Imperfect preparation and Trojan attack on the phase modulator in the decoy-state BB84 protocol

Quantum key distribution (QKD) provides a theoretically secure method for cryptographic key exchange by leveraging quantum mechanics, but practical implementations face vulnerabilities such as Trojan horse attack on phase modulators. This work analyzes the security of QKD systems under such attacks, considering both ideal and imperfect state preparation scenarios. The Trojan attack model is generalized to arbitrary states of probing pulses and conservative bounds of information leakage through side-channel of special form are introduced. The quantum coin imbalance, a critical security parameter, remains low (on the order of ${10}^{-7}$ for ideal state preparation and ${10}^{-5}$ for imperfect preparation) with this new approach and presence additional hardware passive countermeasures. Numerical simulations confirm nonzero secure key rate at distances over 100 km through optical fiber channel.