Counterfactual quantum cryptography based on weak coherent states

Abstract

In the ``counterfactual quantum cryptography'' scheme [T.-G. Noh, Phys. Rev. Lett. 103, 230501 (2009)], two legitimate distant peers may share secret-key bits even when the information carriers do not travel in the quantum channel. The security of this protocol with an ideal single-photon source has been proved by Yin et al. [Z.-Q. Yin, H. W. Li, W. Chen, Z. F. Han, and G. C. Guo, Phys. Rev. A 82, 042335 (2010)]. In this paper, we prove the security of the counterfactual-quantum-cryptography scheme based on a commonly used weak-coherent-laser source by considering a general collective attack. The basic assumption of this proof is that the efficiency and dark-counting rate of a single-photon detector are consistent for any $n$-photon Fock states. Then through randomizing the phases of the encoding weak coherent states, Eve's ancilla will be transformed into a classical mixture. Finally, the lower bound of the secret-key-bit rate and a performance analysis for the practical implementation are both given.