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Continuous variable quantum key distribution with two-mode squeezed states
Lars S. Madsen, Vladyslav C. Usenko, Mikael Lassen, Radim Filip, Ulrik L. Andersen
TL;DR
CV-QKD secure distance is limited by channel loss, excess noise, and imperfect post-processing, while coherent-state protocols are especially affected by excess noise. The paper proposes entangled squeezed states with controlled coherent modulation and optimized data combination, demonstrating secret-key generation in a noisy channel inaccessible to benchmark protocols. The results support improved noise tolerance and secure-communication distance using feasible impure squeezed sources.
Problem
CV-QKD distance is constrained by channel loss, excess noise, and limited reconciliation efficiency, while current channels and post-processing are nearing practical limits.
Method
The protocol combines conditionally squeezed two-mode states with large Gaussian coherent modulation and an optimized gain-weighted combination of Alice’s data.
Results
0.004 ± 0.001 bit per state was generated at 0.45 SNU excess noise and 95% channel transmission, where coherent-state and standard squeezed-state protocols could not generate a key.
Takeaways & Limitations
The protocol improves key rate and channel-noise robustness beyond the optimized coherent-state benchmark for conditionally squeezed states, including impure states.
Takeaways & Limitations
The security analysis includes collective attacks and experimentally studies a proof-of-principle implementation; single-mode squeezed-state variants remain future work.
Abstract
from arXiv · showhide
Quantum key distribution (QKD) enables two remote parties to grow a shared key which they can use for unconditionally secure communication [1]. The applicable distance of a QKD protocol depends on the loss and the excess noise of the connecting quantum channel [2-10]. Several QKD schemes based on coherent states and continuous variable (CV) measurements are resilient to high loss in the channel, but strongly affected by small amounts of channel excess noise [2-6]. Here we propose and experimentally address a CV QKD protocol which uses fragile squeezed states combined with a large coherent modulation to greatly enhance the robustness to channel noise. As a proof of principle we experimentally demonstrate that the resulting QKD protocol can tolerate more noise than the benchmark set by the ideal CV coherent state protocol. Our scheme represents a very promising avenue for extending the distance for which secure communication is possible.
Introduction
CV-QKD extends quantum key distribution to coherent and squeezed or entangled light with continuous-variable measurements. Its secure distance is constrained by channel loss, excess noise, and reconciliation efficiency, motivating a protocol-level alternative.
- QKD lets Alice and Bob generate a shared secret key for one-time-pad communication.
- CV-QKD uses optical quantum states and continuous quadrature measurements to establish correlated data between remote parties.
- Two major obstacles are channel excess noise combined with optical loss and limited classical reconciliation efficiency.
- Around 140 km is the theoretical maximal secure distance for a realistic coherent-state CV-QKD scheme under specified noise, loss, and post-processing conditions.The conditions are 4% excess vacuum noise, 0.2 dB/km loss, and 96.9% post-processing efficiency.
- The proposed entangled-state protocol experimentally demonstrates secret-key generation using 3.5 dB of impure, modulated two-mode squeezing through a noisy, lossy channel unavailable to coherent-state protocols.
Results
The protocol conditionally squeezes one mode of an EPR state, adds controlled Gaussian modulation, and optimizes Alice’s combined data. Theoretical analysis and experiments show improved noise tolerance and key generation in a channel where benchmark protocols fail.
- Our protocol: Alice obtains conditionally squeezed states by homodyne-measuring a randomly selected quadrature of one EPR mode.
- Our protocol: Controlled, independent Gaussian modulation enlarges the squeezed-state distribution before transmission through a potentially noisy and lossy channel.
- Our protocol: Alice forms xM + gxHD, with the gain g optimized according to EPR-state strength and purity; no squeezing reduces the scheme to coherent-state QKD.
- Secret Key: The key-rate analysis considers collective attacks with reverse reconciliation, where β represents post-processing efficiency.
- Secret Key: The protocol increases key rate and tolerable excess noise beyond the squeezed-state protocol without modulation.
- Secret Key: A factor of about 19 increases the maximal secure distance for 3 dB squeezed states compared with the previous squeezed-state protocol.
- Experimental setup and results: Correlations in the experiment arise from both quadrature entanglement and coherent modulation, enabling covariance-based security-limit estimation.
Discussion
The protocol generates secret keys in noisy channels where coherent-state and standard squeezed-state protocols fail, and its performance remains superior under imperfect post-processing. Its performance depends on conditional squeezing and modulation, with single-mode squeezed-state variants left for future work.
- Discussion: 0.45 shot noise units of excess noise and 23.4 SNU modulation yielded a raw key rate of 0.004 ± 0.001 bit per state at 95% transmission.Neither coherent-state nor standard 3.5 dB squeezed-state protocols could generate a key under these conditions.
- Discussion: At 10% transmission, experimentally measured covariance matrices were used to simulate tolerable excess noise and maximum distance across arbitrary channel losses.The analysis compared the protocol with ideal and energy-matched coherent-state benchmarks.
- Discussion: The squeezed-state protocol outperformed coherent-state protocols, including when the compared states used equal channel input energy.Equalizing energy makes the states entering the channel identical, preventing Eve from distinguishing the protocols by that criterion.
- Discussion: With post-processing efficiency β = 0.98 or β = 0.95, the squeezed-state protocol remained superior, and its relative improvement increased.Optimal performance occurred at finite modulation depth; very low β requires a different squeezed-state protocol.
- Discussion: The protocol’s performance is partly parametrized by conditional squeezing, and single-mode squeezed-state implementations were identified for future study.Direct preparation could make conditional squeezing equal to the single-mode squeezing degree.
- Discussion: The key rate and robustness against channel noise improved for any degree of conditional squeezing relative to the idealized optimized coherent-state protocol.The article introduced and experimentally addressed a continuous-variable QKD protocol based on squeezed states.
- Discussion: Relaxing the purity requirement supports security against any channel attack and may allow highly impure squeezed states to operate without coherent modulation.The authors cite fiber Kerr-effect sources producing up to 6.8 dB squeezing and 29.6 dB anti-squeezing.
A. Theory of the protocol
The security analysis derives the reverse-reconciliation key rate from Alice–Bob mutual information and Eve’s Holevo information, evaluated using purified covariance-matrix descriptions. Gaussian-state extremality enables covariance-matrix-based security bounds under collective attacks.
- A. Theory of the protocol: The security proof uses collective attacks and reverse reconciliation, with the asymptotic key rate determined by Alice–Bob mutual information, Eve’s Holevo bound, and β.β denotes post-processing efficiency.
- A. Theory of the protocol: Alice–Bob mutual information is calculated from Alice’s variance and conditional variance as IAB = 1/2 log2(VA/VA|B), with gain optimized afterward.The gain weights Alice’s homodyne results to maximize mutual information.
- A. Theory of the protocol: The Holevo quantity is expressed through von Neumann entropies of Eve’s state and the state conditioned on Bob’s measurement.For untrusted noise, Eve is assumed able to purify Alice and Bob’s system.
- A. Theory of the protocol: Purification introduces trusted modes and an overall covariance matrix, whose symplectic eigenvalues determine the required von Neumann entropies.The resulting calculation uses a pure state shared between Alice and Bob together with explicitly modeled trusted modes.
B. Experimental data security analysis
The experimental security analysis converts measured and processed covariance matrices into security estimates for realistic channels. It combines purification-based analysis, theoretical parameter modeling, and finite-block channel-estimation simulations.
- B. Experimental data security analysis: Measurement and gain optimization produced covariance matrices for different modulation values.These matrices formed the experimental inputs to the subsequent security analysis.
- B. Experimental data security analysis: The experimentally obtained matrices were purified with the Bloch–Messiah reduction theorem because the trusted-mode structure was unknown.The analysis was cross-checked with the entangling-cloner method.
- B. Experimental data security analysis: Theoretical covariance matrices incorporated measured EPR states, detector and channel transmittances, efficiencies, and added coherent modulation.The resulting matrices were tested against specified channel transmission and excess noise.
- B. Experimental data security analysis: Channel-estimation accuracy was simulated as a function of data-block size and reported in Supplementary Figure S3.The simulations address the effect of finite data blocks on estimating the channel.
C. Experimental setup details
The experiment uses a dual-wavelength laser system, bowtie optical parametric oscillators, and homodyne detection to generate and measure squeezed light. The setup operates with specified cavity bandwidths, pump powers, and detection efficiencies.
- C. Experimental setup details: The bowtie OPOs used temperature-controlled type-I periodically poled KTP crystals and operated below threshold with 170 mW pump power each.Their cavity bandwidths were approximately 21 MHz and 24 MHz.
- C. Experimental setup details: The setup produced 8.2 dB of antisqueezing, with total homodyne detection efficiencies of 90% ± 5% at Alice and 85% ± 5% at Bob.Signals were filtered at 90 kHz and digitized at 500 kHz in blocks of approximately 200,000 points.