单光子与强激光场之间的弱非线性Kerr相互作用最近被提出作为量子通信和计算中基于分布式光学的解决方案的基础。在这里，我们分析了上述克尔相互作用，采用连续时间多模模型的输入/输出场的非线性介质。与以前的单模处理这个问题相比，我们的分析考虑了自由场输入光束的全部时间内容以及介质的非瞬时响应。该模型的主要含义是输出端存在相位噪声项，其中一个输入端的交叉克尔相移与另一个输入端的光子通量成比例。我们表明，这些相位噪声项将妨碍Munro、Nemoto和Spiller [New J. Phys. 7，137（2005）]提出的奇偶校验门的满意性能。
英文摘要：
The weak nonlinear Kerr interaction between single photons and intense laser fields has been recently proposed as a basis for distributed optics-based solutions to few-qubit applications in quantum communication and computation. Here, we analyze the above Kerr interaction by employing a continuous-time multi-mode model for the input/output fields to/from the nonlinear medium. In contrast to previous single-mode treatments of this problem, our analysis takes into account the full temporal content of the free-field input beams as well as the non-instantaneous response of the medium. The main implication of this model, in which the cross-Kerr phase shift on one input is proportional to the photon flux of the other input, is the existence of phase noise terms at the output. We show that these phase noise terms will preclude satisfactory performance of the parity gate proposed by Munro, Nemoto, and Spiller [New J. Phys. 7, 137 (2005)].
We provide a security analysis of continuous-variable quantum key distribution (CVQKD) with discrete modulation against general attacks in a realistic finite-size regime. To realize this goal, we first prove security of the continuous-variable quantum key distribution protocol with discrete modulation against collective attacks by using the reliable tomography of the covariance matrix, leading to the reliable and tight error bounds in the derived confidence regions. Combining the proof with de Finetti reduction, the discrete-modulation-based continuous-variable quantum key distribution (DM-CVQ…
A simple proof that Gaussian attacks are optimal among collective attacks against continuous-variable quantum key distribution with a Gaussian modulation
We establish a lower bound on the asymptotic secret key rate of continuous-variable quantum key distribution with a discrete modulation of coherent states. The bound is valid against collective attacks and is obtained by formulating the problem as a semidefinite program. We illustrate our general approach with the quadrature phase-shift keying (QPSK) modulation scheme and show that distances over 100 km are achievable for realistic values of noise. We also discuss the application to more complex quadrature amplitude modulations (QAM) schemes. This work is a major step towards establishing the …
It is known that quantum scissors, as non-deterministic amplifiers, can enhance the performance of Gaussian-modulated continuous-variable quantum key distribution (CV-QKD) in noisy and long-distance regimes of operation. Here, we extend this result to a non-Gaussian CV-QKD protocol with discrete modulation. We show that, by using a proper setting, the use of quantum scissors in the receiver of such discrete-modulation CV-QKD protocols would allow us to achieve positive secret key rates at high loss and high excess noise regimes of operation, which would have been otherwise impossible. This als…
We establish an analytical lower bound on the asymptotic secret key rate of continuous-variable quantum key distribution with an arbitrary modulation of coherent states. Previously, such bounds were only available for protocols with a Gaussian modulation, and numerical bounds existed in the case of simple phase-shift-keying modulations. The latter bounds were obtained as a solution of convex optimization problems and our new analytical bound matches the results of Ghorai et al. (2019), up to numerical precision. The more relevant case of quadrature amplitude modulation (QAM) could not be analy…
Discrete-modulated continuous-variable quantum key distribution protocols are promising candidates for large-scale deployment due to the large technological overlap with deployed modern optical communication devices. The security of discrete modulation schemes has previously analyzed in the ideal detector scenario in the asymptotic limit. In this work, we calculate asymptotic key rates against collective attacks in the trusted detector noise scenario. Our results show that we can thus cut out most of the effect of detector noise and obtain asymptotic key rates similar to those had we access to…
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 …
Continuous-variable quantum key distribution (CV-QKD) has many practical advantages including compatibility with current optical communication technology. Implementation using heterodyne measurements is particularly attractive since it eliminates the need for active phase locking of the remote pair of local oscillators, but the full security of CV QKD with discrete modulation was only proved for a protocol using homodyne measurements. Here we propose an all-heterodyne CV-QKD protocol with binary modulation and prove its security against general attacks in the finite-key regime. Although replac…
We propose a long-distance continuous-variable measurement-device-independent quantum key distribution (CV-MDI-QKD) protocol with discrete modulation. This kind of discrete-modulated schemes have good compatibility with efficient error correction code, which lead to higher reconciliation efficiency even at low signal-to-noise ratio (SNR). Security analysis shows that the proposed protocol is secure against arbitrary collective attacks in the asymptotic limit with proper use of decoy states. And with the using of discrete modulation, the proposed CV-MDI-QKD protocol has simpler implementation a…
Flaws in the process of modulation, or encoding of key bits in the quadratures of the electromagnetic light field, can make continuous-variable quantum key distribution systems susceptible to leakage of secret information. Here, we report such a modulation leakage vulnerability in a system that uses an optical in-phase and quadrature modulator to implement a single sideband encoding scheme. The leakage arises from the limited suppression of a quantum-information-carrying sideband during modulation. Based on the results from a proof-of-concept experiment, we theoretically analyse the impact of …
We use single homodyne detector to accomplish Continuous-Variable quantum key distribution(CV QKD) in a laboratory and urban environment free-space channel. This is based on Gaussian modulation with coherent-states in the polarization degree of freedom. We achieved a QKD distance at 460m, at the repetition rate of 10 kHz. We give the security of this protocol against collective attack in the asymptotic regime. The secure key rate is 0.152 kbps at the typical reconciliation efficiency of 0.95. The experiment setup of this scheme is simplified and the difficulty to realize has been remarkably re…
Multi-carrier continuous-variable quantum key distribution (CV-QKD) is considered to be a promising way to boost the secret key rate (SKR) over the existing single-carrier CV-QKD scheme. However, the extra excess noise induced in the imperfect multi-carrier quantum state preparation process of N subcarriers will limit the performance of the system. Here, a systematic modulation noise model is proposed for the multi-carrier CV-QKD based on the orthogonal frequency division multiplexing (OFDM). Subsequently, the performance of multi-carrier CV-QKD with arbitrary modulation protocol (e.g. QPSK, 2…
Continuous-variable quantum key distribution (CV QKD) protocols with discrete modulation are interesting due to their experimental simplicity and their great potential for massive deployment in the quantum-secured networks, but their security analysis is less advanced than that of Gaussian modulation schemes. In this work, we apply a numerical method to analyze the security of discrete-modulation protocols against collective attacks in the asymptotic limit, paving the way for a full security proof with finite-size effects. While our method is general for discrete-modulation schemes, we focus o…
Continuous-wave (cw) squeezed states of light have applications in sensing, metrology and secure communication. In recent decades their efficient generation has been based on parametric down-conversion, which requires pumping by externally generated pump light of twice the optical frequency. Currently, there is immense effort in miniaturizing squeezed-light sources for chip-integration. Designs that require just a single input wavelength are favored since they offer an easier realization. Here we report the first observation of cw squeezed states generated by self-phase modulation caused by su…
Distributing cryptographic keys over public channels in a way that can provide information-theoretic security is the holy grail for secure communication. This can be achieved by exploiting quantum mechanical principles in so-called quantum key distribution (QKD). Continuous-variable (CV) QKD based on coherent states, in particular, is an attractive scheme for secure communication since it requires only standard telecommunication technology that can operate at room temperature. However, a recently discovered side-channel created in the process of state preparation leads to a leakage of informat…
Continuous-variable quantum key distribution exploits coherent measurements of the electromagnetic field, i.e., homodyne or heterodyne detection. The most advanced security proofs developed so far relied on idealised mathematical models for such measurements, which assume that the measurement outcomes are continuous and unbounded variables. As physical measurement devices have finite range and precision, these mathematical models only serve as an approximation. It is expected that, under suitable conditions, the predictions obtained using these simplified models are in good agreement with the …
We address side-channel leakage in a trusted preparation station of continuous-variable quantum key distribution with coherent and squeezed states. We consider two different scenaria: multimode Gaussian modulation, directly accessible to an eavesdropper, or side-channel loss of the signal states prior to the modulation stage. We show the negative impact of excessive modulation on both the coherent- and squeezed-state protocols. The impact is more pronounced for squeezed-state protocols and may require optimization of squeezing in the case of noisy quantum channels. Further, we demonstrate that…
In this paper we report a continuous-variable quantum key distribution protocol using multimode coherent states generated on subcarrier frequencies of the optical spectrum. To detect the quadrature components of bosonic field we propose a coherent detection scheme where power from a carrier wave is used as a local oscillator. We compose a mathematical model of the proposed scheme and perform its security analysis in the finite-size regime using fully quantum asymptotic equipartition property technique. We calculate a lower bound on the secret key rate for the system under the assumption that t…
We study the non-Gaussian character of quantum optomechanical systems evolving under the fully nonlinear optomechanical Hamiltonian. By using a measure of non-Gaussianity based on the relative entropy of an initially Gaussian state, we quantify the amount of non-Gaussianity induced by both a constant and time-dependent cubic light-matter coupling and study its general and asymptotic behaviour. We find analytical approximate expressions for the measure of non-Gaussianity and show that initial thermal phonon occupation of the mechanical element does not significantly impact the non-Gaussianity. …
Quantum key distribution using weak coherent states and homodyne detection is a promising candidate for practical quantum-cryptographic implementations due to its compatibility with existing telecom equipment and high detection efficiencies. However, despite the actual simplicity of the protocol, the security analysis of this method is rather involved compared to discrete-variable QKD. In this article we review the theoretical foundations of continuous-variable quantum key distribution (CV-QKD) with Gaussian modulation and rederive the essential relations from scratch in a pedagogical way. The…
This study revokes the performance of continuous phase modulation (CPM) able to generate a single-sideband (SSB) spectrum directly. This signal is analyzed in terms of modulation indices, pulse lengths, and pulse widths, all of which affect error probability, bandwidth, SSB property, and receiver complexity. The error probability performance is based on an approximation of the minimum Euclidean distance. A numerical power spectral density calculation for this particular SSB modulation in terms of modulation index is presented. Reasonable tradeoffs in designing modulation schemes have been prop…
Analysis of the frequency shift in coherent population trapping resonance's dynamic continuous-wave spectroscopy at the phase-jump modulation and its comparison with the conventional approach
P. N. Lebedev Physical Institute of the Russian Academy of Sciences（俄罗斯科学院列别杰夫物理研究所）
An attractive implementation for quantum cryptography is the continuous variable variation, as it relies on standard telecommunication components. Modulating the quantum signal using a Gaussian format is attractive since it has been proven to be secure. This work investigates the effect of the roll-off of a root raised cosine pulse shaping and matched filter on the excess noise performance of a Gaussian modulated quantum key distribution system in a simulated back to back configuration. Contrary to intuition, it is found that the roll-off parameter does not significantly impact the performance…
Continuous-variable (CV) measurement-device-independent (MDI) quantum key distribution (QKD) is immune to imperfect detection devices, which can eliminate all kinds of attacks on practical detectors. Here we first propose a CV-MDI QKD scheme using unidimensional modulation (UD) in general phase-sensitive channels. The UD CV-MDI QKD protocol is implemented with the Gaussian modulation of a single quadrature of the coherent states prepared by two legitimate senders, aiming to simplify the implementation compared with the standard, symmetrically Gaussian-modulated CVMDI QKD protocol. Our scheme r…
A numerical security proof technique is used to analyse the security of continuous-variable quantum key distribution (CV-QKD) protocols with phase-shift keying modulation against collective attacks in the asymptotic limit. We argue why it is sufficient to consider protocols with a maximum number of eight signal states and analyse different postselection strategies for protocols with four (QPSK) and eight (8PSK) signal states for untrusted ideal and trusted nonideal detectors. We introduce a \emph{cross-shaped} postselection strategy, and show that both cross-shaped and radial and angular posts…
The possibility of using squeezed states in the recently suggested unidimensional continuous-variable quantum key distribution based on a single quadrature modulation is addressed. It is shown that squeezing of the signal states expands the physicality bounds of the effective entangled state shared between the trusted parties due to the antisqueezing noise in the unmodulated quadrature. Modulation of the antisqueezed quadrature, on the other hand, effectively shrinks the physicality bounds due to the squeezing in the unmodulated quadrature and also provides noise on the reference side of the p…
Despite tremendous theoretical and experimental progress in continuous variable (CV) quantum key distribution (QKD), its security has not been rigorously established for practical systems with multiple imperfections. The idea of tagging is widely used in security proofs of discrete variable quantum key distribution with imperfect devices. In this paper, we generalize the tagging idea from discrete variable to continuous variable. Based on untagged signals, we prove the security of the imperfect quantum key distribution system in the most conservative case. By introducing a generic imperfection…
Discrete-modulation continuous-variable quantum key distribution with probabilistic amplitude shaping over a linear quantum channel
Scuola Superiore Sant’Anna（圣安娜高等学院）；National Laboratory of Photonic Networks, CNIT（光子网络国家实验室，CNIT）；Palacký University（帕拉茨基大学）；Università degli Studi di Milano（米兰大学）；INFN, Sezione di Milano（意大利国家核物理研究所，米兰分部）
We study the sending-or-not-sending (SNS) protocol with discrete phase modulation of coherent states. We first make the security of the SNS protocol with discrete phase modulation. We then present analytic formulas for key rate calculation. We take numerical simulations for the key rate through discrete phase modulation of both the original SNS protocol and the SNS protocol with two way classical communications of active-odd-parity pairing (AOPP). Our numerical simulation results show that only with $6$ phase values, the key rates of the SNS protocol can exceed the linear bound, and with $12$ …
One of the most important practical realizations of the fundamentals of quantum mechanics is continuous-variable quantum key distribution (CVQKD). Here we propose the adaptive multicarrier quadrature division-multiuser quadrature allocation (AMQD-MQA) multiple access technique for continuous-variable quantum key distribution. The MQA scheme is based on the AMQD modulation, which granulates the inputs of the users into Gaussian subcarrier continuous-variables (CVs). In an AMQD-MQA multiple access scenario, the simultaneous reliable transmission of the users is handled by the dynamic allocation …
