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.
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.
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.
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.
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.
We consider discrete-modulation protocols for continuous-variable quantum key distribution (CV-QKD) that employ a modulation constellation consisting of a finite number of coherent states and that use a homodyne- or a heterodyne-detection receiver. We establish a security proof for collective attacks in the asymptotic regime, and we provide a formula for an achievable secret-key rate. Previous works established security proofs for discrete-modulation CV-QKD protocols that use two or three coherent states. The main constituents of our approach include approximating a complex, isotropic Gaussian probability distribution by a finite-size Gauss-Hermite constellation, applying entropic continuity bounds, and leveraging previous security proofs for Gaussian-modulation protocols. As an application of our method, we calculate secret-key rates achievable over a lossy thermal bosonic channel. We show that the rates for discrete-modulation protocols approach the rates achieved by a Gaussian-modulation protocol as the constellation size is increased. For pure-loss channels, our results indicate that in the high-loss regime and for sufficiently large constellation size, the achievable key rates scale optimally, i.e., proportional to the channel's transmissivity.
In comparison to conventional discrete-variable (DV) quantum key distribution (QKD), continuous-variable (CV) QKD with homodyne/heterodyne measurements has distinct advantages of lower-cost implementation and affinity to wavelength division multiplexing. On the other hand, its continuous nature makes it harder to accommodate to practical signal processing, which is always discretized, leading to lack of complete security proofs so far. Here we propose a tight and robust method of estimating fidelity of an optical pulse to a coherent state via heterodyne measurements. We then construct a binary phase modulated CV-QKD protocol and prove its security in the finite-key-size regime against general coherent attacks, based on proof techniques of DV QKD. Such a complete security proof is indispensable for exploiting the benefits of CV QKD.
Abstract In comparison to conventional discrete-variable (DV) quantum key distribution (QKD), continuous-variable (CV) QKD with homodyne/heterodyne measurements has distinct advantages of lower-cost implementation and affinity to wavelength division multiplexing. On the other hand, its continuous nature makes it harder to accommodate to practical signal processing, which is always discretized, leading to lack of complete security proofs so far. Here we propose a tight and robust method of estimating fidelity of an optical pulse to a coherent state via heterodyne measurements. We then construct a binary phase modulated CV-QKD protocol and prove its security in the finite-key-size regime against general coherent attacks, based on proof techniques of DV QKD. Such a complete security proof is indispensable for exploiting the benefits of CV QKD.