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

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, $C_{\mathrm{QBER}}(\tau )=\alpha (N)C_{\mathrm{op}}(\tau )$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>QBER</mml:mi> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>α</mml:mi> <mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>op</mml:mi> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>)</mml:mo> </mml:math> , with $\alpha (N)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> <mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> </mml:math> 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 ( $\mathrm{AUC}=0.9899$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>AUC</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.9899</mml:mn> </mml:math> ). On $181{,}606$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>181</mml:mn> <mml:mo>,</mml:mo> <mml:mn>606</mml:mn> </mml:math> 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 $\Delta \mathrm{QBER}={+}0.5\%$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Δ</mml:mi> <mml:mi>QBER</mml:mi> <mml:mo>=</mml:mo> <mml:mo>+</mml:mo> <mml:mn>0.5</mml:mn> <mml:mi>%</mml:mi> </mml:math> ( $\mathrm{AUC}=0.981$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>AUC</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.981</mml:mn> </mml:math> ). Certification of final key secrecy remains governed by the underlying QKD protocol, parameter estimation, finite-key analysis, and privacy amplification.