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EPJ Quantum Technology

Krylov complexity as a cryptographic resource: eavesdropper detection in quantum key distribution via operator-growth channel fingerprints

Daniel Süß

Published Jul 22, 2026• 0 citations
Publisher

Abstract

Abstract I present a physical-layer monitoring framework for quantum key distribution (QKD) implementations based on Krylov/operator-growth diagnostics. Running alongside standard parameter estimation, privacy amplification, and composable security proofs, it addresses a focused question: whether the temporal correlation structure of an observed quantum bit error rate (QBER) stream carries additional information about physical channel perturbations. Within a specified local-Hamiltonian channel model, the Lanczos coefficients of the channel Liouvillian define an operator-growth autocorrelation template, and deviations from it serve as anomaly scores for implementation monitoring. For the model studied here, the QBER autocorrelation relates to the operator autocorrelation through a proportionality relation, $C_{\mathrm{QBER}}(\tau )=\alpha (N)C_{\mathrm{op}}(\tau )$ C QBER ( τ ) = α ( N ) C op ( τ ) , with $\alpha (N)$ α ( N ) an explicitly specified calibration factor. Local Hamiltonian perturbations generically distort the associated Krylov fingerprint once the perturbation reaches the support of the monitored observable, consistent with the locality of the Lanczos recursion and with Lieb–Robinson bounds. The resulting three-layer pipeline combines adaptive frequency filtering, Krylov-template extraction, and calibrated slope detection. On simulated BB84-like QBER data with realistic hardware noise, the detector achieves high anomaly discrimination under the tested perturbation classes ( $\mathrm{AUC}=0.9899$ AUC = 0.9899 ). On $181{,}606$ 181 , 606 public QBER measurements from a deployed fiber-optic QKD system, it yields a 4.5% clean-test alarm rate against a 5% design target and detects injected perturbations with 96.0% sensitivity at $\Delta \mathrm{QBER}={+}0.5\%$ Δ QBER = + 0.5 % ( $\mathrm{AUC}=0.981$ AUC = 0.981 ). Certification of final key secrecy remains governed by the underlying QKD protocol, parameter estimation, finite-key analysis, and privacy amplification.

DOI
10.1140/epjqt/s40507-026-00541-8

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Abstract I present a physical-layer monitoring framework for quantum key distribution (QKD) implementations based on Krylov/operator-growth diagnostics. Running alongside standard parameter estimation, privacy amplification, and composable security proofs, it addresses a focused question: whether the temporal correlation structure of an observed quantum bit error rate (QBER) stream carries additional information about physical channel perturbations. Within a specified local-Hamiltonian channel model, the Lanczos coefficients of the channel Liouvillian define an operator-growth autocorrelation template, and deviations from it serve as anomaly scores for implementation monitoring. For the model studied here, the QBER autocorrelation relates to the operator autocorrelation through a proportionality relation, CQBER(τ)=α(N)Cop(τ)C_{\mathrm{QBER}}(\tau )=\alpha (N)C_{\mathrm{op}}(\tau )CQBER​(τ)=α(N)Cop​(τ) C QBER ( τ ) = α ( N ) C op ( τ ) , with α(N)\alpha (N)α(N) α ( N ) an explicitly specified calibration factor. Local Hamiltonian perturbations generically distort the associated Krylov fingerprint once the perturbation reaches the support of the monitored observable, consistent with the locality of the Lanczos recursion and with Lieb–Robinson bounds. The resulting three-layer pipeline combines adaptive frequency filtering, Krylov-template extraction, and calibrated slope detection. On simulated BB84-like QBER data with realistic hardware noise, the detector achieves high anomaly discrimination under the tested perturbation classes ( AUC=0.9899\mathrm{AUC}=0.9899AUC=0.9899 AUC = 0.9899 ). On 181,606181{,}606181,606 181 , 606 public QBER measurements from a deployed fiber-optic QKD system, it yields a 4.5% clean-test alarm rate against a 5% design target and detects injected perturbations with 96.0% sensitivity at ΔQBER=+0.5%\Delta \mathrm{QBER}={+}0.5\%ΔQBER=+0.5% Δ QBER = + 0.5 % ( AUC=0.981\mathrm{AUC}=0.981AUC=0.981 AUC = 0.981 ). Certification of final key secrecy remains governed by the underlying QKD protocol, parameter estimation, finite-key analysis, and privacy amplification.
This collection presents three interconnected papers on eavesdropper (Eve) detection in Quantum Key Distribution (QKD) channels, following the chronological development of the framework. Part I Eve Detection in QKD via Dual-Layer Sidereal Filtering and Krylov Complexity Analysis (v4): The empirical starting point. Eve is modeled as a Hamiltonian perturbation at the kappa-coupling interface. Krylov-Statistical Mapping establishes a correlation between b_n deviation and QBER statistical moments (r=0.948, kurtosis; r=0.943, |skewness|) across 50 gamma values. Crossover spectral regime (⟨r⟩=0.366) explicitly documented. Part II: Distinguishing Quantum Scrambling from Eavesdropping via Krylov Temporal Structure and Sidereal Filtering (v2, with Robustness Analysis): The theoretical foundation. A Gaussian autocorrelation template C(t) = exp(−0.5·(slope·t)²) is derived from Krylov-Lanczos b_n coefficients. Under realistic hardware noise (AR(1) + 1/f + afterpulsing + bursts): AUC=0.9899, separation=12.13 sigma. Three methodological killer tests included. All results are simulation-based (N=8 crossover regime; not yet validated on real QKD hardware). Part III: QKD Eve Detector: A Unified Framework (Synthesis): The three-layer detector combining all prior work: (1) Dual-Layer Sidereal Filter for environmental noise removal, (2) Krylov Dynamic Detector for template matching, (3) Multi-Vector Attack Classifier distinguishing six attack types. Self-contained implementation provided in the accompanying demo notebook.Accompanying notebooks reproduce all figures and results. Dependencies: NumPy, SciPy, QuTiP. No special hardware required.
We develop a rigorous mathematical framework proving that the QKD Krylov Detector, a three-layer eavesdropper detection system for Quantum Key Distribution based on Krylov complexity and sidereal filtering, provides information-theoretically grounded security against physically realizable attacks. The central results are: (1) Krylov Locality Theorem: An eavesdropper coupling at distance d from the monitored operator cannot affect the first d Lanczos coefficients. This is an exact algebraic identity, not an approximation. (2) Perturbation Detectability Theorem: The first affected Lanczos coefficient scales as gamma^2, with computable upper and lower bound constants derived from second-order perturbation theory applied to the Lanczos recursion. (3) Information-Distortion Bound: Eve's accessible information (Holevo quantity) scales as O(gamma^2), while the Krylov distortion scales as O(gamma^4), yielding an explicit tradeoff chi <= const * sqrt(Delta_K). Verified numerically using an explicit Eve ancilla model. (4) Krylov Unforgeability Theorem: Any Hamiltonian perturbation that extracts nonzero information from the quantum channel necessarily distorts the Lanczos coefficients beyond the detection threshold. No physically realizable attack can simultaneously extract information and preserve the clean Krylov signature. (5) Coherent Attack Invariance: The Lanczos coefficients are operator-space quantities independent of any quantum state, rendering coherent and entangled attack strategies unable to circumvent detection. (6) Time-Dependent Unforgeability: Time-dependent coupling strategies gamma(t) produce strictly more distortion than constant coupling at the same average strength, by Jensen's inequality applied via a rigorous stroboscopic argument. All results include full proofs and numerical verification for chain lengths N = 6, 8, 10, 12. This work provides the formal security foundation for the QKD Krylov Detector framework developed across six prior publications. Companion code: https://github.com/quantumspiritresearch-crypto/qkd-krylov-detector
This paper closes three theoretical gaps that remained open in the Krylov-based quantum channel authentication framework developed across Papers 1-8 of this series. Physical Bridge (Theorem 1): I derive a formal proof that the QBER autocorrelation is proportional to the operator autocorrelation, C_QBER(τ) = α(N) · C_op(τ), establishing the connection between Krylov theory and observable measurement statistics (numerically confirmed with Pearson r = 0.9997). Physical One-Way Function (Theorem 2): I prove that the Krylov fingerprint map constitutes a physical one-way function with three independent hardness sources: (I) unconditional information-theoretic exponential ill-conditioning of the inverse moment problem (κ_n ~ 32^n), (II) complexity-theoretic reduction to Hamiltonian learning, and (III) physical irreversibility of scrambling dynamics (t_rec ~ 10^765). Hardness source I is unconditional and provides post-quantum security. Universality (Theorem 3): I establish the universality of the detection framework through systematic numerical verification across 8 Hamiltonian families and 10 perturbation types, achieving 8/8 families and 10/10 perturbation types at the physically relevant distance d = 1. All three results are detector-agnostic: they characterize the Krylov framework itself, independent of the specific detection algorithm applied to the Lanczos coefficients. Paper 9 in the Krylov QKD series. See also: qkd-krylov-detector on GitHub.
We propose that quantum scrambling in many-body quantum channels constitutes a cryptographic resource that enables a fundamentally new class of channel authentication protocols. By synthesizing results from three independent lines of investigation, scrambling irreversibility in closed quantum systems, Krylov complexity-based channel detection, and sidereal filtering validated on real astrophysical data, we establish the following chain of reasoning. First, scrambling in symmetry-broken Heisenberg chains produces a practically irreversible dynamical process, with Poincaré recurrence times scaling as τ_rec ~ (1/ε)^{2^N - 1}. Second, the Lanczos coefficients b_n of the channel Hamiltonian determine the functional form of the channel's temporal autocorrelation through the dynamical susceptibility, yielding a Gaussian template for linearly growing b_n. Third, in simulated experiments using template-matching classifiers applied to scrambling-derived correlation traces, compromised channels can be distinguished from clean channels with an area-under-curve (AUC) value of 0.99 under controlled realistic noise conditions. Fourth, as a proof-of-concept for extracting stable temporal fingerprints from noisy real-world signals, the sidereal filtering layer has been validated on 59,389 pulsar time-of-arrival measurements from the NANOGrav 15-year dataset. We argue that the combination of these results implies that scrambling defines a physical one-way signature: a measurable, predictable, and practically unforgeable channel fingerprint whose security rests on quantum-mechanical irreversibility rather than computational hardness assumptions. This work synthesizes and extends the author's prior publications on sidereal filtering, scrambling dynamics, and the QKD Eve Detector framework, connecting them into a unified cryptographic paradigm.
Quantum Key Distribution (QKD) provides information-theoretic security by exploiting the principles of quantum mechanics. Among QKD protocols, the BB84 scheme remains the most widely adopted for both theoretical research and practical implementation. A critical parameter determining the reliability and security of BB84 is the Quantum Bit Error Rate (QBER), which quantifies errors in the sifted key arising from channel noise or potential eavesdropping. This paper presents a systematic review and analysis of QBER within the BB84 protocol, examining its calculation, statistical estimation methods, and role in detecting eavesdropping activity. Simulation results, corroborated by reported experimental observations, reveal a near-linear relationship between eavesdropping intensity and QBER, with values approaching 25% under full intercept-resend attacks. Four confidence interval estimation methods, Wald, Wilson, Clopper-Pearson, and Hoeffding's inequality, are compared for robust QBER analysis in finite-key scenarios. Protocol enhancements, including decoy-state methods, hybrid cryptographic models, and quantum-resistant authentication, are discussed as mechanisms to mitigate errors and strengthen resilience across fiber, free-space, underwater, and satellite QKD systems. Open challenges in distinguishing noise-induced errors from malicious eavesdropping, and the role of adaptive error correction and machine-learning-assisted QBER estimation in future quantum networks, are identified as key directions for further research.
Paper [8] in the QKD Krylov Detector research series. We validate the Calibrated Slope Detector on 181,606 experimental QBER measurements from a real fiber-optic QKD system. The detector achieves a false-positive rate of 4.5% and detects simulated intercept-resend attacks from ΔQBER ≥ 0.5% with 96.0% sensitivity (AUC = 0.981) — an order of magnitude more sensitive than conventional QBER threshold monitoring. Supplementary validation on the NIST Loophole-Free Bell Test dataset (152.6 million trials) confirms specificity on clean quantum data. This completes the validation triangle: theoretical unforgeability (Paper [7]), simulation-based sensitivity (Papers [5]–[6]), and real-data validation (this work). Software: https://github.com/quantumspiritresearch-crypto/qkd-krylov-detector (v1.3.0)