantum communication protocols, including MDI-CV-QKD, where independent light sources are used by different users.The discussion frames this as an additional application of locally generated-LO technology.

Appendix A: Laser phase noise

The appendix derives how independent signal and LO laser phase noise contributes to phase-recovery noise and describes measurements of both lasers. It also relates the measured recovery-noise parameter to secure-key-rate simulations under finite-size limitations.

  • Laser phase noise: The appendix derives the noise variance of the phase-recovery scheme from independent signal- and LO-laser noise terms.The derivation establishes the contribution quantified by Eq. (11).
  • Laser phase noise: Phase recovery estimates the intermediate phase difference φ1 from reference-pulse phases measured at t0 and t2.The signal and LO phases are modeled at three times, with φ0, φ1, and φ2 denoting their differences.
  • Laser phase noise: The signal-laser phase evolves across delays Td with independent Gaussian noises NS,1 and NS,2 of variance ⟨(∆θS(Td))2⟩.The LO laser follows an analogous evolution with independent noises NL,1 and NL,2 of variance ⟨(∆θL(Td))2⟩.
  • Laser phase noise: The phase-noise experiments split each laser into two fiber paths and measure their phase difference with an optical hybrid, balanced photodetectors, and an oscilloscope.Measurements were taken at Td = 5ns, 20ns, and 25ns.
  • Laser phase noise: At Td = 20ns, the measured phase noises are 0.035 ± 0.001 and 0.044 ± 0.001 for the two lasers.The observed laser phase noise depends linearly on Td, as predicted by Eq. (9).
  • Simulation of secure key rate: The secure-key-rate simulations use σφ = 0.04 alongside realistic channel, detector, reconciliation, and modulation parameters.The simulations indicate that the proposed LO phase-recovery scheme can achieve efficient QKD; the finite-size discussion notes that Fig. 9 assumes infinitely many pulses.
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