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High-Dimensional Intra-City Quantum Cryptography with Structured Photons
Alicia Sit, Frédéric Bouchard, Robert Fickler, Jérémie Gagnon-Bischoff, Hugo Larocque, Khabat Heshami, Dominique Elser, Christian Peuntinger, Kevin Günthner, Bettina Heim, Christoph Marquardt, Gerd Leuchs, Robert W. Boyd, Ebrahim Karimi
TL;DR
The paper addresses the lack of experimental verification of high-dimensional single-photon QKD outside the laboratory. It implements a 4-dimensional BB84 system using polarization and OAM over a 0.3 km intra-city free-space link, and numerically evaluates secure key rates from experimental constraints. The system achieves laboratory-validated MUB performance and supports secret-key-rate analysis under the measured conditions.
Problem
High-dimensional single-photon QKD had not been experimentally verified outside the laboratory.
Method
The system combines polarization and OAM state encoding with experimentally measured detection matrices and dual optimization for secret-key-rate calculation.
Results
Laboratory bit error rates were 0.83% in dimension 2 and 1.83% in dimension 4 for ℓ = 2.
Takeaways & Limitations
The numerical approach provides a lower bound on achievable secure key rates and agrees well with theoretical rates for the tested dimension-4 MUBs.
Abstract
from arXiv · showhide
Quantum key distribution (QKD) promises information-theoretically secure communication, and is already on the verge of commercialization. Thus far, different QKD protocols have been proposed theoretically and implemented experimentally [1, 2]. The next step will be to implement high-dimensional protocols in order to improve noise resistance and increase the data rate [3-7]. Hitherto, no experimental verification of high-dimensional QKD in the single-photon regime has been conducted outside of the laboratory. Here, we report the realization of such a single-photon QKD system in a turbulent free-space link of 0.3 km over the city of Ottawa, taking advantage of both the spin and orbital angular momentum photonic degrees of freedom. This combination of optical angular momenta allows us to create a 4-dimensional state [8]; wherein, using a high-dimensional BB84 protocol [3, 4], a quantum bit error rate of 11\% was attained with a corresponding secret key rate of 0.65 bits per sifted photon. While an error rate of 5\% with a secret key rate of 0.43 bits per sifted photon is achieved for the case of 2-dimensional structured photons. Even through moderate turbulence without active wavefront correction, it is possible to securely transmit information carried by structured photons, opening the way for intra-city high-dimensional quantum communications under realistic conditions.
Part 1: MUTUALLY UNBIASED BASIS
The paper defines mutually unbiased bases (MUBs) for encoding quantum states and specifies their use in dimensions 2 and 4. Laboratory measurements compare theoretical and experimental detection probabilities for these bases.
- For dimension d equal to a prime power, d + 1 mutually unbiased bases can be found.
- In dimension 2, photons are represented by compound states combining polarization Π with orbital angular momentum ℓ.
- Laboratory bit error rates were 0.83% in dimension 2 and 1.83% in dimension 4 for ℓ = 2.
- In dimension 4, the natural basis combines horizontal or vertical polarization with OAM values ±ℓ.
- The dimension-4 MUBs are generated from matrices defining the {|ψ⟩i} and {|ϕ⟩j} state sets.
Part 2: GENERATION OF IMPLEMENTED MUBS IN DIMENSIONS 2 AND 4
The implemented MUBs are generated with q-plates and waveplate sequences tailored to the encoding dimension. Bob mirrors Alice’s optical transformations to project received photons onto selected states.
- In dimension 2, a half-wave plate followed by a q-plate generates the MUBs with ℓ = 2q.
- In dimension 4, a q-plate is sandwiched between either half-wave plates or quarter-wave plates to generate the two MUB sets.
- Bob uses Alice’s waveplate angles in reverse sequence to project received photons onto particular states.
Part 3: EXPERIMENTAL DATA
The experiment constructs probability-of-detection matrices from repeated coincidence measurements across the intra-city link, with target correction used for the reported data. A separate turbulent-night dataset tests dimension-4 performance under stronger turbulence.
- Coincidence counts are accumulated for 200 ms, with fifty data points recorded for each Bob measurement.
- Probability-of-detection matrices are obtained by averaging each measurement’s data points and normalizing over every state sent by Alice.
- The dimension-2 dataset includes normalized raw data and target-corrected data measured across the intra-city link.
- The dimension-4 dataset includes normalized raw data and target-corrected data measured across the intra-city link.
- A separate normalized raw matrix records dimension-4 performance on a turbulent night.
Part 4: NUMERICAL APPROACH FOR THE SECRET KEY RATE CALCULATION
The paper computes secret-key rates numerically from experimental constraints using a dual optimization formulation. The optimization agrees with theoretical rates and provides a lower bound on the achievable secure key rate.
- The secret-key-rate calculation uses a dual optimization problem developed as an efficient numerical approach for unstructured QKD.
- Alice’s and Bob’s raw-key measurements define ZA and ZB, while γi values are determined from averages of experimental measurements.
- For generalized BB84 in dimension 4 with two MUBs, experimental constraints are expressed through coarse-grained error operators in the MUBs.
- The numerical approach may be extended to two-way classical communication to tolerate higher qubit error rates.
- The numerical optimization saturates the secure-key-rate bound and agrees well with theory for different average error rates.