Quantum Cryptography with Finite Resources: Unconditional Security Bound for Discrete-Variable Protocols with One-Way Postprocessing

We derive a bound for the security of quantum key distribution with finite resources under one-way postprocessing, based on a definition of security that is composable and has an operational meaning. While our proof relies on the assumption of collective attacks, unconditional security follows immediately for standard protocols such as Bennett-Brassard 1984 and six-states protocol. For single-qubit implementations of such protocols, we find that the secret key rate becomes positive when at least $N\sim {10}^5$ signals are exchanged and processed. For any other discrete-variable protocol, unconditional security can be obtained using the exponential de Finetti theorem, but the additional overhead leads to very pessimistic estimates.

Figure 1

Lower bound for the key rate $r$ as a function of the number of exchanged quantum signals $N$, for the BB84 (full lines) and the six-states protocol (dashed lines); values: $\epsilon ={10}^{-5}$, ${\epsilon }_{\mathrm{EC}}={10}^{-10}$, ${\mathrm{\text{leak}}}_{\mathrm{EC}}/n=1.2h(Q)$, and several $Q=e_0=e_1$.