Decoy-state quantum key distribution with a leaky Mach-Zehnder receiver

Abstract

We prove the security of phase-encoded decoy-state BB84 quantum key distribution for a Mach-Zehnder receiver with imbalanced beam splitters and two threshold detectors that leak fine-grained information about the detection patterns. For this, we derive a lower bound on Bob's single-photon contributions and combine it with the decoy-state method and a recent phase-error framework for leaky receivers. The leakage-induced correction to the phase-error rate is independent of channel loss, providing a loss-tolerant treatment of receiver leakage even in the presence of beam-splitter imbalance. Our results provide a practical route toward loss-tolerant security proofs for imperfect QKD receivers in the presence of receiver leakage.