Demonstration of microwave single-shot quantum key distribution

Quantum key distribution (QKD) enables secure key exchange between authenticated remote parties based on quantum mechanical principles rather than computational hardness. Continuous-variable (CV) QKD is technologically compatible with existing communication platforms and can deliver high secret key rates over large distances with less demanding experimental requirements than discrete-variable schemes. In optics, CV-QKD has been deployed in networks with high secure bit rates, while superconducting circuits at microwave frequencies have advanced rapidly, making microwaves naturally compatible with superconducting quantum processors. Recent theory indicates microwave CV-QKD can operate in open-air conditions and is resilient to weather compared to optical links. For squeezed-state protocols where information is encoded in a single quadrature, Bob must perform single-shot quadrature measurements (SQMs). In the optical domain this is done via homodyne detection; here, an equivalent microwave detection is realized using superconducting phase-sensitive amplifiers. In this work the authors implement a one-way CV-QKD protocol using Gaussian modulation of propagating squeezed microwave states in a cryogenic environment, employing Josephson parametric amplifiers (JPAs) for generation and SQMs, and analyze unconditional security including finite-size effects and channel parameter estimation.

The study builds on extensive CV-QKD developments demonstrating compatibility with classical platforms and high secure rates in optical networks. It contrasts CV-QKD’s homodyne/heterodyne detection with more demanding single-photon detection in discrete-variable protocols. Prior work in superconducting circuits at microwave frequencies shows strong potential for scalable quantum computing and microwave quantum communication, including displacement of propagating squeezed microwaves and quantum links between cryogenic systems. Theoretical analyses suggest feasibility of microwave CV-QKD over open-air channels with resilience to weather. Security foundations rely on the optimality of collective Gaussian attacks and entangling cloner models, and practical CV-QKD considers finite-size analyses and trusted noise models.

- Protocol and states: Alice prepares displaced squeezed microwave states at carrier frequency ω/2π = 5.48 GHz. For each symbol, she randomly chooses to squeeze along the q- or p-quadrature using a flux-driven JPA operated phase-sensitively (pump at ωp = 2ω), then displaces the state via a cryogenic directional coupler. Symbols a are drawn from a Gaussian codebook with variance σa^2, forming key KA = {a}. The encoding axis matches the squeezing axis. For maximal security, the ensemble obeys σs^2 + σa^2 = σ0^2 to conceal the encoding basis from Eve. The average squeezing level is kept at S = 3.6(4) dB. - Quantum channel: Implemented by a second cryogenic directional coupler (highly asymmetric microwave beam splitter) with fixed power transmissivity 1 − ε = 0.9885 (loss ε) and a tunable coupled Gaussian noise characterized by an average photon number ñ (simulating different thermal backgrounds). This provides an untrusted noisy-loss channel under Eve’s control. - Detection (SQMs): Bob employs a second JPA in a phase-sensitive configuration to strongly amplify one quadrature and deamplify the conjugate, enabling single-shot quadrature measurements with high quantum efficiency without averaging. The amplified single-mode covariance is modeled as Vout = J V J + N with J = [[0, 1/√G],[1/√G, 0]], degenerate gain G, and N a diagonal added-noise matrix. For G ≫ 1 the deamplified quadrature becomes inaccessible, enforcing Heisenberg-limited single-quadrature access. Quadrature quantum efficiency is η = 1/(1 + 2Nx). Example operating point: G = 19.1(4) dB and η = 65(2)%. Signals are digitized on an FPGA to extract single I/Q points per symbol to obtain βi, yielding Bob’s key KB. - Post-processing: Basis sifting (Alice reveals basis after all SQMs; half the data retained) and direct reconciliation (DR) are considered. In asymptotic secret key rate plots, sifting is accounted for by a 50% factor. Error correction is assumed classically with Alice’s key as reference. - Modeling and calibration: A full operator model of the chain T = H L4 SB L3 CE L2 CA L1 S1 transforms input states to the output, where S1 and SB are (noisy) squeeze operators for source and measurement JPAs, CA/CE are beam splitters for displacement and channel injection with transmissivities TA and TE, H models the HEMT amplifier with added noise N, and Li are path-loss beam splitters. Parameters are obtained from independent calibration (Planck spectroscopy photon-number calibration and Gaussian Wigner tomography). The single-shot quadrature distributions are compared to the calibrated model (no free fitting), and agreement is quantified via the Bhattacharyya coefficient. - Performance metrics: Mutual information I(KA:KB) = (1/2) log2(1 + SNR) computed from experimental data for amplified (XB) and deamplified (YB) quadratures. Eve’s information is upper-bounded via the Holevo quantity XE under optimal collective Gaussian (entangling cloner) attacks. Secret key per symbol Ksec satisfies I − XE ≤ Kexp, with finite-size corrections included (finite key length N and channel parameter estimation).

- Single-shot detection validation: For amplified quadrature, substantial mutual information between Alice and Bob is observed; for the deamplified quadrature, MI is near zero, consistent with single-quadrature access. Model-experiment agreement is excellent with Bhattacharyya coefficients B(P(XB), Pm(XB)) = 99.98(1)% and B(P(YB), Pm(YB)) = 99.87(2)%. - Security (asymptotic): Positive secret key per symbol is obtained, demonstrating unconditional security. Secret key remains positive up to a coupled noise level of ≈0.062(2) photons for the measured configuration. - Finite-size security: Secure communication is experimentally demonstrated for a finite key length N = 16,665 symbols. Finite-size terms lower the tolerable noise cutoff compared with the asymptotic case; modeling indicates viable keys also for larger N (e.g., 10^6). - Channel and distance feasibility: Based on experimental parameters (ε with 1 − ε = 0.9885; noise η = ηth/2), the maximally tolerable losses versus thermal background photon number ηth are mapped. Estimated secure distances: up to d ≈ 1190 m in a cryogenic environment at ≈30 mK with superconducting coax losses γ ≈ 1.0 × 10^−3 dB/m; up to d ≈ 84 m in open air at 300 K around 5 GHz with atmospheric absorption γ = 6.3 × 10^−6 dB/m (path losses −80 dB could be compensated by a pair of ~2 m parabolic antennas at ~40 dB gain each). - Rates: Extrapolated experimental secret key rate ≈42 kbit/s; Shannon-Hartley upper bound on raw rate up to ≈152 kbit/s for the 2nd run with 400 kHz measurement bandwidth. - Protocol enhancements: Adding finite trusted preparation noise on Alice’s side improves security performance; increased codebook size, squeezing level, and measurement quantum efficiency further enhance key rates within device limits.

The findings confirm that single-shot quadrature measurements with phase-sensitive JPAs provide a microwave equivalent of optical homodyne detection, enabling high-fidelity extraction of displacement information from propagating squeezed states. Unconditional security is achieved under collective Gaussian attacks, both asymptotically and for finite-size keys, validating the practicality of microwave CV-QKD with current technology. The key limiting factor is total noise (untrusted coupled channel noise plus trusted detection/amplification noise). Codebook size and squeezing can improve performance but are constrained by JPA compression at higher input powers. The demonstrated approach also supports quantum state tomography of itinerant microwave fields and could extend to non-Gaussian states. Estimated secure distances in cryogenic links (∼1.2 km) and short-range open-air (∼84 m) indicate strong potential for secure local microwave quantum networks, provided low-loss interfaces to antenna systems can be realized. Enhancements such as higher-efficiency or traveling-wave parametric amplifiers, improved phase stabilization and filtering, and multiplexing (time/frequency/code) could substantially increase achievable secure key rates.

This work experimentally realizes a one-way continuous-variable QKD protocol at microwave frequencies using propagating displaced squeezed states and single-shot quadrature measurements via superconducting JPAs. The protocol achieves unconditional security against collective Gaussian attacks, including finite-size effects, and demonstrates accurate channel parameter estimation. Modeling and measurements agree closely, validating genuine single-shot detection. The results indicate feasibility of secure microwave quantum communication over ∼1.2 km in cryogenic links and ∼84 m in open air, with extrapolated secret key rates in the tens of kbit/s and potential raw rates exceeding 100 kbit/s. Future research should focus on increasing codebook size and squeezing within device limits, improving measurement quantum efficiency, adopting traveling-wave parametric amplifiers for higher saturation power and bandwidth, implementing multiplexing to boost rates, and developing low-loss cryogenic-to-antenna interfaces for open-air links.

- Total noise budget: Security and key rates are limited by coupled channel noise and measurement chain noise (JPA added noise, intrinsic losses, pump-induced noise, higher-order nonlinearities). - Device constraints: JPA compression near −130 dBm limits usable codebook size and displacement magnitude; measurement quantum efficiency <100% reduces performance. - Finite-size effects: Finite key length (e.g., N = 16,665) and channel parameter estimation induce penalties that lower tolerable noise compared to the asymptotic limit. - Assumptions: Security analysis assumes trusted preparation/detection noise and optimal collective Gaussian (entangling cloner) attacks; deviations or additional side channels are not treated. - Practical links: Open-air estimates neglect uncompensated path loss (assumed mitigable by high-gain antennas) and focus on unavoidable absorption; cryogenic and antenna interface losses/bandwidth constraints remain technological challenges. - Phase stability and bandwidth: Phase stabilization of JPAs and limited measurement bandwidth (400 kHz) currently constrain achievable secret key rates.