Beaming Data: Making Quantum Teleportation Practical

Author: Denis Avetisyan

New research explores how to reliably transmit quantum information over realistic communication channels, despite noise and signal loss.

A quantum teleportation protocol leverages a shared two-mode squeezed state-distributed via a low-loss quantum channel-and a feedforward mechanism to reconstruct an input state at a distant location, relying on a Bell-type measurement and local unitary operations informed by classical communication, where the ensemble of input states is defined by a truncated Gaussian distribution, acknowledging the inherent decay of fidelity through noisy channels.

This study demonstrates secure continuous-variable quantum teleportation using finite-energy codebooks and analyzes performance under realistic channel conditions.

While quantum communication promises unconditionally secure information transfer, practical implementations face limitations imposed by channel imperfections and finite resources. This is addressed in ‘Practical quantum teleportation with finite-energy codebooks’, which investigates continuous-variable quantum teleportation under realistic conditions, including feedforward losses, noise, and truncated Gaussian codebooks. Our analysis demonstrates that these impairments can be mitigated-and secure communication maintained-by optimizing feedforward gain and carefully considering codebook configurations, revealing modified no-cloning thresholds and security criteria. Could these findings pave the way for robust and secure microwave quantum networks operating with limited resources?

The Fragility of Classical Security

Conventional encryption methods, the bedrock of modern digital security, paradoxically depend on a principle that simultaneously exposes their inherent weakness: the No-Cloning Theorem. This theorem, originating in quantum mechanics, states that an unknown quantum state cannot be perfectly duplicated. While seemingly unrelated, classical security protocols leverage the difficulty of copying information – the computational cost of breaking encryption – as their defense. However, this is not a fundamental limit, but rather a reliance on technological constraints. An adversary with sufficient computational power could, in theory, overcome these barriers. Consequently, classical systems are always vulnerable to eavesdropping, because the information itself can be copied, even if doing so is currently impractical. This fundamental limitation drives the search for truly secure communication methods, like those offered by quantum cryptography, which rely on the laws of physics, rather than computational complexity, to guarantee confidentiality.

The bedrock of classical encryption relies on the computational difficulty of certain mathematical problems, but quantum communication proposes a fundamentally different approach to security, rooted in the laws of physics themselves. Central to this potential is the No-Cloning Theorem, a principle of quantum mechanics stating that an unknown quantum state cannot be perfectly duplicated. This isn’t a limitation of technology, but an inherent property of the universe; any attempt to copy the state inevitably disturbs it, alerting legitimate parties to the presence of an eavesdropper. Unlike classical bits which can be copied endlessly without alteration, any interception of a quantum message leaves a detectable trace. This feature allows for the creation of cryptographic protocols where the very act of observation compromises security, offering a pathway to communication demonstrably secure against even the most powerful computational attacks. $|\psi\rangle$ cannot be perfectly replicated, ensuring that intercepted information is inherently unreliable to an unintended recipient.

Despite the theoretical promise of quantum communication, practical implementation faces significant hurdles stemming from the realities of physical channels. The transmission of quantum states – the very foundation of secure communication – is inherently susceptible to noise and signal degradation. Photons, often used as quantum carriers, can be scattered, absorbed, or otherwise altered during transit, leading to errors in the received information. Furthermore, imperfections in the devices used to create, manipulate, and detect these quantum states introduce additional errors. These challenges necessitate sophisticated error correction protocols and advanced technological refinements to maintain the integrity of quantum information and ensure genuine security – a pursuit that remains at the forefront of quantum information science.

This model comprehensively describes analog continuous-variable quantum teleportation, accounting for both signal loss and added noise.

State Transfer: The Essence of Quantum Teleportation

Quantum teleportation is a process by which the exact quantum state of a particle, described by its wavefunction, is transferred from one location to another. Critically, this does not involve the physical transmission of the particle itself. Instead, information describing the quantum state is transferred, and a new particle at the receiving location is manipulated to assume that identical state. This is accomplished by leveraging quantum entanglement and classical communication; the original particle’s state is destroyed in the process of measurement at the sending location, adhering to the no-cloning theorem. The transferred state is not a copy, but a recreation of the original state on a different particle.

Quantum teleportation relies on a pre-shared entangled state between the sender (Alice) and the receiver (Bob). This entangled resource, typically a Bell state such as $\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ , establishes a correlation between the two qubits, regardless of the distance separating them. Alice performs a Bell measurement on the qubit to be teleported and her half of the entangled pair, collapsing the entanglement. The result of this measurement-one of four possible outcomes-is then communicated to Bob via a classical channel. Bob then applies a specific unitary transformation, determined by Alice’s classical message, to his half of the entangled pair, recreating the original unknown quantum state. The entanglement is consumed in the process, meaning a new entangled pair is required for each teleportation event.

The successful completion of quantum teleportation necessitates classical communication through a dedicated channel, termed the Classical Feedforward Channel. While quantum entanglement establishes the correlation required for state transfer, it does not, on its own, transmit information. The sender, after performing a Bell state measurement on their particle and the entangled particle, obtains two bits of classical information detailing the measurement result. These two bits must be transmitted to the receiver via the Classical Feedforward Channel. The receiver then uses this classical information to apply a specific unitary transformation to their entangled particle, effectively reconstructing the original unknown quantum state. Without this classical communication, the receiver’s particle remains in a mixed state, and teleportation fails; the classical channel therefore represents a fundamental component of the protocol, limiting the speed of teleportation to the speed of light.

Teleportation fidelity, calculated with realistic experimental parameters, demonstrates a quantum advantage-maintained up to liquid helium temperatures with feedforward channel loss up to 7 dB and at room temperature with a feedforward coupling of approximately -24 dB-as influenced by feedforward channel loss, noise temperature, feedforward coupling strength, and measurement gain, with asymptotic limits defined by the classical <span class="katex-eq" data-katex-display="false">F_{cl}=1/2</span> and no-cloning <span class="katex-eq" data-katex-display="false">F_{nc}=2/3</span> bounds. — Teleportation fidelity, calculated with realistic experimental parameters, demonstrates a quantum advantage-maintained up to liquid helium temperatures with feedforward channel loss up to 7 dB and at room temperature with a feedforward coupling of approximately -24 dB-as influenced by feedforward channel loss, noise temperature, feedforward coupling strength, and measurement gain, with asymptotic limits defined by the classical $F_{cl}=1/2$ and no-cloning $F_{nc}=2/3$ bounds.

Continuous Variables: Bridging Theory and Practicality

Continuous-variable quantum teleportation encodes quantum information using coherent states, which are quantum states that possess a definite amplitude and phase. Unlike qubit-based teleportation relying on discrete variables, this approach leverages the continuous degrees of freedom of electromagnetic fields. This offers practical advantages due to the relative ease of generating, manipulating, and measuring coherent states with high fidelity using standard optical components. Coherent states also exhibit Poissonian statistics, simplifying the characterization of signal and noise. Furthermore, continuous-variable teleportation is inherently more robust against certain types of noise and loss, as the information is distributed across the entire waveform rather than being localized to a single quantum entity. The use of homodyne or heterodyne detection allows for the complete reconstruction of the quantum state via measurement of quadrature amplitudes.

Displacement operations and Bell-type measurements are fundamental to continuous-variable quantum teleportation. A displacement operation shifts the quadrature of a coherent state by a defined amplitude and phase, enabling the input state to be superimposed onto the entangled resource. Bell-type measurements, specifically homodyne or photon-number detection, project the combined input and entangled states onto one of four Bell states $\ket{\Phi^{+}}, \ket{\Phi^{-}}, \ket{\Psi^{+}}, \ket{\Psi^{-}}$ . These measurements do not directly reveal the teleported state, but rather provide classical information necessary for reconstructing the state at the receiver through appropriate displacement operations based on the measurement outcome. The efficiency of teleportation is directly linked to the fidelity of these displacement operations and the precision of the Bell-type measurement apparatus.

The two-mode squeezed vacuum state ( $| \psi \rangle = \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle)$ ) is a fundamental entangled state utilized as a key resource in continuous-variable quantum teleportation. Its generation commonly employs a Josephson Parametric Amplifier (JPA), a nonlinear superconducting device that amplifies quantum signals while preserving quantum coherence. The JPA operates by modulating the capacitance of a superconducting circuit, effectively creating correlated photon pairs in the squeezed state. The degree of squeezing, and thus the entanglement quality, is determined by the pump amplitude and frequency of the JPA, influencing the achievable fidelity of the teleportation protocol. Alternative generation methods include optical parametric oscillators, but JPAs offer advantages in terms of integration and control for practical applications.

The mutual information between Alice and Bob <span class="katex-eq" data-katex-display="false">I(A:B)</span> and Eve’s information <span class="katex-eq" data-katex-display="false">\chi_E</span> decrease with feedforward channel temperature <span class="katex-eq" data-katex-display="false">T_{ff}</span>, establishing secure teleportation fidelity thresholds <span class="katex-eq" data-katex-display="false">F_s</span> that are further influenced by resource squeezing <span class="katex-eq" data-katex-display="false">S</span> and measurement gain <span class="katex-eq" data-katex-display="false">G</span>, with security unattainable in the shaded parameter regime. — The mutual information between Alice and Bob $I(A:B)$ and Eve’s information $\chi_E$ decrease with feedforward channel temperature $T_{ff}$ , establishing secure teleportation fidelity thresholds $F_s$ that are further influenced by resource squeezing $S$ and measurement gain $G$ , with security unattainable in the shaded parameter regime.

Codebooks and Loss: Navigating Imperfect Channels

Truncated codebook schemes represent a strategic approach to communication in challenging environments by deliberately limiting the number of possible encoded states. This reduction in complexity isn’t a sacrifice of information, but rather a bolstering of resilience against channel noise. Traditional codebooks, while capable of representing a wider range of data, become increasingly vulnerable as noise introduces errors; a small disturbance can easily map a signal to an incorrect, but still valid, state. By intentionally curtailing the codebook, the probability of mistranslation due to noise is diminished, as fewer states exist to be confused with the intended signal. Essentially, the system trades a slight reduction in potential data transmission rate for a substantial gain in reliability – a pragmatic compromise particularly valuable in $lossy$ channels where data corruption is frequent. This technique allows for more accurate reconstruction of the original message, even when significant noise is present, effectively prioritizing clarity over sheer data volume.

Employing Gaussian distributions when constructing codebooks for digital communication presents a compelling trade-off between maximizing the amount of information transmitted and maintaining reliability in the face of transmission errors. This approach leverages the central limit theorem, allowing for efficient packing of codewords while simultaneously providing a degree of inherent robustness to noise. Unlike deterministic or overly sparse codebooks, Gaussian distributions distribute codeword energy more evenly, minimizing the impact of individual bit errors and reducing the probability of misdecoding. The effectiveness stems from the distribution’s ability to smoothly cover the signal space, ensuring that even noisy received signals can be accurately mapped back to the intended codeword – a crucial characteristic when operating under $lossy$ channel conditions where some signal degradation is inevitable. Consequently, Gaussian codebooks offer a practical solution for achieving high throughput and dependable communication, particularly in environments characterized by imperfect or unreliable transmission channels.

The efficacy of truncated and Gaussian codebooks in noisy communication systems is profoundly shaped by the characteristics of lossy channels. These channels, unlike simple additive noise scenarios, introduce data loss – bits are not merely corrupted, but disappear entirely – demanding a more nuanced approach to codebook design. Studies reveal that as the probability of bit erasure increases within a lossy channel, even carefully constructed codebooks experience a dramatic decline in performance, with error rates escalating rapidly. The challenge lies in balancing the need for sufficient codebook diversity to represent information accurately against the inherent vulnerability to complete data loss. Researchers are actively investigating adaptive codebook strategies, where the codebook’s structure is dynamically adjusted based on real-time channel conditions, to mitigate these effects and maintain reliable communication in the face of significant data loss. $P_{loss}$ -the probability of loss-becomes a critical parameter in determining the achievable communication rate and overall system reliability.

The probability distributions of Gaussian, truncated uniform, and truncated Gaussian codebooks, visualized as functions of complex displacement amplitude α and in phase space spanned by field quadratures <span class="katex-eq" data-katex-display="false">{p, q}</span>, demonstrate how truncation affects the shape of these distributions. — The probability distributions of Gaussian, truncated uniform, and truncated Gaussian codebooks, visualized as functions of complex displacement amplitude α and in phase space spanned by field quadratures ${p, q}$ , demonstrate how truncation affects the shape of these distributions.

Secure Fidelity: Defining the Threshold of Trust

Secure fidelity represents a critical benchmark in quantum communication, establishing the lowest acceptable signal quality necessary to ensure information remains confidential. This concept acknowledges that no real-world channel is perfect; some signal degradation is inevitable. Research indicates that a secure fidelity threshold of approximately two-thirds is required to actively counteract eavesdropping attempts and maintain a secure connection. Below this level, the potential for an attacker to glean meaningful information increases dramatically, compromising the integrity of the communication. Consequently, system designs prioritize maintaining signal fidelity above this threshold through techniques like error correction and optimized transmission protocols, effectively defining a practical limit for secure data transfer rates.

The limits of an eavesdropper’s knowledge in quantum communication are fundamentally constrained by information theory, specifically through the concepts of Mutual Information and Holevo Quantity. These quantities establish upper bounds on how much information an adversary can glean about the transmitted message. Mutual Information, in this context, is quantified as $\ln(1 + 4\sigma^2/(1+\cosh(2r)))$ , reflecting the relationship between the transmitted signal and the information available to the eavesdropper, where σ represents noise and $r$ is a parameter related to signal strength. Similarly, the Holevo Quantity, expressed as $\ln(1 + k\sigma^2/v_{out})$ , provides another, related limit, considering the quantum nature of the signal and incorporating $k$ and $v_{out}$ to account for channel characteristics and output variance. These bounds are not absolute guarantees of security, but rather benchmarks that define the maximum information leakage possible, allowing communication protocols to be designed to minimize this leakage and ensure secure transmission.

Establishing truly secure communication hinges on meticulous system design, demanding optimized codebooks tailored to the specific characteristics of the transmission channel. Recent research demonstrates that simply achieving a functional signal isn’t enough; a minimum level of quantum squeezing – quantified at 2.39 dB – is crucial for bolstering security against potential eavesdroppers. This squeezing effectively reduces noise and enhances signal clarity, directly impacting the ability to reliably transmit information while minimizing the information leakage accessible to an attacker. The interplay between codebook design and channel properties determines the overall secure fidelity, and deviations from this optimal balance can significantly compromise the confidentiality of the transmitted data, highlighting the need for careful calibration and continuous monitoring of the communication system.

Fidelity <span class="katex-eq" data-katex-display="false">F_s</span> for a truncated Gaussian codebook converges to the results of truncated uniform and Gaussian codebooks as the truncation-to-variance ratio <span class="katex-eq" data-katex-display="false">N/\sigma^2</span> approaches 0 and infinity, respectively, as demonstrated by the numerical calculations (squares) and derived results (solid lines) from equation (27). — Fidelity $F_s$ for a truncated Gaussian codebook converges to the results of truncated uniform and Gaussian codebooks as the truncation-to-variance ratio $N/\sigma^2$ approaches 0 and infinity, respectively, as demonstrated by the numerical calculations (squares) and derived results (solid lines) from equation (27).

The pursuit of practical quantum teleportation, as detailed in this study, reveals an inherent tension between ideal theory and the constraints of physical systems. Each truncation of Gaussian states, each accommodation for channel loss, represents a necessary decay of perfection. It is a recognition that information, like all physical entities, is subject to the erosive forces of time. As Werner Heisenberg observed, “The ultimate values are not the values that are measured, but the values that are predicted.” This prediction, born from mathematical models, must constantly adapt to the signals received from a decaying system-a system where the fidelity of teleportation is intrinsically linked to the limitations imposed by finite resources and inevitable noise. Refactoring the codebooks, in essence, becomes a dialogue with the past, acknowledging the imperfect nature of transmission and striving for graceful degradation rather than absolute preservation.

What Lies Ahead?

This work, like all logging of quantum states, is a chronicle of a fleeting moment. The demonstrated conditions for secure continuous-variable quantum teleportation, while promising, exist within a landscape defined by inevitable decay. Channel losses and noise aren’t mere technical hurdles; they represent the system’s fundamental interaction with time-a constant erosion of initial conditions. The truncation of Gaussian states, a pragmatic concession to finite energy, introduces a subtle but critical distortion – a deliberate forgetting of higher-order correlations.

Future investigations will likely focus on extending the lifespan of these ephemeral states. Not necessarily through brute-force error correction-a Sisyphean task-but by embracing the inherent limitations. Perhaps the true metric isn’t fidelity of replication, but rather the graceful degradation of information. The development of codebooks tailored to specific noise profiles, anticipating the system’s decline, could yield more robust and practical communication protocols.

Deployment of such systems marks a point on the timeline, not an arrival. The question isn’t whether these quantum links will eventually fail – they will – but rather how elegantly they do so. The pursuit of perfect teleportation is a phantom; the challenge lies in engineering resilience in the face of entropy, accepting that all systems, even those leveraging the strangeness of quantum mechanics, are ultimately subject to the relentless march of time.

Original article: https://arxiv.org/pdf/2512.23388.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/