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High-dimensional quantum cryptography with twisted light
Mohammad Mirhosseini, Omar S. Magaña-Loaiza, Malcolm N. O'Sullivan, Brandon Rodenburg, Mehul Malik, Martin P. J. Lavery, Miles J. Padgett, Daniel J. Gauthier, Robert W. Boyd
TL;DR
Polarization-based QKD is limited by qubit capacity and tight error bounds, motivating higher-dimensional spatial-mode encoding. This proof-of-principle experiment uses OAM and mutually unbiased ANG modes with fast DMD generation and mode sorting. It achieves 2.05 bits per sifted photon with a 7-dimensional alphabet, while its security analysis assumes an infinite key and remains vulnerable to photon-number-splitting attacks.
Problem
Polarization-based QKD uses qubit encoding, limiting information per photon, while spatial-mode QKD requires practical high-dimensional implementations and robust security analysis.
Method
The experiment uses a 7-dimensional alphabet in OAM and mutually unbiased ANG bases, generated with a DMD and measured using an efficient mode-sorting technique.
Results
2.05 bits per sifted photon are achieved, with 93% mode-sorting separation efficiency and a 10.5% QBER below security bounds for intercept-resend and coherent attacks.
Takeaways & Limitations
Spatial-mode encoding demonstrates practical high-dimensional QKD with increased information capacity and a path toward real-world multilevel quantum communication.
Takeaways & Limitations
The secure-key analysis assumes an infinite key, while the finite-key experiment remains vulnerable to photon-number-splitting attacks.
Abstract
from arXiv · showhide
Quantum key distributions (QKD) systems often rely on polarization of light for encoding, thus limiting the amount of information that can be sent per photon and placing tight bounds on the error that such a system can tolerate. Here we describe a proof-of-principle experiment that indicates the feasibility of high-dimensional QKD based on the transverse structure of the light field, allowing for the transfer of more than 1 bit per photon. Our implementation uses the orbital angular momentum (OAM) of photons and the corresponding mutually unbiased basis of angular position (ANG). Our experiment uses a digital micro-mirror device for the rapid generation of OAM and ANG modes at 4 kHz, and a mode sorter capable of sorting single photons based on their OAM and ANG content with a separation efficiency of 93\%. Through the use of a 7-dimensional alphabet encoded in the OAM and ANG bases, we achieve a channel capacity of 2.05 bits per sifted photon. Our experiment shows that, in addition to having an increased information capacity, QKD systems based on spatial-mode encoding will be more tolerant to errors and thus more robust against eavesdropping attacks.
1. Introduction
Conventional polarization-based QKD is limited by qubit capacity, while spatial-mode encoding with OAM and ANG modes offers higher-dimensional transmission and improved propagation properties. The experiment demonstrates a 7-dimensional OAM/ANG QKD implementation using fast mode generation and sorting.
- Motivation: QKD conventionally encodes information in photon polarization, limiting systems to qubit state spaces.Spatial degrees of freedom and multilevel quantum states are identified as resources for increasing information capacity and robustness.
- Motivation: Diffraction mixes transverse spatial modes, increasing QBER and limiting secure key rates in long-haul links.OAM modes can alleviate this effect because their rotational symmetry helps preserve orthogonality during propagation.
- Encoding: The experiment encodes information in 7-dimensional OAM and mutually unbiased ANG bases.ANG modes are formed from equal-amplitude combinations of OAM modes with |ℓ|≤3 and remain mutually unbiased with respect to OAM.
- Implementation: Alice uses a DMD and AOM to generate attenuated spatial-mode pulses, while Bob’s mode sorter maps OAM and ANG modes to separated fiber-coupled spots.The communication link uses a 4f telescope, and the setup supports preparation and measurement in complementary bases.
- Key generation: The sifted key is formed after Alice and Bob publicly compare bases and discard measurements made in different bases.Symbols are converted to binary, randomized, and processed through error correction and privacy amplification.
Transmission and Detection Efficiency
The experiment quantifies detection performance through mode-sorting errors, background and detector effects, and mutual information between transmitted and received symbols. Measurements use conditional-probability matrices built from repeated transmissions of each symbol.
- Optical transmission: 85% power transmission efficiency was measured for the mode sorter’s refractive elements.
- Detection chain: 18% coupling efficiency was measured between the transformed modes and the seven-fiber array, while APD quantum efficiency was η = 0.65.
- Detector noise: 50 c/s typical dark-count rate and 0.3% after-pulsing probability characterize each detector.
- Conditional detection: The conditional-probability matrix compares Bob’s detected modes with Alice’s sent modes using theoretical ideal predictions and experimental results.The experiment sends 14 million pulses with µ = 0.1 photon per pulse to construct the matrix.
- Measurement procedure: Repeating one symbol many times per matrix row eliminates mode-switching overhead and improves error-rate measurement accuracy.
- Error sources: The total measured error rate is 10.5%, with δ = 0.065 from imperfect sorting and ϵ = 0.04 from dark counts and background effects.The error probability ϵ is attributed to stray light, thermally induced dark counts, and after-pulsing.
Secure key rate
The experiment evaluates secure key generation against individual, intercept-resend, and coherent attacks. Its high-dimensional encoding supports positive secure-key performance, while finite-key security remains unresolved.
- Key-rate performance: 6.8 bits/sec is the calculated net key rate in burst mode, based on a sifted rate of 4 photons/sec at frep = 4 kHz.The DMD’s burst mode is limited by its internal memory, while long-key generation is reduced to approximately 1 Hz by hologram loading and synchronization.
- Security analysis: The measured QBER lies below theoretical security bounds for both intercept-resend and coherent attacks.For coherent attacks, the allowed error rate increases with system dimension, while increasing MUBs beyond two lowers key rate without improving coherent-attack security.
- Security analysis: 0.35 bits of information per sifted photon is estimated for Eve in cloning-based individual attacks.The estimate uses the experimental error rate and treats cloning-based eavesdropping as optimal for individual qudit attacks.
- Security analysis: 1.7 secure bits per sifted photon remain after privacy amplification removes Eve’s estimated information.
- Limitations: Finite-key security is not established because the secure-key analysis assumes an infinite key, whereas the experiment produces only a finite key.A finite-key security analysis is identified as future work.
- Key-rate performance: 6.5 bits per second is the estimated secure key rate after basis reconciliation and privacy amplification in burst mode.The measured raw key generation rate is 16.4 bits per second, and the estimate exceeds previous spatial-mode protocols by more than three orders of magnitude.
Photon number splitting attacks
Photon-number-splitting attacks threaten this implementation because weak coherent pulses can contain multiple photons, especially when channel loss is present.
- Photon number splitting attacks: PNS attacks are more serious in lossy links because Eve can replace the channel and selectively transmit signals to Bob.The paper describes this as a threat when imperfect single-photon sources are used.
- Photon number splitting attacks: The security condition requires Bob’s detection probability to exceed the multiphoton-pulse probability.The condition is expressed as pdetection > pmulti.
- Photon number splitting attacks: The detection probability combines signal detections and detector dark counts.The paper states that total detection events include both contributions.
- Photon number splitting attacks: 5 × 10^-3 multiphoton probability exceeded approximately 2 × 10^-3 detection probability, so PNS security could not be guaranteed.These are the experimental values reported for pmulti and pdetection.
4. Steps towards practical QKD with OAM modes
The paper identifies several requirements for moving OAM-based QKD toward practical deployment, including secure key rate, transmission performance, and implementation robustness.
- 4. Steps towards practical QKD with OAM modes: Practical OAM-based QKD requires addressing several challenges beyond demonstrating improved secure key rate.The paper frames secure key rate and free-space transmission as practical concerns.
Fast key generation:
Fast mode generation is necessary for practical key rates, but the demonstrated and proposed technologies span kHz to GHz switching regimes with important implementation constraints.
- Fast key generation:: GHz key rates are required for practical QKD, whereas DMD holography currently provides kHz mode switching.The paper reports 4 kHz operation in this experiment and identifies faster technologies under development.
- Fast key generation:: MHz or potentially GHz OAM generation is possible with on-chip resonators or Q-plates, but ANG states require amplitude and phase modulation.Free-space holography is described as well suited to generating ANG states.
- Fast key generation:: A static-hologram and AOM approach reached MHz switching, but wavelength changes could enable spectrum-based side-channel attacks.The wavelength shift prevents applying that method directly to QKD in the paper’s assessment.
- Fast key generation:: Higher-efficiency spatial light modulators or custom anti-reflection-coated refractive elements could increase transmission efficiency six-fold.The paper presents this as a way to improve the detection-system throughput.
- Fast key generation:: Near-infrared operation can reduce scattering loss in air.This is identified as a method for minimizing atmospheric scattering loss.
Turbulence mitigation:
Atmospheric turbulence mixes neighboring OAM and ANG modes, while larger mode alphabets increase information per photon but introduce aperture and security-analysis constraints.
- Turbulence mitigation:: Atmospheric turbulence degrades spatial mode profiles and mixes neighboring OAM and ANG modes during propagation.The paper lists using every other mode and adaptive optics as mitigation strategies.
- Larger dimensionality:: Sorting 25 OAM and ANG modes previously achieved 4.17 bits per detected photon on average.The paper links larger mode sets to increased information capacity.
- Larger dimensionality:: Increasing the number of modes raises the information carried by each photon and the secure bit rate.The supported number of modes is ultimately limited by transmitting and receiving aperture sizes.
- Security assumptions:: Security analysis assumes an infinitely long key and uniform error rate and transmission efficiency across modes.Finite-key post-processing would need to be measured for a more rigorous secure-key-rate calculation.
- Security scope:: The implementation is secure against intercept-resend and cloning-based attacks but remains vulnerable to photon-number-splitting attacks.The paper proposes reducing system loss or using decoy states to avoid this limitation.
- Experimental platform:: The experiment uses a 7-dimensional OAM/ANG alphabet, 4 kHz DMD generation, and 93% mode-measurement accuracy.These results establish the proof-of-principle spatial-mode implementation described in the conclusion.
Appendix A. Diffraction and radial modes
The appendix considers whether OAM modes’ differing radial profiles allow Eve to gain information from photons outside Bob’s receiving aperture. It also identifies intercept and intercept as a denial-of-service attack without security loss.
- Diffraction and radial modes: Larger OAM mode indices ℓ produce larger far-field rms radii, raising concern about photons falling outside Bob’s aperture.The concern arises from the different radial profiles of OAM modes.
- Aperture geometry: Figure A1 identifies Bob’s and Eve’s receiving apertures.
- Attack implication: Intercept and intercept is also called denial of service because Eve detects a photon without retransmitting it.Alice and Bob discard frames with no photons during basis reconciliation.
Intercept and intercept attack
This attack models Eve detecting photons outside Bob’s apertures in a random basis, then resending states into Bob’s aperture. The aperture geometry is treated generally, with all listed radii ranging from zero to infinity.
- Intercept and resend attack: Eve detects a photon outside Bob’s apertures in a random basis and resends a measurement-determined state into Bob’s aperture.The geometry of Bob’s and Eve’s apertures is shown in Fig. A1.
- Intercept and resend attack: The analysis considers the most general aperture geometry because RB, RE1, and RE2 are each bounded between zero and infinity.
Intercept and resend attack
The appendix calculates Eve’s detection probabilities for OAM and ANG modes under random mode selection. It concludes that neither basis reveals information about the other, even for photons detected outside Bob’s aperture and at any propagation plane.
- OAM propagation: The optical field of an OAM mode is represented at an arbitrary propagation plane z.
- OAM probabilities: For uniformly random OAM inputs, where d = 2N + 1, the appendix evaluates Eve’s probabilities for detecting individual and arbitrary OAM modes.Each OAM mode is selected with probability 1/d.
- ANG probabilities: The appendix similarly evaluates Eve’s probabilities for detecting individual and arbitrary ANG modes and simplifies the result using the ANG-mode definition.
- Security consequence: Eve is equally likely to detect a photon from either basis at any radial range, so photons outside Bob’s aperture provide no information.This result remains valid at every plane z from near field to far field.