The security of classical data encryption is threatened by the potential of quantum computers to easily solve computationally hard problems. Quantum communication, leveraging quantum physics for secure information exchange, offers a solution. Quantum key distribution (QKD) is a powerful tool for unconditionally secure quantum communication. Microwave quantum communication is particularly promising for future quantum networks due to its compatibility with superconducting quantum processors and existing near-distance communication standards. This work experimentally demonstrates a continuous-variable (CV) QKD protocol based on displaced squeezed microwave states, generated and detected using superconducting parametric devices. Unlike discrete-variable QKD, CV-QKD protocols typically use less demanding measurement techniques like homodyne or heterodyne detection. Optical CV-QKD has shown success in large networks, achieving high secure bit rates. The advancements in superconducting circuit-based quantum information processing at microwave frequencies make microwave CV-QKD highly attractive. Its frequency compatibility with superconducting processors allows for secure communication, and the use of Gaussian states (coherent or squeezed) simplifies generation and control. Theoretical studies suggest the viability of microwave CV-QKD in open-air conditions, potentially complementing existing short-range communication protocols. A general CV-QKD protocol involves a sender (Alice) and a receiver (Bob) exchanging information using coherent or squeezed states. A malicious eavesdropper (Eve) attempts to intercept the key encoded in the states' quadratures. The security relies on the single use of each state, preventing Eve from gaining information through averaging. For protocols using squeezed states, single-shot quadrature measurements (SQMs) are crucial. In the microwave domain, superconducting phase-sensitive amplifiers provide the equivalent of optical homodyne detection. This experiment implements a one-way CV-QKD protocol using Gaussian modulation of squeezed microwave states in a cryogenic environment, assessing its practical limitations using superconducting Josephson parametric amplifiers (JPAs) for SQMs. The focus is on trusted preparation losses and detection noise, with the quantum channel modeled as a lossy channel with added Gaussian noise. Security is assessed against collective Gaussian attacks by Eve, considering both conventional finite-size effects and quantum channel parameter estimations.

The paper extensively reviews existing literature on quantum key distribution (QKD), highlighting the advantages of continuous-variable (CV) protocols over discrete-variable protocols. It cites several key papers demonstrating the success of optical CV-QKD in achieving high secure key rates over long distances. The authors also cite significant advancements in superconducting circuit-based quantum information processing, emphasizing the potential of microwave frequencies for quantum communication. Theoretical work predicting the feasibility of microwave CV-QKD in open-air conditions is referenced, along with studies on microwave attenuation in different atmospheric conditions. Several publications on the development and characterization of Josephson parametric amplifiers (JPAs) for quantum measurement are reviewed, particularly those demonstrating their capability for phase-sensitive amplification and single-shot quadrature measurements. The literature on security analysis of CV-QKD protocols, including the treatment of Gaussian attacks and finite-size effects, is also reviewed. Specific references are made to papers detailing the theory of practical CV-QKD implementations, methods for secure key rate calculation considering finite key lengths, and various reconciliation techniques used in CV-QKD.

The experiment implements a one-way CV-QKD protocol using displaced squeezed microwave states. Alice generates squeezed states using a superconducting flux-driven JPA, operating in a phase-sensitive regime. The states are displaced in phase space using a cryogenic directional coupler to encode a symbol from a Gaussian distribution. The chosen quadrature (q or p) is random for each symbol. The symbols constitute Alice's key. To maintain security, the variance of the squeezed and displaced quadratures is fixed to prevent Eve from determining the encoding basis. A constant average squeezing level is maintained throughout. The displaced squeezed states are sent through a quantum channel simulated using a second directional coupler introducing fixed losses and tunable Gaussian noise (simulating thermal background). Bob uses a second JPA for single-shot quadrature measurements (SQMs), with the quantum efficiency dependent on the added JPA noise. The amplified quadrature is measured, with information on the deamplified quadrature being inaccessible. The measurement uses a superconducting Josephson parametric amplifier (JPA), achieving high quantum efficiency and phase-sensitive amplification. The single-shot measurements are obtained without signal averaging. The process includes sifting where Alice discloses the encoding basis, discarding half the data. Direct reconciliation is used for error correction. The single-shot measurements' strong phase-sensitive amplification is analyzed using the covariance matrix formalism, considering the amplification gain and added noise. The quadrature amplification noise is characterized by the quantum efficiency. The mutual information (MI) between Alice and Bob's keys is calculated to assess correlations, with the MI expressed in terms of signal-to-noise ratio. The accuracy of the model is validated using the Bhattacharyya coefficient. Security analysis is performed using the Holevo quantity, which bounds the information leaked to Eve, assuming a collective Gaussian attack. Both asymptotic and finite-size security analyses are conducted. Finite-size effects are considered by adding finite-size induced terms to the security analysis. The effect of a realistic sifting procedure is included by multiplying the secret key rates by 50% in the asymptotic limit. The experimental setup is described in detail, including the superconducting JPAs, directional couplers, and measurement techniques. Independent calibration measurements are used to extract experimental parameters for the theoretical model. Wigner tomography is used to obtain these parameters, with photon number calibration via Planck spectroscopy. The Holevo quantity is calculated using an integral over the ensemble of states.

The experiment successfully demonstrates a microwave CV-QKD protocol with single-shot quadrature measurements (SQMs). High-fidelity SQMs are achieved using superconducting JPAs, showing a quantum efficiency well above 50%. The experimental results exhibit strong correlations between Alice's and Bob's keys in the amplified quadrature, with negligible correlations in the deamplified quadrature, consistent with the Heisenberg uncertainty principle. The mutual information between Alice and Bob's keys is accurately modeled using a theoretical model, validated by a high Bhattacharyya coefficient (near unity). Unconditional security is demonstrated in the asymptotic regime, showing a positive secret key. The performance is improved by adding finite trusted noise on the preparation side. The experiment achieves unconditional security up to a specific level of coupled noise photons. Analysis considering the finite key length demonstrates secure communication for a key length of 16,665 symbols. Accurate statistical estimations of channel losses and coupled noise are achieved experimentally. The results suggest that unconditionally secure microwave communication is feasible in a cryogenic environment (up to 1190 m) and in open-air conditions (up to 84 m). The main limiting factor is identified as the total noise, composed of coupled noise and amplification noise. The experiment demonstrates that superconducting phase-sensitive amplifiers provide a microwave equivalent of optical homodyne detection. The potential for using these SQMs in quantum state tomography is highlighted. An experimental secret key rate of 42 kbit/s is achieved, with an upper bound estimated at 152 kbit/s based on the Shannon-Hartley theorem. Phase stabilization of the JPAs is identified as a key factor limiting the key rate.

The successful demonstration of microwave CV-QKD with single-shot measurements addresses the challenge of achieving unconditionally secure quantum communication at microwave frequencies. The findings validate the feasibility of secure microwave communication using current technology, paving the way for secure local area quantum networks. The results' implications for secure short-range open-air communication are significant, particularly given the robustness of microwave signals to weather conditions. The experimental setup and analysis provide a valuable benchmark for future developments in microwave QKD. The identification of noise as a primary limiting factor guides future research towards improved phase-sensitive amplifiers and noise reduction techniques. The demonstrated SQMs offer a promising approach for quantum state tomography applications. The achievement of a substantial secret key rate, though limited by phase stabilization and bandwidth, points towards further improvements achievable through technical enhancements.

This work demonstrates the experimental feasibility of unconditionally secure microwave CV-QKD using single-shot measurements. The protocol achieves secure communication over significant distances in both cryogenic and open-air conditions. Key limitations are identified, including noise and phase stabilization, providing directions for future improvements focusing on enhanced phase-sensitive amplifiers, noise reduction, and bandwidth optimization. This achievement contributes to the development of secure local area microwave quantum networks.

The main limitations of this study are related to the total noise, primarily originating from the coupled noise and amplification noise of the JPAs. The codebook size, limited by JPA compression effects, also affects the performance. Phase stabilization of the JPAs is another factor limiting the achievable secret key rates. While the open-air communication distance is estimated, practical implementation may face challenges related to the development of low-loss and broadband interfaces between the cryogenic system and antennae. The analysis relies on the assumption of a collective Gaussian attack, and the generalization to more complex attacks requires further investigation.