Software-enhanced simultaneous quantum-classical communication protocol with Gaussian post-selection

TL;DR

The paper addresses the vulnerability of simultaneous quantum-classical communication (SQCC) in CV-QKD to fluctuating channels by introducing Gaussian post-selection to adapt the modulation variance after channel estimation. This software-based filtering, implemented via a Gaussian filter with gain $g$ and success probability $P_A$, effectively yields an adjusted modulation variance $\tilde{V}_{mod}$ and allows post-transmission optimisation without hardware changes. The authors develop a finite-size composable security analysis for the post-selected SQCC, deriving an asymptotic key rate $K^ exists = P_A(\beta I_{AB}-I_E)$ and a finite-size key rate $K^{fs}_{ps}$ that account for post-selection, detector imperfections, and renormalisation effects. Results show substantial improvements in transmission distance and robustness for both fibre and satellite-to-ground channels, including extended communication windows and higher duty cycles under varying weather, bringing performance close to optimised-variance SQCC while remaining hardware-free. These findings demonstrate a practical path to real-world quantum-secure communication over terrestrial and space-based links using software-based adaptation to time-varying channels.

Abstract

Simultaneous quantum-classical communication (SQCC) protocols offer a practical approach to continuous-variable quantum key distribution (CV-QKD) by encoding quantum and classical signals onto the same optical pulse. However, like most QKD protocols, their performance is limited when experimental parameters, such as modulation variance, are optimised based on stationary channel assumptions. In fluctuating environments, such as free-space links, this can result in sub-optimal key rates and reduced transmission distances. In this work, we introduce Gaussian post-selection into the SQCC framework, enabling a software-based optimisation of the modulation variance after channel estimation. This passive approach enhances key rates in both asymptotic and finite-size regimes without requiring hardware modifications and remains effective even when receiver imperfections are taken into account. We demonstrate that our protocol significantly improves the transmission distance and robustness of SQCC across both fibre and free-space channels. In particular, we show that the protocol enables full communication windows under ideal weather conditions and maintains higher duty cycles during adverse weather in satellite-to-ground scenarios. These results highlight the practicality of post-selection based SQCC for real-world quantum communication over both terrestrial fiber networks and satellite-based free-space links.

Figures (5)

• Figure 1: Schematic of the filtering process in the SQCC protocol. (a) Prepare-and-measure model where Alice encodes coherent states $\ket{(x_a+ip_a)/2}=\ket{\Tilde{\alpha}}$ from a Gausssian distribution with zero mean and variance, $V_{mod}$, using white noise generated by function generators. She samples $x_a$ and $p_a$ simultaneously, then applies a large phase displacement of $\Tilde{d}$ chosen from her alphabet $d_i$. The encoded states are then sent through a quantum channel with transmittivity $T$ and thermal noise $W$. Bob performs heterodyne detection to measure $x_b$ and $p_b$, simultaneously and re-displaces his measurement outcomes by $\Tilde{d}$. If deemed necessary, he applies an electronic gain to rescale his data to make the joint distribution between Alice and Bob Gaussian. After data acquisition and once the states have been sent to Bob, Alice applies a Gaussian filter to her quantum data kept on her computer. (b) Entanglement-based model where Alice prepares a two-mode squeezed vacuum (TMSV) state, retains mode $A_1$ and performs a heterodyne measurement. Mode $A_2$ is sent to Bob through a quantum channel. A classical displacement operation is applied after the channel, under the assumption that Eve has full knowledge of the classical communication. Bob performs a heterodyne measurement on the received mode, re-displaces the outcomes, and applies a renormalisation procedure to restore a Gaussian distribution. Alice and Bob then estimate the channel parameters. If Alice’s modulation variance is found to be sub-optimal for the estimated channel, she applies a Gaussian filter to her heterodyne outcomes $x_a$ and $p_a$, effectively emulating an optimally prepared state. PM/AM: Electro-optic phase/amplitude modulators. LO: Local oscillator.
• Figure 2: The asymptotic results assuming a fibre loss of 0.2 dB/km and thermal noise $\xi = 0.05$, expressed in shot-noise units (SNU).(a) The orange lines depict the SQCC protocol results with a fixed thermal variance of $V = 10$ SNU. The navy blue lines correspond to the SQCC protocol where the modulation variance is optimised for each distance, while the light blue lines show the results using a Gaussian modulation variance with the filter gain optimised per distance.(b) Solid lines represent the mutual information between Alice and Bob, while dashed lines show Eve’s Holevo information. The orange lines correspond to the SQCC protocol with $V = 10$, and the blue lines to the protocol with Gaussian post-selection. The dots indicate the point at $d = 41$ km, which is shown in detail in the panel. From left to right, we plot Alice and Bob’s mutual information, Eve’s information, and the key rate for both protocols: orange boxes show the original SQCC results, where the key rate is negative at this distance, while blue boxes show the SQCC protocol with Gaussian post-selection, where the key rate becomes positive with the optimal filter gain $g=0.25$. Inset: columns representing $\beta \mathrm{I_{AB}}$, $\mathrm{I_{E}}$ and $\mathrm{KR}$, respectively. $\mathrm{I_{AB}}$: Mutual information between Alice and Bob, $\mathrm{I_E}$: Eve's information and $\beta$: Reconciliation efficiency. For these simulations, the reconciliation efficiency is fixed at $\beta = 95\%$, while Bob’s classical bit-error rate, $\mathcal{W}$, varies with the channel parameters. The classical displacement is kept constant at $d=60$. We assume the detection efficiency is $\eta=0.95$ and the detector noise is $v_{el}=0.01$ (Refer to Appendix \ref{['sec:parameters_general']} for the simulation parameters).
• Figure 3: Finite-size key rate performance of the SQCC protocol over a thermal channel with excess noise $\xi=0.05$ and clasical displacement of $d=60$, shown for different block sizes. Dashed lines represent a block size of $N=10^{10}$, while the dot-dashed lines correspond to a block size of $N=11$. The orange line shows the original SQCC protocol with $V=10$ SNU, the light blue line includes Gaussian post-selection, and the dark blue line shows the SQCC protcol with optimised variance.
• Figure 4: Key rate of satellite-to-ground communication as a function of the elevation angle, shown for both the original SQCC protocol and the SQCC protocol with Gaussian post-selection. The satellite is assumed to be in Low Earth Orbit at an altitude of 500 km, with the optical ground station located at sea level ($0$ km elevation). We adopt the satellite-to-ground channel model from Sayat et al.sayat2024satellite, assuming a reconciliation efficiency of $\beta = 92\%$ (see Appendix \ref{['sec:parameter_satellite']} for details). (a) The orange lines correspond to key rates using a fixed thermal variance of $V = 8$ SNU, while the blue lines show the performance of the SQCC protocol with Gaussian post-selection applied. Solid lines indicate the asymptotic key rates, while the dot-dashed and dashed lines represent finite-size results with block sizes of $N=10^{12}$ and $N=10^{11}$, respectively. The simulation assumes good weather conditions, with a visibility of $V_{visib} = 200$ km and low atmospheric turbulence characterised by $C_n^2 = 10^{-16}$ m$^{-2/3}$. At each elevation angle, the post-selection gain is optimised to maximise the key rate. (b) Key rates under adverse weather conditions, assuming a visibility of $V_{visib} = 20$ km and strong atmospheric turbulence with $C_n^2 = 10^{-13}$ m$^{-2/3}$. (c) Duty cycle comparison between the SQCC protocol with and without post-selection under good weather conditions, based on the key rate threshold of $10^{-4}$ from (a). (d) Duty cycle comparisonbetween the SQCC protocol with and without post-selection under adverse weather conditions, using the same key rate threshold of $10^{-4}$ from (b).
• Figure 5: Asymptotic key rates in the untrusted-detector scenario. Alice’s modulation variance is set to $V = 10$ SNU, with channel noise $\xi = 0.05$ SNU. (a) Detector efficiency $\eta = 95\%$, and electronic noise $v_{el} = 0.01$ SNU. (b) Detector efficiency $\eta = 99\%$, and electronic noise $v_{el} = 0.01$ SNU. (c) Detector efficiency $\eta = 95\%$, and electronic noise $v_{el} = 0.001$ SNU. (d) Detector efficiency $\eta = 99\%$, and electronic noise $v_{el} = 0.001$ SNU.