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Overcoming Noise in Entanglement Distribution

Sebastian Ecker, Frédéric Bouchard, Lukas Bulla, Florian Brandt, Oskar Kohout, Fabian Steinlechner, Robert Fickler, Mehul Malik, Yelena Guryanova, Rupert Ursin, Marcus Huber

arXiv:1904.01552v2quant-phphysics.optics

TL;DR

The paper asks whether high-dimensional entanglement’s theoretical noise resistance can be demonstrated under realistic experimental constraints. It implements two pathways—fine-graining and additional mutually unbiased measurements—in energy-time and OAM photon pairs, certifying entanglement at noise fractions up to 92% and 72%, respectively. The results support high-dimensional encodings as a route toward noise-tolerant quantum communication.

  • Problem

    Whether high-dimensional entanglement’s theoretical noise resistance can manifest as a practical advantage under implementation-specific constraints remains an open question.

  • Method

    Two experiments use energy-time and OAM entangled photons to implement fine-graining and additional mutually unbiased measurement bases under externally added noise.

  • Results

    Entanglement is certified up to noise fractions of 0.92 for energy-time Pathway I and 0.72 for OAM Pathway II.

  • Takeaways & Limitations

    High-dimensional encodings can reveal entanglement obscured by physical noise and may be adopted for long-distance or free-space quantum communication.

Abstract

from arXiv · show

Noise can be considered the natural enemy of quantum information. An often implied benefit of high-dimensional entanglement is its increased resilience to noise. However, manifesting this potential in an experimentally meaningful fashion is challenging and has never been done before. In infinite dimensional spaces, discretisation is inevitable and renders the effective dimension of quantum states a tunable parameter. Owing to advances in experimental techniques and theoretical tools, we demonstrate an increased resistance to noise by identifying two pathways to exploit high-dimensional entangled states. Our study is based on two separate experiments utilising canonical spatio-temporal properties of entangled photon pairs. Following these different pathways to noise resilience, we are able to certify entanglement in the photonic orbital-angular-momentum and energy-time degrees of freedom up to noise conditions corresponding to a noise fraction of 72 % and 92 % respectively. Our work paves the way towards practical quantum communication systems that are able to surpass current noise and distance limitations, while not compromising on potential device-independence.

I. INTRODUCTION

High-dimensional entanglement is theoretically noise-resistant, but whether this advantage survives the operational constraints of real implementations remains unresolved. The paper addresses this gap with experiments using photonic spatio-temporal encodings.

  • Entanglement is a key resource for quantum information processing and device-independent quantum cryptography.
  • Photon loss and environmental noise threaten entanglement distribution over long distances outside protected laboratories.
  • SPDC naturally produces high-dimensional entanglement in photon-pair properties including energy-time, angle-angular momentum, and position-momentum.
  • High-dimensional encodings offer practical benefits such as natural availability in down-conversion and more bits per communicated photon.
  • Whether high-dimensional entanglement provides practical noise-resilience advantages depends on physical implementation and operational constraints.
  • The study conducts two experiments to test increased noise resilience using energy-time and transverse position-momentum entanglement.

II. PATHWAYS TO NOISE RESILIENCE

The paper identifies two routes for recovering certifiable entanglement from noisy high-dimensional states: finer discretisation or additional mutually unbiased measurements. These routes exploit either diluted noise or extra information about non-classical correlations.

  • Ideal bipartite entangled states can be represented in a Schmidt basis, but experimental imperfections and background photons degrade the state.
  • For isotropic states, the critical noise threshold before separability scales as pc = 1/(d+1).
  • The experiments do not assume a noise model when analysing entanglement; the white-noise model only motivates expectations.
  • Finite-resolution temporal and spatial measurements require discretisation, while increasing dimension can introduce crosstalk and additional noise channels.
  • Pathway I fine-grains the existing state space to dilute noise, whereas Pathway II uses additional mutually unbiased bases to obtain more state information.
  • Both pathways can recover entanglement in an assumption-free manner when standard two-basis measurements fail to certify it.

III. EXPERIMENTAL IMPLEMENTATION

Two photonic experiments implement the noise-resilience pathways using energy-time and orbital-angular-momentum entanglement generated by SPDC. Both use continuous-wave pumping, optical background noise, and photon-coincidence detection.

  • The energy-time experiment uses a narrow-bandwidth pump to obtain a large Schmidt number and ancillary polarization entanglement for time-domain interference.
  • Both experiments generate photon pairs in a nonlinear crystal pumped by a continuous-wave diode laser and expose them to an external noise source.
  • External noise is introduced with adjustable light sources near detectors or under daylight-like ambient conditions, and detection uses avalanche photodiodes.
  • The OAM experiment exploits momentum conservation in SPDC to generate photon pairs anti-correlated in orbital angular momentum.
  • Energy-time states are discretised by dividing a fixed time frame into d time bins, while OAM dimensionality is set by the selected spatial modes.

IV. ENERGY-TIME ENTANGLEMENT (PATHWAY I)

Pathway I increases energy-time entanglement’s certifiable noise tolerance by fine-graining a fixed time frame into more bins. The experiment observes steadily higher thresholds as dimension increases, despite increasing crosstalk.

  • IV. ENERGY-TIME ENTANGLEMENT (PATHWAY I): The entanglement witness is measured in two bases: direct time-bin detection and a superposition basis implemented with delayed interference.
  • IV. ENERGY-TIME ENTANGLEMENT (PATHWAY I): A postselection-free Franson interferometer preserves noninterfering events that would otherwise be discarded from the high-dimensional state space.
  • IV. ENERGY-TIME ENTANGLEMENT (PATHWAY I): The experiment fine-grains a fixed 320-clock-cycle frame into d = 10, 20, 40, and 80 time bins.
  • IV. ENERGY-TIME ENTANGLEMENT (PATHWAY I): Fine-graining increases crosstalk between time bins because of fundamental and technical limitations.

V. ORBITAL ANGULAR MOMENTUM ENTANGLEMENT (PATHWAY II)

The OAM experiment uses higher-dimensional mutually unbiased bases to increase noise tolerance, certifying entanglement at progressively larger noise fractions as dimension increases.

  • Pathway II: Higher dimensions provide more mutually unbiased bases, enabling Pathway II to increase the noise threshold through additional OAM-mode measurements.For prime-power dimensions, complete sets contain d + 1 mutually unbiased bases.
  • Certification criterion: Entanglement certification uses correlation visibilities measured across all d + 1 mutually unbiased bases, with the separable-state boundary given by a visibility sum exceeding 2.The experiment uses intensity flattening to measure correlations in the mutually unbiased bases.
  • Experimental trade-off: The OAM measurements fully characterize the generated states through high-dimensional projective measurements, but reach lower overall dimensions than the energy-time experiment.Higher dimensions may require custom-tailored phase matching or inclusion of radial as well as azimuthal modes.
  • Experimental results: 0.24 is the maximum certified noise fraction for d = 2, increasing to 0.48, 0.63, and 0.72 for d = 3, 5, and 7.The threshold begins to saturate at higher dimensions because measurement fidelity decreases and the state departs further from an ideal maximally entangled state.

VI. DISCUSSION

The experiments show that high-dimensional encoding can reveal entanglement obscured by realistic environmental noise, while practical gains remain a balance between robustness and added noise.

  • VI. DISCUSSION: The two experiments identify separate pathways in which increasing dimension improves noise resistance but also introduces additional noise.The authors describe a practical optimum where enhanced resistance outweighs dimension-dependent noise.
  • VI. DISCUSSION: Both pathways could in principle be combined using multiple time-bin MUB measurements or multi-outcome spatial measurements, potentially increasing measurable dimensionality and noise thresholds.The combined implementation is presented as a possibility rather than an demonstrated result.
  • VI. DISCUSSION: The external-noise model approximates long-distance communication dominated by detector dark counts and free-space communication affected by daylight background photons.The setup uses nearby constant-luminosity light sources to reduce the signal-to-noise ratio and create accidental detections.
  • VI. DISCUSSION: The energy-time setup could be directly adopted for long-distance or free-space communication, whereas the OAM implementation requires multi-outcome measurement technology.With current single-outcome OAM measurements, added modes experience the same environmental noise, worsening the robustness–noise trade-off.
  • VI. DISCUSSION: Developing protocols that directly use high-dimensional noisy entanglement is identified as the next challenge, including possible applications to QKD and other quantum-information tasks.The paper notes that noisy entangled states can provide advantages in entanglement-assisted communication and channel discrimination.

APPENDIX A: ENERGY-TIME ENTANGLEMENT EXPERIMENT

The energy-time experiment generates and measures time-bin entanglement with a Franson-interferometer setup, reconstructing a density matrix and applying a dimension-dependent witness.

  • Setup: The source combines SPDC in a ppKTP crystal with a hyperentangled photon-pair source, matched imbalanced PMZIs, and time-tagged detection.The crystal is designed for type-II quasi-phase matching, and the interferometers have a 2.67 ns arm imbalance.
  • Measurement: Energy-time measurement bases are selected by changing polarization measurements after the PMZIs, switching between computational and Franson bases.The polarization choice erases or reveals interferometer path information.
  • Data processing: Detection events are binned into dimension-dependent intervals of duration t_d = F/d, with only frames containing exactly one detection on each side retained.The fixed interferometer imbalance restricts the usable dimensions to d ∈ {10, 20, 40, 80} for a 320-clock-cycle frame.
  • Entanglement certification: The entanglement witness is computed from density-matrix elements reconstructed from experimental count matrices, including polarization-dependent information.For each dimension, W(ρ_ET, d) > 0 certifies that the state is not separable.

APPENDIX B: OAM ENTANGLEMENT - EXPERIMENT

The OAM experiment produces photon pairs with spatial-mode coherence and measures their orbital angular momentum independently after deterministic polarization-based separation.

  • Photon-pair generation: OAM-entangled photon pairs are generated by type-II SPDC in a 5 mm ppKTP crystal pumped by a 405 nm diode laser.Single-mode-fiber coupling is used to improve transverse coherence and the mode profile needed for high-dimensional entanglement.
  • Photon separation: The crystal is tuned for wavelength-degenerate, orthogonally polarized photon pairs at 810 nm, whose polarization separates the photons at a beam splitter.The spatial mode of each photon can then be measured independently.

1. Pathway I - Entanglement witness derivation

The Pathway I analysis uses an entropy-vector entanglement witness whose positivity certifies bipartite entanglement, with density-matrix terms reconstructed or bounded from experimental count matrices.

  • Witness criterion: The entropy-vector witness sums selected off-diagonal density-matrix terms while penalising diagonal terms.If all entropy-vector components are non-zero, the state cannot be expressed as a convex combination of separable states.
  • Witness criterion: A positive W1(ρAB, C, {A}) is a necessary condition for certifying entanglement in the bipartite state.For the bipartite case, the relevant entropy-vector witness has one component.
  • Witness construction: The witness set C is fixed to neighbouring index pairs rather than optimised, assuming the produced state is close to maximally entangled.The chosen set uses adjacent indices, while the maximally entangled state gives the witness its maximum value.
  • Experimental reconstruction: The energy-time density matrix is reconstructed from count matrices measured in computational and Franson bases, with polarization used as a proxy for time-bin detection.Diagonal elements are obtained by eliminating polarization; non-directly measured terms are bounded from count combinations.
  • Experimental reconstruction: The witness is adapted to the interferometer imbalance through an integer shift f and evaluated using measured diagonal terms plus bounded off-diagonal contributions.The derivation combines count matrices to bound the first witness term, while the square-root term is directly measured.

3. Pathway I - Discretization and correlation measurements

Pathway I increases the effective time-bin dimension by dividing a fixed-duration frame into shorter bins. Measurements show that higher dimensions can turn a negative witness positive, while timing crosstalk increases noise.

  • Discretization: The fixed 26.2 ns frame is divided into d time-bins, so increasing d decreases the bin duration td while keeping noise per frame constant.The experiment uses d ∈ {10, 20, 40, 80}.
  • Correlation measurements: At the same high external-noise condition, d = 10/20 gives a negative witness whereas d = 40/80 gives a positive witness.The positive values certify entangled states, while the negative values correspond to separable states under the witness criterion.
  • Correlation measurements: Increasing dimension spreads noise quadratically across off-diagonal elements while correlated diagonal events spread linearly, but timing-jitter crosstalk rises at higher d.The crosstalk is visible for d = 40 and d = 80.

4. Pathway I - Error analysis

Error bars for the entanglement-detection plot are generated by Poisson resampling of the experimental count matrices.

  • Error analysis: Poisson resampling treats photon detections in each count matrix as the distribution mean for estimating error bars.This assumes the detection probability remains constant during the experiment.

5. Pathway II - Correlation measurements

Pathway II increases OAM entanglement’s noise resilience by adding mutually unbiased basis measurements. For d = 3, certification extends from 0.36 noise with two MUBs to 0.48 with all four.

  • Two MUBs: 0.36 noise fraction remains certifiable with measurements in two MUBs, B0 and B3.The visibility sum remains larger than 4/3.
  • Three MUBs: 0.45 noise fraction remains certifiable when three MUBs, B0, B2, and B3, are measured.The visibility sum remains larger than 5/3.
  • Four MUBs: 0.48 noise fraction remains certifiable when measurements include all four MUBs.The visibility sum remains larger than 2, demonstrating the largest noise resilience reported for this pathway.
  • Measurement strategy: Increasing noise shifts the verification threshold so that more MUB measurements are required to certify entanglement.High-dimensional entangled states enable measurements in more than three MUBs.
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