A Formal Approach to Unconditional Security Proofs for Quantum Key Distribution

We present an approach to automate Shor-Preskill style unconditional security proof of QKDs. In Shor-Preskill’s proof, the target QKD, BB84, is transformed into another QKD based on an entanglement distillation protocol (EDP), which is more feasible for direct analysis. We formalized heir method as program transformation in a quantum programming language, QPL. The transform is defined as rewriting rules which are sound with respect to the security in the semantics of QPL. We proved that rewriting always terminates for any program and that the normal form is unique under appropriate conditions. By applying the rewriting rules to the program representing BB84, we can obtain the corresponding EDP-based protocol automatically. We finally proved the security of the obtained EDP-based protocol formally in the quantum Hoare logic, which is a system for formal verification of quantum programs. We show also that this method can be applied to B92 by a simple modification.