The finite key effect of side-channel-secure quantum key distribution beyond post-selection technique

Abstract

By applying the framework of entropic uncertainty relation (EUR) and the Quantum Leftover Hash Lemma (QLHL), we introduce a security-proof method for variable-length side-channel-secure (SCS) quantum key distribution (QKD) against coherent attacks. This method reframes composable security as a statistical fluctuation problem of phase errors, enabling direct proofs against coherent attacks through observables and virtual observables. It yields tight key rates for the SCS protocol and reduces pulse requirements by over two orders of magnitude compared to prior works that employ the post-selection technique. {Utilizing the characteristic of the SCS protocol that it has no untagged-bit bit-flip errors, we prove that the secure key length for the SCS protocol can be determined after Cascade error correction, using its actual parity disclosures. We also explain how the same argument extends to protocols with compatible bit-error-pattern and phase-error measurements.} We further identify sufficient conditions under which the final key length may be determined after error correction in a broader class of QKD protocols. Under the framework of EUR and QLHL, we clarify the applicability of several commonly used concentration bounds to variable-length QKD and the appropriate manner of their implementation. This work enhances the practical value of the SCS protocol and clarifies the security justification of key-rate formulas used in practical variable-length QKD implementations.