A Survey on Continuous Variable Quantum Key Distribution for Secure Data Transmission: Toward the Future of Secured Quantum-Networks

Quantum teleportation [58] enables the transfer of an undisclosed quantum state between distant qubits or qumodes. It relies on classical communication and quantum entanglement, playing a crucial role in quantum information protocols. It also has applications in distributed information processing for quantum computation.

Quantum teleportation circuits operate on a consistent fundamental principle. Two observers, Alice and Bob, situated apart, share a maximally entangled quantum state [59]. This state can be one of the four Bell states for discrete variables or a maximally entangled state with a fixed energy level for continuous variables. They also have access to a classical communication channel as shown in Figure 2.

Figure 2.

Illustration of continuous-variable quantum teleportation. An arbitrary coherent state |ψ〉 (top rail) is to be teleported from Alice to Bob. Two infinitely squeezed vacuum states (middle and bottom rails) serve as the shared resource between them. First, these two squeezed states pass through a 50:50 beam splitter (left 50:50 block) to create an entangled EPR-like pair, with one mode held by Alice and the other mode sent to Bob. Next, Alice couples her unknown state |ψ〉 with her half of the entangled pair using another 50:50 beam splitter (right 50:50 block). She then performs homodyne measurements of both the amplitude (x) and phase (p) quadratures on the combined output. The measured values, 〈x〉 and 〈p〉, are sent via a classical channel to Bob, who applies corresponding displacement operations (the X and Z boxes) to his mode. This “corrects” his half of the entangled pair into an identical copy of |ψ〉. The inset (lower box) shows the 50:50 beam splitter’s action, splitting or combining two input modes in equal proportions.

To transfer an unknown state from Alice to Bob, Alice conducts a joint measurement on her portion of the entangled state and the unknown state. This measurement involves projecting onto the Bell basis. Alice subsequently communicates the measurement results to Bob. Armed with this information, Bob can convert his half of the entangled state into an accurate copy of the original unknown state. This transformation may entail a conditional phase flip for qubits or displacement for qumodes.

The initial concept of quantum teleportation focused on discrete-variable quantum communication with qubits. However, its versatility also applies to continuous-variable qumodes, especially when they are spatially distant. In Figure 2, we can see the circuit specifically designed for continuous-variable quantum teleportation, where BS stands for beam splitter, and X and Z are position displacement and momentum displacement which are defined as D^(x0)=e−iℏx0p^ and D^(p0)=eiℏp0x^ , respectively, to recover the state |ψ〉 exactly (The impact of these operators are D^(x0)ψ(x)=ψ(x−x0) and D^(p0)φ(x)=φ(p−p0) , respectively).

The Strawberry Fields Library in Python was used to effectively simulate the squeezed states and their quantum state evolution through the proposed algorithm. This library facilitates the simulation of such photonic devices for quantum applications.

Assuming we aim to teleport an arbitrary coherent state such as Figure 1a. As shown in Figure 2, The entangled states of Bob and Alice should be a combination of states as depicted. They are considered entangled due to the beam splitter transformation applied to the initial infinite squeezed states. By performing the third and fourth steps on the states, Bob will achieve the same state as depicted in [60].

The process of continuous variables quantum teleportation depicted in Figure 2(for simplicity we call the coherent state ate the beginning of the circuit mode 0 and Alice and Bob become mode 1 and mode 2 respectively) can be paraphrased as follows: (a)

Initially, the qumodes |0〉_p and |0〉_x (Alice and Bob) are set up as infinitely squeezed vacuum states in momentum and position space, respectively. First, we express the meaning of vacuum squeezed state which is: 3|ξ〉=S^(ξ)|0〉 where Ŝ(ξ) is defined as Equation (1). So, one can calculate the squeezed position wave function as follows [61]: 4〈ξ〉=ψr(x)=(λπ)14exp exp(−λx22) where λ = e^2r, which is called the squeezing parameter. By taking the squeezing parameter into infinite the squeezed position wave function becomes Dirac delta function and the uncertainty in position vanishes. Moreover, same calculations could be done for infinite momentum vacuum squeezed state which is defined as 〈ξ〉=ψr(p)=(1λπ)1/4exp(−x22λ) .

Figure 3.

Wigner function of quadrature diagram of Bob and Alice entangled states after going through beamsplitters.

As pointed out in [62], any unitary transformation on individual qubits can be depicted as a series of three rotations, one around the y^ axis and two around the z^ axis. Motivated by this fact, a Mach-Zehnder integrated cell is appropriate for implementing an arbitrary single-qubit unitary transformation [63]. An alteration to the Mach-Zehnder interferometer cell can be made to align with the structure illustrated in Figure 2. The updated integrated cell is shown in Figure 4a.

Figure 4.

(a) A modified integrated unit cell inspired by the Mach-Zehnder Cell. It can act as an arbitrary Measurement gate for identifying state’s position or momentum by changing the variable θ of the phase shifter. (b) Bent directional coupler with labeled parameters: R_in and R_out represent the inner and outer bending radii, respectively, while θ indicates the bending angle. The structure is designed to facilitate efficient light coupling between waveguides while minimizing losses and cross-talk effects.

Elements of the structure shown in Figure 2 can be easily reconfigured using two of these newly adapted cells, two homodyne detectors, and two directional couplers. The resulting circuit will resemble Figure 5. The two directional couplers are necessary to create entanglements between the states of Alice and Bob. They function similarly to the beamsplitters in Figure 2. This type of circuit will also require two homodyne detectors to measure the average value of arbitrary quadrature components. It’s important to note that in this setup, the quantum states are not formed using single photons, but rather continuous variable quantum states that have the capability to be squeezed. Moreover, bent directional couplers [64,65] as depicted in Figure 4b play a pivotal role in the design and optimization of integrated photonic circuits, particularly within Mach-Zehnder interferometers (MZIs). These couplers facilitate efficient light propagation and interaction by guiding optical signals through curved paths with minimal losses.

Figure 5.

A 3D representation of a Photonic Integrated Circuit (PIC) designed to implement quantum teleportation in a continuous quantum variable circuit. The input states are introduced into the circuit from the left where (a) is with bent directional couplers and (b) is with conventional directional couplers, and at the conclusion (right side), Bob will possess Alice’s initial state of |ψ〉. The two blue cones symbolize homodyne detectors that measure the average value of arbitrary quadrature components. The outcomes of these measurements are utilized as controls for X and Z which are position displacement and momentum displacement, respectively, to recover the state |ψ〉 exactly.

Their inclusion in the system enhances the compactness and scalability of the overall architecture, which is essential for quantum communication systems. Furthermore, the use of bent directional couplers ensures precise splitting and combining of optical signals, thereby maintaining the coherence and stability critical for continuous variable quantum key distribution (CV-QKD). By enabling tighter integration of components, these couplers support the development of high-performance and miniaturized photonic devices for secure data transmission which can be seen as a new avenue.

Quantum Key Distribution (QKD) has undergone significant advancements with the development of Continuous Variable (CV) protocols, starting from Ralph’s pioneering proposal to encode key information using the amplitude and phase quadratures of the light field. Initially, discrete modulation schemes focused on binary-encoded keys [66–68]. However, continuous modulation protocols, such as Gaussian modulation, soon emerged, utilizing coherent and squeezed states of light [69–71].

Modulation plays a central role in CV-QKD [72], encoding classical information into quantum states for secure key exchange. In Gaussian modulation, Alice prepares coherent or squeezed states by modulating their quadrature components—amplitude and phase—using Gaussian-distributed random variables. Electro-optic modulators driven by high-speed random number generators achieve these modulations, enabling high-dimensional encoding that improves key rates and resilience against noise.

For discrete modulation, Alice selects from a finite set of states, such as quadrature phase-shift keying (QPSK) or other constellation formats. While this approach simplifies state preparation and error correction, it often requires more sophisticated security proofs.

Alice transmits the modulated quantum states over an insecure quantum channel—optical fiber or free space— to Bob, who measures the quadratures using homodyne or heterodyne detection. Homodyne detection involves Bob actively selecting and measuring either the amplitude or phase quadrature, leading to data sifting where half of the data is discarded to ensure matched bases. Heterodyne detection, by contrast, measures both quadratures simultaneously, eliminating sifting and enhancing the key rate.

The no-switching protocol utilizes heterodyne detection with Gaussian-modulated coherent states, simplifying implementation and improving efficiency. Furthermore, protocols leveraging Einstein–Podolsky–Rosen (EPR) entangled states demonstrate that modulation can be avoided altogether, relying on quantum correlations instead [73].

Modulation techniques have evolved to address challenges such as excess noise, channel loss, and implementation complexity [74]. Doubly modulated coherent states reduce inefficiencies by requiring only Alice to discard half of her data during sifting. Additionally, modulating entangled states of light has improved robustness to channel noise, resulting in higher key rates and extended distribution distances [75,76].

Digital signal processing (DSP) and hardware advancements have further enhanced modulation precision and system performance. Modern CV-QKD systems utilize high-speed modulators and advanced reconciliation algorithms to maximize key rates and minimize errors. For example, heterodyne-based Gaussian-modulated coherent state protocols eliminate active basis choice and sifting, enhancing the key rate [77,78].

The secret key rate in CV-QKD is given by: 6K=βI(A:B)−χ(E) where β is Reconciliation efficiency which is in [0,1], I(A : B) is Mutual information between Alice and Bob, computed as: 7I(A : B) = H(A) + H(B) − H(A, B) where H(·) represents the Shannon entropy, and χ(E) is Holevo bound, representing Eve’s maximum accessible information.

For Gaussian-modulated coherent states, the mutual information is: 8I(A:B)=12(1+VsVN) where V_s is the signal variance, and V_N is the noise variance.

The Holevo bound is calculated using the symplectic eigenvalues of the covariance matrices describing the quantum states. It represents the upper bound of information accessible to an eavesdropper.

By leveraging quantum correlations, CV-QKD protocols utilizing EPR entangled states have demonstrated improved robustness and higher key rates, even without direct signal modulation [73]. For instance, a protocol using squeezed states and heterodyne detection proposed in 2009 achieved higher secret key rates than earlier Gaussian protocols [79].

CV-QKD can be implemented in two frameworks: the prepare-and-measure (PM) scheme and the entanglementbased (EB) scheme. The PM scheme is typically simpler to implement, while the EB scheme offers a unified theoretical framework for key rate calculations. These two approaches are equivalent for Gaussian protocols [80,81].

In summary, advancements in modulation methods, hardware, and theoretical models have enabled CV-QKD to achieve higher key rates, extended distribution distances, and enhanced robustness against noise, solidifying its position as a viable solution for secure quantum communication.

Figure 6 illustrates an experimental setup for long-distance Continuous Variable Quantum Key Distribution (CV QKD) [22]. This process is achieved by utilizing two cascaded intensity modulators to generate high-extinction-ratio optical pulses with a width of 100 ns and a repetition rate of 500 kHz from a 1550-nm continuous-wave (CW) single-frequency fiber laser. These optical pulses are then split into a weak signal field and an intense local oscillator (LO) using a fiber coupler (99/1). The pulsed coherent states undergo bivariate Gaussian modulation in the phase space through a combination of an intensity modulator and a phase modulator. The signal and LO are directed through a 50-km single-mode fiber spool using time multiplexing and polarization multiplexing, achieved by an 80-m fiber and a polarizing beam splitter (PBS), respectively. When Bob receives the quantum states sent by Alice, he randomly measures either the amplitude or phase quadrature of the signal using a pulsed balanced homodyne detector (BHD). The random basis switch and relative phase locking between the signal and LO are implemented with a phase modulator in the LO path. To accurately measure the BHD output’s peak voltage and synchronize the system, a small portion of the LO beam is extracted to recover the system clock, which is then precisely delayed to the desired value.

Figure 6.

This is the experimental configuration proposal for a 50-km Continuous Variable Quantum Key Distribution (CV QKD). In this setup, DAQ refers to the data acquisition module, AM is the amplitude modulator, PM stands for the phase modulator, and PBS is the polarizing beam splitter. DPC is the dynamic polarization controller, MVOA denotes the motor variable optical attenuator, and BHD is the balanced homodyne detector.

In another study [82], Roberts et al. address a significant challenge in high-rate decoy-state Quantum Key Distribution (QKD) systems: the instability of intensity modulators (IMs), which can lead to security vulnerabilities known as side channels. These side channels arise when the intensity of a light pulse depends on the preceding pulse, a phenomenon termed the “patterning effect.” This effect can inadvertently leak information, compromising the security of the QKD system. To mitigate this issue, the researchers propose the use of a tunable extinction ratio Sagnac-based intensity modulator. This design offers several advantages: A.

Patterning Effect Mitigation: By operating at specific modulation points, the Sagnac-based IM ensures that the output intensity of each pulse is independent of previous pulses, effectively eliminating the patterning effect.

By integrating this Sagnac-based IM into QKD systems, a practical solution to enhance security and performance is achieved, addressing critical issues associated with traditional intensity modulators.

Excess noise significantly impacts the performance of the system, affecting both the secure key rate and transmission distance. Ideally, excess noise results solely from Eve’s eavesdropping behavior. However, in practice, technical noises may arise even without eavesdropping. These technical noises include LO pulse leakage due to finite optical pulse extinction ratios, quantum state preparation, non-ideal phase locking between the LO and signal, nonlinear optical effects in the fiber, and BHD stability.

In their comprehensive review, Zhang et al. [22] emphasize the pivotal role of laser stability in Continuous-Variable Quantum Key Distribution (CV-QKD) systems. They highlight those fluctuations in the laser’s frequency and intensity can introduce excess noise, which adversely affects the system’s performance and security. To mitigate these issues, the authors recommend implementing precise temperature control and effective vibration isolation. By maintaining a stable operating environment, these measures can significantly reduce frequency drift and intensity noise, thereby enhancing the reliability and security of CV-QKD systems.

While Eve typically cannot manipulate these noises effectively, they can vary over time, necessitating real-time monitoring to characterize system parameters. In practical QKD systems, such noises are often attributed to Eve’s attack when assessing security.

Although some possible challenges were mentioned in this part, further and deeper discussions about security concerns of CV-QKD and possible solutions such as measurement-device-independent [83] methods are covered in a further section of this paper.

As discussed in the Introduction, fiber-based DV-QKD has been successfully demonstrated over ultra-low-loss fibers, achieving key rates at the kbps level for distances up to 100 km. Several DV-QKD protocols, including encoding based on spatial dimensions [84], photon polarization [85], and time bins [86], have been realized on silicon wafers. For on-chip photon detection, superconducting nanowire-based single-photon detectors with detection efficiencies of up to 90% are integrated [87,88]. Notably, CV-QKD is particularly suitable for photonic chip integration due to its compatibility with existing telecom technologies [24,89].

In 2013, a fiber-based Continuous Variable Quantum Key Distribution (CV-QKD) system achieved a secure key rate of approximately 1 kbps over an 80 km transmission distance [90]. By effectively managing excess noise, the distance was subsequently extended to over 100 km [91].

Recently, there have been reports of on-chip quantum entropy sources based on detecting vacuum fluctuations [92] and phase fluctuations [93–95]. Notably, a chip-based homodyne detector demonstrated a gain of 4.5 kVA⁻¹ with a 150-MHz bandwidth [92].

Despite these advancements, integrating all necessary components for CV-QKD systems into a single chip presents significant challenges due to the diverse material requirements and fabrication processes for each component [96]. One way to overcome this challenge is by modular integration. This strategy involves fabricating distinct functional blocks—such as modulators, detectors, and couplers—separately, and subsequently interconnecting them. This method simplifies fabrication, enhances scalability, and facilitates easier upgrades and maintenance [97].

Existing techniques supporting modular integration include heterogeneous and hybrid integration. Heterogeneous integration combines different materials, such as III-V semiconductors with silicon photonics [98], enabling the integration of active components like lasers and amplifiers onto silicon-based platforms. Hybrid integration involves the optical or electrical coupling of separate chips, each optimized for specific functions, to form a cohesive photonic system. Packaging techniques, such as photonic wire bonding [99], facilitate seamless communication between these integrated modules by providing low-loss interconnections.

Applying modular integration to CV-QKD architectures can significantly enhance system performance. For instance, components like Mach-Zehnder interferometers and bent directional couplers can be fabricated independently and then interconnected, allowing for optimized performance of each module. This approach not only reduces fabrication complexity but also allows for the selection of the most suitable materials and fabrication processes for each component, thereby improving the overall efficiency and reliability of the CV-QKD system [97].

The optimization of Photonic Integrated Circuits (PICs) for the application of squeezed light presents a challenge, as most PICs are specifically designed for coherent light or single photons. A significant hurdle is integrating the squeezed light source with other components of the PIC. Despite these challenges, efforts are being made to address them, such as an experimental chip designed by Xanadu [106]. This device is a programmable nanophotonic chip capable of performing quantum circuit tasks using a squeezed light beam.

As per Figure 8a, the core component of Xanadu’s programmable device is a photonic chip measuring 10 mm × 4 mm. This chip can generate squeezed light in up to eight optical modes. To achieve this, the chip is initialized into four separate two-mode squeezed vacuum states. The squeezing happens between specific bichromatic mode pairs, each occupying one of the four spatially separated waveguide modes. An interferometer, consisting of a network of beam splitters and phase shifters, is used to effectively manipulate these spatial modes. This setup allows for a user-programmable gate sequence, corresponding to an SU(4) transformation applied to the spatial modes, synthesizing an eight-mode Gaussian state. This state is then measured on the Fock basis using eight separate photon-numberresolving detectors. For a visual representation of the machine’s operation, refer to the equivalent quantum circuit diagram in Figure 8b.

Figure 8.

(a) The chip is being reintroduced, as shown in a micro-graph of the actual device. (b) The provided quantum circuit diagram serves as a comparable representation of the photonic hardware, emphasizing its functional features similar to [69].

Photon loss is a notable phenomenon in quantum optics that can impact quantum systems in various ways. When photons are lost or absorbed by the environment, it can result in quantum state degradation, such as Decoherence and State Dephasing, Reduced Fidelity, etc. However, this protocol demonstrates robustness [107], functioning effectively even in the presence of photon loss, non-ideal sources, and imperfect detection. The protocol’s ability to scale to handle a large number of photons provides strong evidence of its resilience, highlighting its practical application potential. This scalability is a significantly simpler task compared to building a universal quantum computer [108].

Experimental demonstrations of CV-QKD over metropolitan networks and field deployments highlight its potential for large-scale secure communication systems. Advancements in modulation techniques and protocol designs continue to bridge the gap between theoretical innovation and practical implementation, ensuring secure information transmission in the quantum era.

Despite these advancements, CV-QKD systems remain susceptible to specific vulnerabilities, such as quantum attacks and side-channel attacks. Quantum attacks, including collective Gaussian attacks, involve eavesdroppers performing collective measurements on transmitted quantum states to extract key information. Side-channel attacks exploit imperfections in system components, like the saturation attack on homodyne detectors, which can bias noise estimation and compromise security.

To enhance the robustness of CV-QKD systems, several mitigation strategies are proposed: A.

Enhanced Component Design: Improving the linearity and dynamic range of detectors reduces susceptibility to saturation attacks.

Incorporating machine learning (ML) and tensor networks into CV-QKD systems offers promising avenues for enhancing both functionality and security. ML techniques have been applied to various stages of CV-QKD protocols, including phase error estimation, excess noise estimation, state discrimination, parameter estimation, key sifting, and key rate estimation. These applications aim to alleviate implementation complexities and improve system performance [35].

For instance, ML-assisted phase estimation can enhance carrier recovery processes, leading to more accurate phase compensation and reduced excess noise. This improvement is crucial for maintaining high secret key rates over long distances [36,40].

Moreover, tensor networks, which efficiently represent and manipulate high-dimensional data, can optimize the processing of quantum information in CV-QKD systems. Their application can lead to more efficient data handling and improved security analyses, particularly in complex quantum networks [109–111].

Measurement-device-independent quantum key distribution (MDI-QKD) [83] enhances security by eliminating all detector side-channel vulnerabilities [112,113], making it a robust solution for secure communications across various platforms. In fiber-based systems, MDI-QKD has achieved significant milestones, including successful key distribution over 404 km of ultralow-loss optical fiber, demonstrating its potential for long-distance applications [15]. For free-space channels, recent advancements have addressed challenges such as atmospheric turbulence. Notably, a study demonstrated MDI-QKD over a 19.2 km urban atmospheric link, marking a crucial step toward satellite-based quantum communications [114,115]. In the realm of integrated photonics, the development of chip-based MDI-QKD systems has been pivotal. Researchers have successfully implemented an all-chip MDI-QKD system using silicon photonic technology, achieving a per-pulse key rate of 2.923 bits [116]. Furthermore, new methods can provide ease of use for MDI-QKD devices, for example by introducing plug-and-play capability [117]. These advancements collectively underscore the versatility and security of MDI-QKD across diverse communication infrastructures using different modulation methods, even with the presence of some imperfections such as the fast-fading channel [118–126].