Theoretical development of discrete-modulated continuous-variable quantum key distribution

1 Introduction

Quantum private communication is a practical direction of the application of quantum mechanics. It offers chances to communicate securely against the art-of-state techniques and predictably powerful technologies, such as quantum computers. Various forms of communication are carried out according to cryptographic tasks (Gu et al., 2021; Li et al., 2021a), such as quantum private query (Liu B. et al., 2022) and quantum digital signatures (Lu et al., 2021). The most basic one is quantum key distribution (QKD) (Zhou et al., 2016; Pirandola et al., 2020; Wang et al., 2020; Xu et al., 2020; Kwek et al., 2021) that does not involve the transmission of information, but only shares a string of identical and secure keys for both parties against quantum attacks. For this symmetric encryption, the one-time pad (Shannon, 1949) ensures that if each key generated from QKD is used only once, the theoretical security of information can be achieved.

Continuous-variable QKD (CV-QKD) (Su et al., 2009; Diamanti and Leverrier, 2015; Guo et al., 2021; Su et al., 2022; Zhao et al., 2022) is an important direction of QKD, which encodes keys in continuous degrees of freedom. Another notable choice is to encode keys in discrete degrees of freedom on single photon, which is called discrete-variable QKD (Yin et al., 2021; Xie et al., 2022; Zeng et al., 2022). In general, CV-QKD has advantages in experimental implementation (Karinou et al., 2018). The intensity of signals in CV-QKD is usually stronger to resist channel loss, increasing the amount of measurement outcomes and improving the key rates. Measurement devices are homodyne detectors or heterodyne detectors (Liu J. et al., 2022) that are compatible with classical optical communication and inexpensive to manufacture and maintain, compared to the single-photon detector in discrete-variable QKD (Zhao et al., 2006; Yin et al., 2020; Jiang et al., 2021).

For key encoding, discrete modulation (Ralph, 1999) is proposed earlier than Gaussian modulation, but Gaussian modulation (Grosshans and Grangier, 2002; Grosshans et al., 2003) pushes the CV-QKD to the hot spot. The Gaussian modulation scheme achieves security against general attacks (Leverrier et al., 2010; Leverrier, 2015; Pirandola, 2021), which is based on U(N) symmetry by sampling and estimating quadratures from a Gaussian distribution. However, the U(N) symmetry of the Gaussian distribution requires signal continuity, while we can only prepare finite kinds of signal states (Lupo, 2020). It is necessary to propose a stricter security proof for the discreteness of prepared states. The preparation step can be further simplified by sending a few coherent states and 1 bit value for each coherent state. This encoding manner is similar to discrete variables, which facilitates simplified postprocessing.

Discrete-modulated continuous-variable quantum key distribution (DM-CV-QKD) is now easy to realize in terms of preparation and measurement (Aguiar et al., 2022; Wang H. et al., 2022). However, many problems remain to be solved, such as security (Shao et al., 2022), key rate calculation, and fast postprocessing methods (Gyongyosi and Imre, 2018; Zhou et al., 2021). This work reviews the theoretical achievements that have been made in DM-CV-QKD, and openly discusses the issues worthy of further research.

2 Protocol and variants

We discuss the prepare-and-measure type protocols with two users Alice and Bob and an adversary Eve. The first binary-modulated scheme (Silberhorn et al., 2002), which sends states with a Gaussian distribution but interprets a state as logical 0 or one according to the displacement direction (Heid and Lütkenhaus, 2007), is the early form of discrete modulation. Later, schemes with different signal states or different probability distributions evolved (Zhao et al., 2020a; Dias and de Assis, 2021; Kaur et al., 2021).

The phase-shift-keying (PSK) type prepares different signal states by changing phases. Concretely, the M-PSK type scheme (Sych and Leuchs, 2010; Papanastasiou et al., 2018) prepares M kinds of signals represented by $\left|\alpha e^{2\pi ik/M}\right.⟩$, where k ∈ {0, 1, …, M − 1}. Signals have the same amplitude α > 0 but different phases. The value of M can be taken as 2 (Zhao et al., 2009), 3 (Brádler and Weedbrook, 2018), 4 (Leverrier and Grangier, 2009) or 8 (Becir et al., 2012; Guo et al., 2018, 2020). M = 4 is the most popular choice in the investigation (Xuan et al., 2009; Hirano et al., 2017; Liao et al., 2018; Ghorai et al., 2019; Lin et al., 2019; Liu et al., 2021).

The quadrature amplitude modulation (QAM) type (Denys et al., 2021) is a practical implementation of Gaussian modulation, using limited kinds of signal states to approximate the Gaussian distribution. Each quadrature has m choices, and there are M = m² signal states in total. The state that takes the kth $\widehat{q}$ quadrature and the lth $\widehat{p}$ quadrature can be denoted by $\left|{\alpha }_{k,l}\right.⟩$, where ${\alpha }_{k,l}=\frac{\alpha \sqrt{2}}{\sqrt{m-1}}\left(k-\frac{m-1}{2}\right)+i\frac{\alpha \sqrt{2}}{\sqrt{m-1}}\left(l-\frac{m-1}{2}\right)$ and α > 0 is the amplitude. The probability distribution is $P_{k,l}=\frac{1}{2^{2(m-1)}}\left(\genfrac{}{}{0ex}{}{m-1}{k}\right)\left(\genfrac{}{}{0ex}{}{m-1}{l}\right)$. Signal states are arranged in a coordinate system formed by two quadratures as a square lattice.

The amplitude and phase shift keying (APSK) type (Almeida et al., 2021) is likely a multi-ring of PSK type. The states in each ring have the same amplitude, and adjacent states have the same phase difference. It has 4, 12, 16, 32, 64, 128, and 256 states in the first ring to the seventh ring, respectively. For instance, 16-APSK has two rings, and 32-APSK has three rings. To approach the Gaussian modulation, the probability of a state in ring p is P_p = 1/(RM_p), where R is the total number of rings and M_p is the total number of states in ring p.

For detector selection, homodyne detection is more precise and simpler than heterodyne detection (Caves and Drummond, 1994; Laudenbach et al., 2018) in principle. However, if there are too many types of signal states, it is better to use heterodyne detection to distinguish them accurately. It is worth noting that more states can improve key rates but complicate postprocessing, especially because error correction is difficult due to the high error rate caused by the difficulty of distinguishing different states.

3 Security analysis

Security is the deciding factor in whether a protocol should exist or not. In this section, we briefly summarize some art-of-state security analysis methods. In general, security analysis starts from the Devetak-Winter formula (Igor and Andreas, 2005), which reads

$K=I\left({\mathbf{S}}_A;{\mathbf{S}}_B\right)-\max \chi \left({\mathbf{S}}_B:\mathbf{E}\right),(1)$

where I(S_A; S_B) is the mutual information between Alice and Bob’s measurement outcomes S_A and S_B that are used to distil keys, and χ(S_B: E) is the Holevo bound between Bob’s outcome S_B and the quantum register E of Eve. The search range for the maximal solution of χ(S_B: E) is any possible attack that matches the statistics of experimental parameters obtained by Alice and Bob.

Discrete modulation schemes cannot be directly applied to Gaussian attack optimization and quantum de Finetti theory due to insufficient symmetry of the prepared signal states. Estimation of the Holevo bound χ(S_B: E) is an admittedly difficult problem.

3.1 Analytical method

Based on physical conditions, analytical methods scale the objective function by various inequalities until a bound that can be directly calculated with known experimental parameters is found. Therefore, the key rate formula obtained by the analytical method has the advantage of being easy to compute.

Considering universality, one method (Denys et al., 2021) is constructing the covariance matrix according to the security analysis of Gaussian modulation schemes. χ(S_B: E) can be explicitly bounded by taking the value of the Holevo information for a Gaussian state with the covariance matrix

$\left[\begin{array}{cc}\hfill V{\mathbb{I}}_2\hfill & \hfill Z{\sigma }_Z\hfill \\ \hfill Z{\sigma }_Z\hfill & \hfill W{\mathbb{I}}_2\hfill \end{array}\right](2)$

where V and W can be directly given by experimental parameters, but Z is linearly related to the density operator of both parties and is influenced by Eve. σ_Z is the pauli-Z matrix and ${\mathbb{I}}_2$ is two-dimensional identity matrix. Under constraints of the experimental outcomes, we can find a scaling that gives upper and lower bounds on the parameter Z of the covariance matrix. The lower bound of the secret key rate of CV-QKD with an arbitrary modulation is secure against collective attacks under the asymptotic regime. (Almeida et al., 2021) applies this method to the M-QPSK type considering finite-size effects.

It is worth noting that a binary-modulated CV-QKD protocol has achieved the highest security against general coherent attacks in the finite-key-size regime (Matsuura et al., 2021). The security is based on the fidelity of an optical pulse to a coherent state. For this purpose, a tight and robust method of estimating fidelity via heterodyne detection has been proposed. The drawbacks of this method are not tight and not universal, since the key rate is small, and the transmission distance is short.

3.2 Numerical method

Numerical methods usually have the advantage of being tight, since the lower bound of the key rate can be calculated directly according to the experimental parameters with less scaling. One method (Ghorai et al., 2019) is based on the covariance matrix and is the predecessor of the universal analytical method (Denys et al., 2021) mentioned above. The difference is how to calculate the parameter Z of the covariance matrix. In the numerical method, we directly search the optimal value of Z under the constraints using semidefinite programming. The numerical solution and the analytical solution of the key rate are consistent, but neither is as high as the numerical solution of the nonlinear method (Lin et al., 2019).

In this nonlinear method, the secret key rate K against collective attacks in the asymptotic limit from Devetak-Winter formula (Igor and Andreas, 2005) is rewritten as

$K=\underset{\rho \in \mathbf{S}}{\min }f\left(\rho \right)-{\text{leak}}_{\text{obs}}^{\text{EC}},(3)$

where the first term is associated with privacy amplification and ${\text{leak}}_{\text{obs}}^{\text{EC}}$ is information leakage during error correction. ρ is the density operator shared by Alice, Bob, and possibly other parties involved in the protocol. S = {ρ ∈ H₊|Tr(Γ_iρ = γ_i, ∀i)} denotes the set of states satisfying linear constraints from asymptotic experimental data, where H₊ is the set of positive semidefinite operators, {Γ_i} is the set of Hermitian operators and {γ_i} is the corresponding set of expectation values. χ(S_B: E) is implicit in f(ρ) (Winick et al., 2018)

$f\left(\rho \right)=D\left(\mathcal{G}\left(\rho \right)\Vert \mathcal{Z}\left(\mathcal{G}\left(\rho \right)\right)\right),(4)$

where D(σ‖τ)≔Tr(σ log  σ) − Tr(σ log  τ) is the quantum relative entropy, $\mathcal{G}$ is a completely positive map related to the postselection and $\mathcal{Z}$ is a completely positive trace preserving map related to the key map. Since the relative entropy is jointly convex, $\mathcal{G}$ and $\mathcal{Z}$ are linear maps, S is convex, f(ρ) is convex in ρ and min_ρ∈Sf(ρ) is a convex optimization problem. A two-step numerical method is developed for calculating a reliable lower bound on the convex optimization problem min_ρ∈Sf(ρ) (Winick et al., 2018). We first need to find a ρ close to the minimum ρ* of Eq. 4 with feasible convex optimization methods such as the Frank-Wolfe method (Frank and Wolfe, 1956). Subsequently, we solve the dual problem of the linearization of f about ρ and obtain the reliable lower bound.

This method has been used for 4-PSK type schemes (Lin et al., 2019; Lin and Lütkenhaus, 2020; Liu et al., 2021), in which the constraints S include the statistics of experimental outcomes, the definition of density operator and the requirement that Eve cannot modify Alice’s system A. The optical mode Bob received is in an infinite-dimensional Hilbert space; thus, a photon-number cutoff assumption is imposed (Lin et al., 2019) and latter removed by (Upadhyaya et al., 2021).

3.3 Neural network method

Although numerical methods offer a tight key rate bound of DM-CV-QKD, the high requirement of time and computational resources in solving semidefinite programming remains a key challenge (Hu et al., 2021). Machine learning (Zdeborová, 2017; Mehta et al., 2019) can efficiently learn complex patterns, thus potentially speeding up the computation of key rates. Recently, several works have taken a solid step in this direction (Liu Z.-P. et al., 2022; Zhou et al., 2022).

A brief workflow of the neural network method is illustrated in Figure 1. The architectures of neural networks are chosen by considering the complexity of protocols and the scale of the data sets collected. The data set composed of the training set and test set is collected from numerical simulations or experiments. Each entry in the data set consists of the input features and an output label. Data preprocessing is necessary to significantly improve the prediction accuracy of the key rates. Then, the neural network is trained on the training set and evaluated on a validation set. A critical ingredient is a loss function including two adjustable hyperparameters designed specifically, which keeps the predicted key rates reliable and tight. It usually costs several trials to find the hyperparameters that make the neural network perform best. After completing this stage, the trained-well neural network is tested on the test set and can be deployed on certain devices in quantum networks to infer key rates online in real-time.

FIGURE 1

FIGURE 1. Schematic diagram of the brief workflow of the neural network method.

With the set of values {γ_i} (Lin et al., 2019) mentioned above as the input features and the corresponding key rates as the output labels, this method can be introduced into DM-CV-QKD to reduce the time and computational resource waste of solving semidefinite programming (Zhou et al., 2022). It yields a speedup of approximately six or seven orders of magnitude compared with numerical methods, which means it can infer key rates in milliseconds when given new inputs.

In addition, (Liu Z.-P. et al., 2022), introduced Bayesian optimization into the neural network method to automatically search the best structure and hyperparameters. This improved automatic neural network method has calculated the key rates of two variants of DM-CV-QKD protocols (Lin et al., 2019; Liu et al., 2021) with high reliability, considerable tightness and great efficiency.

4 Satellite-to-ground DM-CV-QKD

In the past few decades, the communication distance of QKD has been extended to several hundreds of kilometers (Yin et al., 2016; Chen et al., 2021; Pittaluga et al., 2021; Wang S. et al., 2022) due to progress in experimental technology and protocol design. Unfortunately, the amount of secret bits distributed through a lossy ground-based channel per use cannot exceed the repeaterless bound proposed by Pirandola et al (Pirandola et al., 2017). Quantum repeaters, relying on entanglement distribution, entanglement swapping and quantum memories, are a solution to mitigate the problem of fundamental limits in the lossy channel. However, such technology is currently far from practical for the large-scale deployment of quantum networks. Satellites provide an alternative opportunity for long distance QKD due to less decibels of loss in a satellite-to-ground channel compared with ground-based fiber connections. The milestone in this field is the first complete satellite-to-ground QKD experiment realized with Micius (Liao et al., 2017a; Liao et al., 2017c; Yin et al., 2017b). In the same year, QKD was implemented in a small payload on-board of the Tiangong-2 space laboratory (Liao et al., 2017b). Apart from QKD, entanglement distribution (Yin et al., 2017a) and quantum teleportation (Ren et al., 2017) have been realized.

An important aspect in a global QKD network is developing stable, high-throughput QKD links from a constellation of satellites to a network of ground stations. CV-QKD exploits the heritage of classical optical communication in terms of high-speed components and space qualification and may bring a breakthrough in this field. Currently, the progress in satellite CV-QKD mainly focuses on the investigation of feasibility theoretically considering the fluctuation effects in realistic satellite-to-ground links and calculating the secret key rate of Gaussian modulation CV-QKD (Kish et al., 2020; Villasenor et al., 2020; Dequal et al., 2021). The preliminary experimental study was implemented in terms of signal measurement in satellite-to-ground links (Günthner et al., 2017). Further efforts can be made to investigate the theoretical feasibility of DM-CV-QKD (Wang T.-L. et al., 2019), and the demonstration of satellite CV-QKD is expected.

5 Measurement-device-independent DM-CV-QKD

If one wants to extend the advantages of CV QKD to the condition of longer transmission distance, constructing a measurement-device-independent (MDI) type variant (Pirandola et al., 2015; Wang P. et al., 2019; Ye et al., 2020; Tian et al., 2022) is a reasonable choice. With efficient reconciliation error correction, an MDI type DM-CV-QKD protocol can transmit longer than the MDI type Gaussian-modulated CV-QKD protocol (Ma et al., 2019; Zhao et al., 2020b). At the same time, MDI protocols (Huang et al., 2022) close the loophole of detectors (Ma et al., 2014; Zhang et al., 2014), which offers higher security than protocols trusting both senders and detectors. Additionally, MDI protocols naturally facilitate scenarios in which two users Alice and Bob are in different media, such as fiber-to-water (Yu et al., 2022) and space-to-water (Peng et al., 2022). The intermediate party, Charlie, can be placed at the junction of the two media, so that each signal is transmitted in only one medium.

6 Discussion

DM-CV-QKD is a vibrant research direction with great potential for development. Compared with traditional discrete-variable systems, continuous-variable systems have indisputable low-cost advantages. In terms of preparation, discrete modulation is also simpler than Gaussian modulation, and eases the interpretation, error correction and privacy amplification of measurement outcomes during the postprocessing. Relaxing the requirements for experimental conditions inevitably increases the difficulty of security analysis.

The quality of security analysis should be evaluated from four aspects: security levels, tightness, universality and computation complexity. Most security analysis methods can only resist collective attacks under the asymptotic regime, which is still some distance from security against coherent attacks under the finite-size regime. Recently, analytical and numerical methods that can be widely applied to different discrete modulation schemes have been proposed. Analytical methods are usually easy to compute but not tight enough for long distance transmission. Numerical methods, especially nonlinear methods, can find the tight lower bound of the key rate but consume considerable computational resources and time. There is still room for improvement in terms of tightness of key rate bounds, security level and postprocessing speed.

Imposing the idea of machine learning in DM-CV-QKD is a promising way to enhance the practicality (Li et al., 2018). For instance, the neural network can reduce time consumption in predicting the secret key rate (Liu Z.-P. et al., 2022; Zhou et al., 2022). Ensemble learning has been used to predict communication failure caused by channel disturbance (Li et al., 2021b). Additionally, some technical means can be applied to further improve the performance (Li and Cvijetic, 2018), such as quantum catalysis (Guo et al., 2020; Ye et al., 2021) and quantum scissors (Ghalaii et al., 2020; Li Y. et al., 2021), multicarrier (Gyongyosi, 2020).

While mastering the basic protocol of DM-CV-QKD, its application range can be expanded according to its characteristics. It has the potential for communications in free-space channels (Hill et al., 2017; Mélen et al., 2017), satellite-to-earth channels (Liao et al., 2017a) and seawater channels (Ruan et al., 2019). In addition, MDI protocols against side-channel attacks and device-independent protocols with higher security are also worthy of further study.

Author contributions

C-LL arranged and wrote the section of numerical method and the satellite-to-ground protocol. Z-PL and M-GZ organized and wrote the section on the neural network method. W-BL wrote other parts and integrated all the content. H-LY and Z-BC supervised the project.

Funding

We gratefully acknowledge support from the Natural Science Foundation of Jiangsu Province (No. BK20211145), the Fundamental Research Funds for the Central Universities (No. 020414380182), the Key Research and Development Program of Nanjing Jiangbei New Area (No. ZDYD20210101), and the Program for Innovative Talents and Entrepreneurs in Jiangsu (No. JSSCRC2021484).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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