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Postselection Strategies for Continuous-Variable Quantum Key Distribution Protocols with Quadrature Phase-Shift Keying Modulation

Florian Kanitschar, Christoph Pacher

Published Apr 19, 2021• 4 citations
OpenAlex

Abstract

We use a recent numerical security proof technique to analyse three different postselection strategies for continuous-variable quantum key distribution protocols (CV-QKD) with quadrature phase-shift keying modulation. For all postselection strategies studied we provide novel analytical results for the operators that define the respective regions in phase space. In both, the untrusted and trusted detector model, a new, cross-shaped postselection strategy, clearly outperforms a state-of-the-art radial postselection scheme for higher transmission distances and higher values of noise. Motivated by the high computational effort for the error-correction phase we also studied the case when a large fraction of the raw key is eliminated by postselection: We observe that the secure key rate in case that only $20\%$ of the raw key passes the cross-shaped postselection is still roughly $80\%$ of the secure key rate without postselection for low values of excess noise and roughly $95\%$ for higher values of excess noise. Furthermore, we examine a strategy with radial and angular postselection that combines both the advantages of state-of-the-art radial postselection schemes and our cross-shaped strategy at the cost of higher complexity. Employing the cross-shaped postselection strategy, that can easily be introduced in the data processing, both new and existing CV-QKD systems can improve the achievable secure key rates significantly.

Related papers

Security analysis of quantum key distribution protocols

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Quantum Key Distribution, or short QKD, aims to establish a secure key withoutmaking any additional assumptions about the abilities or computational power of an adversary who is only limited by the laws of nature. The process of finding a mathematical expression for, or at least a lower bound on, the secure keyrate of some QKD protocol, given relevant system parameters, is called securityproof. In this thesis, we use a recent numerical security proof technique to examine different post selection strategies for continuous-variable quantum key distribution(CV-QKD) protocols with quadrature phase-shift keying modulation and four or eight signal states. CV-QKD protocols use coherent states to encode information and measure the field-quadrature components by homodyne or heterodyne detection. The basic idea of the used numerical security proof technique is to solve thekey rate finding problem in a two-step process. In the first step, the problem is solved approximately, using a numerical algorithm, which yields an upper bound on the secure key rate. This is followed by step two, where the obtained upperbound is converted into a lower bound, using a sequence of theorems and taking numerical errors into account. Postselection aims to increase the secure key rate by removing those parts of the key, where a potential adversary might have gained more information than the communicating parties. The investigations are carried out both for the untrusted ideal and the trusted non-ideal detector scenario and we provide novel analytical results for the operators related to the postselection map.For four-state protocols, we demonstrate that a new cross-shaped postselectionstrategy out performs the state-of-the-art radial postselection scheme clearly in regions of medium to high transmission distances and medium to high values of noise and performs comparable to a more complicated radial&angular scheme. As the error-correction phase is a known bottleneck for many real QKD systems, we examined the secure key rate when a large fraction of the raw key is removed by postselection. We observe that a smart choice of the postselection strategy can increase the key rate while lowering the raw key rate, hence the computational effort in the error-correction phase. One can think this even further: For cross-shaped postselection we showed that the secure key rate is roughly 80% of the secure keyrate without postselection when only 20% of the raw key passes the postselection.For high noise levels this can be increased even further. The cross-shaped post-selection strategy can easily be implemented in the data processing of both new and existing CV-QKD systems, hence can increase the achievable secure key rate significantly. Furthermore, we examined the radial&angular postselection strategy which combines the advantages of the radial postselection scheme for low transmission distances and those of the cross-shaped postselection scheme for medium and high transmission distances on the cost of higher complexity.Additionally, we examine an eight-state phase-shift keying protocol and compare the obtained key rates to the key rates for the four-state protocol and investigate a radial postselection scheme. We observe that the 8PSK protocol yields higher secure key rates and reaches higher maximal achievable transmission distances than a QPSK protocol with a comparable postselection strategy, in particular for medium to high values of excess noise. Furthermore, we explore the relation between the achievable secure key rate and the probability to pass the postselectionphase for various noise-levels and two different practically relevant reconciliation efficiencies. This leads to similar strategies as found for the QPSK protocols to reduce the raw key rate significantly while decreasing the secure key rate only moderately. For very high values of excess noise, the raw key rate can be increased further compared to the key rate achieved without performing postselection whileremoving a large fraction of the raw key rate.

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Continuous variable quantum key distribution via discrete modulations ensures information-theoretic security using standard telecom technologies, providing affordable and scalable quantum communications with simplified classical postprocessing. However, existing security proofs against general attacks often rely on restrictive assumptions, such as a bounded dimension for coherent states, or require impractically large block sizes. In this work, we develop a finite-size security analysis that removes these limitations while incorporating realistic experimental features. Our approach combines an optimization under infinite dimensions, a security proof against general attacks, a trusted detector model accounting for the receiver imperfections, and postselection strategies to reuse discarded measurement outcomes. We report positive key rates in the finite-size regime for relevant block sizes of the order of $10^7$. These results contribute to narrowing the gap between theoretical security proofs and practical implementations of discrete-modulated continuous variable quantum key distribution protocols.

quant-pharXiv:2606.20346•Jun 19, 2026•0 citations
We use a recent numerical security proof technique to analyse three different postselection strategies for continuous-variable quantum key distribution protocols (CV-QKD) with quadrature phase-shift keying modulation. For all postselection strategies studied we provide novel analytical results for the operators that define the respective regions in phase space. In both, the untrusted and trusted detector model, a new, cross-shaped postselection strategy, clearly outperforms a state-of-the-art radial postselection scheme for higher transmission distances and higher values of noise. Motivated by the high computational effort for the error-correction phase we also studied the case when a large fraction of the raw key is eliminated by postselection: We observe that the secure key rate in case that only 20%20\%20% of the raw key passes the cross-shaped postselection is still roughly 80%80\%80% of the secure key rate without postselection for low values of excess noise and roughly 95%95\%95% for higher values of excess noise. Furthermore, we examine a strategy with radial and angular postselection that combines both the advantages of state-of-the-art radial postselection schemes and our cross-shaped strategy at the cost of higher complexity. Employing the cross-shaped postselection strategy, that can easily be introduced in the data processing, both new and existing CV-QKD systems can improve the achievable secure key rates significantly.
Quantum Key Distribution, or short QKD, aims to establish a secure key withoutmaking any additional assumptions about the abilities or computational power of an adversary who is only limited by the laws of nature. The process of finding a mathematical expression for, or at least a lower bound on, the secure keyrate of some QKD protocol, given relevant system parameters, is called securityproof. In this thesis, we use a recent numerical security proof technique to examine different post selection strategies for continuous-variable quantum key distribution(CV-QKD) protocols with quadrature phase-shift keying modulation and four or eight signal states. CV-QKD protocols use coherent states to encode information and measure the field-quadrature components by homodyne or heterodyne detection. The basic idea of the used numerical security proof technique is to solve thekey rate finding problem in a two-step process. In the first step, the problem is solved approximately, using a numerical algorithm, which yields an upper bound on the secure key rate. This is followed by step two, where the obtained upperbound is converted into a lower bound, using a sequence of theorems and taking numerical errors into account. Postselection aims to increase the secure key rate by removing those parts of the key, where a potential adversary might have gained more information than the communicating parties. The investigations are carried out both for the untrusted ideal and the trusted non-ideal detector scenario and we provide novel analytical results for the operators related to the postselection map.For four-state protocols, we demonstrate that a new cross-shaped postselectionstrategy out performs the state-of-the-art radial postselection scheme clearly in regions of medium to high transmission distances and medium to high values of noise and performs comparable to a more complicated radial&angular scheme. As the error-correction phase is a known bottleneck for many real QKD systems, we examined the secure key rate when a large fraction of the raw key is removed by postselection. We observe that a smart choice of the postselection strategy can increase the key rate while lowering the raw key rate, hence the computational effort in the error-correction phase. One can think this even further: For cross-shaped postselection we showed that the secure key rate is roughly 80% of the secure keyrate without postselection when only 20% of the raw key passes the postselection.For high noise levels this can be increased even further. The cross-shaped post-selection strategy can easily be implemented in the data processing of both new and existing CV-QKD systems, hence can increase the achievable secure key rate significantly. Furthermore, we examined the radial&angular postselection strategy which combines the advantages of the radial postselection scheme for low transmission distances and those of the cross-shaped postselection scheme for medium and high transmission distances on the cost of higher complexity.Additionally, we examine an eight-state phase-shift keying protocol and compare the obtained key rates to the key rates for the four-state protocol and investigate a radial postselection scheme. We observe that the 8PSK protocol yields higher secure key rates and reaches higher maximal achievable transmission distances than a QPSK protocol with a comparable postselection strategy, in particular for medium to high values of excess noise. Furthermore, we explore the relation between the achievable secure key rate and the probability to pass the postselectionphase for various noise-levels and two different practically relevant reconciliation efficiencies. This leads to similar strategies as found for the QPSK protocols to reduce the raw key rate significantly while decreasing the secure key rate only moderately. For very high values of excess noise, the raw key rate can be increased further compared to the key rate achieved without performing postselection whileremoving a large fraction of the raw key rate.
A numerical security proof technique is used to analyze the security of continuous-variable quantum key distribution (CV-QKD) protocols with phase-shift keying (PSK) modulation against collective attacks in the asymptotic limit. We argue that it is sufficient to consider protocols with a maximum of eight signal states and analyze different postselection strategies for protocols with four (QPSK) and eight (8PSK) signal states for untrusted ideal and trusted nonideal detectors. We introduce a cross-shaped postselection strategy, and show that both cross-shaped and radial and angular postselection clearly outperform a radial postselection scheme (and no postselection) for QPSK protocols. For all strategies studied, we provide analytical results for the operators defining the respective regions in phase space. We outline several use cases of postselection, which can easily be introduced in the data processing of both new and existing CV-QKD systems. Motivated by the high computational effort for error correction, we study the case when a large fraction of the raw key is eliminated by postselection and observe that this can be achieved while increasing the secure key rate. Postselection can also be used to partially compensate the disadvantage of QPSK protocols over 8PSK protocols for high transmission distances, while being experimentally less demanding. Finally, we highlight that postselection can be used to reduce the key rate gap between scenarios with trusted and untrusted detectors while relying on less assumptions on Eve's power.
We use a recent numerical security proof approach to calculate tight secure key rates in the presence of collective attacks for a continuous-variable (CV) eight-state phase-shift keying (8PSK) protocol with heterodyne detection in the asymptotic limit. The results are compared to achievable secure key rates of a QPSK protocol obtained with the same security proof technique. Furthermore, we investigate the influence of radial postselection on the secure key rate and examine a recently suggested strategy to reduce the computational requirements on the error-correction phase for four-state phase-shift-keying protocols. Based on our investigations, we suggest different strategies to reduce the raw key of 8PSK protocols significantly, only on the cost of a slightly lower secure key rate. This can be used to lower the computational effort of the error-correction phase, a known bottleneck in many practical implementations, considerably.
Security analysis of continuous variable quantum key distribution (CV-QKD) remains a topic of extensive academic investigation. In this work, we propose a coherent-state CV-QKD protocol integrating a displacement-based quantum receiver and post-selection strategies. We establish a framework of the protocol under both individual and collective attacks, incorporating practical constraints such as noises and device imperfections. To enhance performance of the protocol, the postselection strategy and the transmitted signal power are optimized for different channel attenuation levels. Numerical results offer insights into the strategy of employing displacement receivers in CV-QKD protocols, revealing that larger transmitted signal power is required under low channel attenuation or severe receiver-side imperfections and noises to achieve higher the secret key rate.
Discrete-modulated (DM) continuous-variable quantum key distribution (CV-QKD) protocols are promising candidates for commercial implementations of quantum communication networks due to their experimental simplicity.While tight security analyses in the asymptotic limit exist, proofs in the finitesize regime are still subject to active research.We present a composable finite-size security proof against independently and identically distributed collective attacks for a general DM CV-QKD protocol.We introduce a new energy testing theorem to bound the effective dimension of Bob's system and rigorously prove security within Renner's -security framework and address the issue of acceptance sets in protocols and their security proof.We want to highlight that our method also allows for nonunique acceptance statistics, which is necessary in practise.Finally, we extend and apply a numerical security proof technique to calculate tight lower bounds on the secure key rate.To demonstrate our method, we apply it to a quadrature phase-shift keying protocol for both untrusted, ideal and trusted, nonideal detectors.The results show that our security proof method yields secure finite-size key rates under experimentally viable conditions up to at least 72 km transmission distance.
Continuous variable quantum key distribution via discrete modulations ensures information-theoretic security using standard telecom technologies, providing affordable and scalable quantum communications with simplified classical postprocessing. However, existing security proofs against general attacks often rely on restrictive assumptions, such as a bounded dimension for coherent states, or require impractically large block sizes. In this work, we develop a finite-size security analysis that removes these limitations while incorporating realistic experimental features. Our approach combines an optimization under infinite dimensions, a security proof against general attacks, a trusted detector model accounting for the receiver imperfections, and postselection strategies to reuse discarded measurement outcomes. We report positive key rates in the finite-size regime for relevant block sizes of the order of 10710^7107. These results contribute to narrowing the gap between theoretical security proofs and practical implementations of discrete-modulated continuous variable quantum key distribution protocols.