Finite-Size Security Bounds in Semi-Quantum Key Distribution: Spectral, Operator-Theoretic, and Entropic Perspectives

We study Semi-Quantum Key Distribution (SQKD) with a focus on finite-size security bounds, developed through three complementary perspectives. (i) Spectral disturbance: wrong-basis L\"uders updates produce closed-form spectra and purity loss, which serve as basis-independent indicators of disturbance. (ii) Operator-theoretic reduction: in Z/Z-sifted rounds, intercept-resend attacks can be represented as an effective depolarizing channel, characterized by a fidelity-QBER relation. (iii) Entropic trade-offs: Maassen-Uffink and memory-assisted uncertainty relations certify security through X tests and reflection rounds, even when the sifted QBER is low. The exposition provides step-by-step derivations supported by physically interpretable figures, and the framework concludes with finite-size estimates based on concentration inequalities that are suited for practical parameter estimation.