Security proof for quantum key recycling with noise

Quantum Key Recycling aims to re-use the keys employed in quantum encryption and quantum authentication schemes. QKR protocols can achieve better round complexity than Quantum Key Distribution. We consider a QKR protocol that works with qubits, as opposed to high-dimensional qudits. A security proof was given by Fehr and Salvail [1] in the case where there is practically no noise. A high-rate scheme for the noisy case was proposed by Škorić and de Vries [2], based on eight-state encoding. However, a security proof was not given. In this paper we introduce a protocol modification to [2]. We provide a security proof. The modified protocol has high rate not only for 8-state encoding, but also 6-state and BB84 encoding. Our proof is based on a bound on the trace distance between the real quantum state of the system and a state in which the keys are completely secure. It turns out that the rate is higher than suggested by previous results. Asymptotically the rate equals the rate of Quantum Key Distribution with one-way postprocessing.