Security-proof framework for two-way Gaussian quantum-key-distribution protocols

Two-way Gaussian protocols have the potential to increase quantum key distribution (QKD) protocols' secret-key rates by orders of magnitudes [Phys. Rev. A 94, 012322 (2016)2469-992610.1103/PhysRevA.94.012322]. Security proofs for two-way protocols, however, are underdeveloped at present. In this paper, we establish a security proof framework for the general coherent attack on two-way Gaussian protocols in the asymptotic regime. We first prove that coherent-attack security can be reduced to collective-attack security for all two-way QKD protocols. Next, we identify two different constraints that each provide intrusion parameters which bound an eavesdropper's coherent-attack information gain for any two-way Gaussian QKD protocol. Finally, we apply our results to two such protocols.