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.
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.
Extension of the Physical Bridge from closed to open quantum systems governed by Lindblad master equations. The proportionality C_meas(τ) ∝ C_op^open(τ) is proven to hold under arbitrary Markovian dissipation and verified numerically (Pearson r > 0.99) across three noise channels (dephasing, amplitude damping, depolarizing), two Hamiltonian families (Heisenberg, SYK₄), and dissipation strengths spanning two orders of magnitude. A key new result is that the Krylov framework provides qualitatively distinct signatures for coherent errors (Lanczos coefficient shifts) versus decoherence (autocorrelation envelope damping), enabling error-type discrimination for quantum computer benchmarking. Part 12 of the Krylov QKD research series.
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 QBER ( τ ) = α ( N ) C op ( τ ) , with 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 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 ΔQBER=+0.5% Δ QBER = + 0.5 % ( 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.
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 review examines the preprint by Daniel Süß titled "Quantum Scrambling as a Cryptographic Resource: From Krylov Complexity to Physical Channel Authentication," which proposes quantum scrambling in many-body quantum channels as a novel cryptographic primitive for channel authentication in quantum key distribution (QKD). Key postulates include the irreversibility of scrambling dynamics, the role of Lanczos coefficients in determining channel autocorrelation signatures, and the practical unforgeability protected by doubly exponential Poincaré recurrence times. Hypotheses center on the deterministic, measurable, and secure nature of scrambling-derived fingerprints, validated through simulations and real astrophysical data from the NANOGrav dataset. Results demonstrate high detection accuracy (AUC = 0.99) under realistic noise. Similar research in quantum chaos, Krylov complexity, and QKD security is surveyed, highlighting ongoing trends toward physics-based cryptography amid quantum threats. Algebraic interpretations frame Krylov subspaces as orthogonal bases for operator evolution, while geometric views depict scrambling as diffusive spreading in operator space. The field is advancing toward hybrid quantum-classical protocols resilient to quantum attacks, potentially revolutionizing secure communications. Mamun, SM (2026)……………………………………1 A Comprehensive Review of Quantum Scrambling as a Cryptographic Resource in Quantum Key Distribution Also available on Academia.edu: https://www.academia.edu/attachments/132019982/download_file?s=portfolio
Abstract In a quantum many-body system, autocorrelation functions can determine linear responses nearby equilibrium and quantum dynamics far from equilibrium. In this letter, we bring out the connection between the operator complexity and the autocorrelation function. In particular, we focus on a particular kind of operator complexity called the Krylov complexity. We find that a set of Lanczos coefficients {bn} { b n } computed for determining the Krylov complexity can reveal the universal behaviors of autocorrelations, which are otherwise impossible. When the time axis is scaled by b1 b 1 , different autocorrelation functions obey a universal function form at short time. We further propose a characteristic parameter deduced from {bn} { b n } that can largely determine the behavior of autocorrelations at the intermediate time. This parameter can also largely determine whether the autocorrelation function oscillates or monotonically decays in time. We present numerical evidences and physical intuitions to support these universal hypotheses of autocorrelations. We emphasize that these universal behaviors are held across different operators and different physical systems.