University of Vigo and University of Toyama

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Correlations between consecutive pulses in quantum key distribution (QKD) systems, introduced by practical modulator limitations, pose a significant threat to established security proofs. The research team has developed a new mathematical framework to address this critical issue. Their work extends existing phase‑error‑estimation‑based security proofs to encompass encoding correlations, offering a more robust analysis for QKD systems utilising correlated sources. This framework overcomes limitations in previous approaches, bringing theoretical security guarantees closer to the realities of implementing QKD technology and bolstering confidence in its practical application. It represents a substantial step towards securing QKD against imperfections present in real‑world devices.

The team’s framework extends existing phase‑error‑estimation‑based security proofs to encompass sources exhibiting these correlations—a problem that has long hindered the translation of theoretical security into real‑world QKD systems. This breakthrough offers both greater generality and increased rigor in security assessments. The study rigorously establishes that phase‑error‑rate bounds, traditionally calculated for independent events, can be directly combined to provide an overall bound for the entire sifted key, even when encoding correlations are present.

This allows for a streamlined security analysis, eliminating the need for separate privacy‑amplification steps for individual protocol partitions and resolving concerns regarding the composability of security proofs. Experiments show the framework successfully incorporates unbounded correlations directly within the phase‑error proof, accounting for long‑range dependencies through a minor adjustment to the failure probability of the phase‑error‑rate bound. By constructing a general mathematical framework, the scientists provide a versatile tool applicable across various QKD implementations.

The team focuses on prepare‑and‑measure protocols, where Alice sends states to Bob, but confirms the framework’s adaptability to interference‑based protocols involving an untrusted middle node. The research details a source‑replacement scheme considering a general QKD protocol where Alice selects settings in each round and emits a state dependent on previous choices. This framework accounts for the impact of correlations, where the photonic system in round k depends on the setting from round k‑1, and the setting in round k influences the encoding of the pulse in round k+1. By treating these effects as encoding flaws and side‑channel leakage, the team establishes a robust method for securing QKD systems against realistic imperfections and paving the way for more secure communication networks.

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Correlated Sources Enhance QKD Security Proofs

The research addresses limitations in quantum key distribution (QKD) security proofs when practical modulators introduce correlations between emitted pulses. Scientists developed a mathematical framework to extend phase‑error‑estimation‑based security proofs to encompass imperfect, correlated sources, significantly improving the alignment between theoretical security and real‑world implementations. This work overcomes previous approaches that required separate privacy amplification for sub‑keys, increasing complexity and potential failure points. The study pioneered a method for directly combining phase‑error‑rate bounds for individual key partitions into a single bound for the full sifted key.

Experiments employ a prepare‑and‑measure protocol where Alice prepares and sends states to Bob, represented by the global source‑replacement state (|\Psi_N\rangle_{A N 1 T N 1}). This state accounts for correlations arising from the history of previous settings, influencing the current state. Uncorrelated sources are treated as a special case where this state simplifies, allowing for factorization. Crucially, the team engineered a phase‑error estimation technique that defines a scenario where Alice and Bob pre‑determine key‑generating rounds before extracting key bits. Alice measures qubits in the computational basis, while Bob utilizes a two‑outcome positive‑operator‑valued measure (POVM) on his sifted‑key systems.

The phase‑error rate is then defined as the fraction of incorrect predictions, with the goal of proving that the probability of exceeding a certain error rate is less than a defined failure probability. This approach directly incorporates unbounded correlations within the phase‑error proof, accounting for long‑range correlations through a slight increase in the failure probability. The framework’s adaptability extends beyond prepare‑and‑measure protocols to interference‑based QKD systems, demonstrating its broad applicability.

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Correlated Pulse Sources Enhance QKD Security

Scientists have developed a mathematical framework to extend security proofs for quantum key distribution (QKD) protocols to encompass sources exhibiting correlations between consecutively emitted pulses. This work addresses a critical limitation in existing security analyses, which typically assume uncorrelated pulses—a condition often violated by practical modulator devices. The research demonstrates that the fundamental security framework, deriving secure key lengths from phase‑error‑rate bounds, naturally applies even when sources are correlated. Experiments revealed a method to extend phase‑error‑rate upper bounds from uncorrelated to correlated sources, significantly narrowing the gap between theoretical security guarantees and real‑world implementations.

The team established that as long as the family of multi‑round correlated states satisfies the single‑round conditions of the original uncorrelated proof, the protocol rounds can be partitioned and analyzed individually. Results demonstrate that the overall phase‑error rate can be bounded by a weighted average of the phase‑error rates of each partition, allowing for the determination of key length and secrecy parameters. Measurements confirm that if correlations extend up to length (l_c), the phase‑error rate in the correlated scenario satisfies a specific inequality, providing a quantifiable upper bound crucial for establishing secure key generation. Further tests prove the framework’s adaptability by extending security proofs that incorporate fidelity bounds to reference states to also include encoding correlations. Specifically, the research shows that by identifying appropriate reference states and accounting for residual contributions from past settings, the phase‑error rate bound remains valid even with correlated sources. Considering a linear time‑invariant (LTI) phase modulator, scientists obtained an exponential bound on the correlation strength.

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Correlated Pulse Sources, Enhanced QKD Security Proofs

Researchers have developed a novel mathematical framework to rigorously assess the security of quantum key distribution (QKD) protocols when encoding correlations exist between consecutively emitted pulses. This work directly extends existing security proofs, traditionally based on phase‑error estimation, to encompass scenarios with imperfect, yet correlated, sources—a crucial step towards bridging the gap between theoretical security and practical implementations. The framework overcomes limitations found in previous approaches by eliminating the need for multiple privacy‑amplification steps and addressing concerns regarding composability. The established framework handles unbounded correlation lengths systematically and can be applied to any existing phase‑error‑estimation‑based analysis, provided certain admissibility conditions on the emitted states are met.

This advancement is significant because practical modulators inevitably introduce these correlations due to bandwidth constraints, which previously complicated security analyses. The authors acknowledge a limitation in that the framework relies on appropriate admissibility conditions for the emitted states, a standard requirement in QKD security proofs. Future research may explore extending the framework to other areas involving temporally correlated sources beyond QKD, potentially broadening its applicability. Their research extends existing phase‑error‑estimation‑based security proofs to encompass encoding correlations, offering a more robust analysis for QKD systems utilising correlated sources. This work overcomes limitations in previous approaches, bringing theoretical security guarantees closer to the realities of implementing QKD technology and bolstering confidence in its practical application. The framework represents a substantial step towards securing QKD against imperfections present in real implementations.

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Reference

Rigorous phase‑error‑estimation security framework for QKD with correlated sources – arXiv: 2601.08417

https://arxiv.org/abs/2601.08417