Preprocessing noise in finite-size quantum key distribution

It is known that preprocessing noise may boost quantum key distribution by expanding the range of values of tolerated noise. For BB84, adding trusted noise may allow the generation of secret keys even for qubit error rate (QBER) beyond the 11% threshold in the asymptotic regime. Here we study the effect of preprocessing noise in the finite-size regime where only a limited number of signals are exchanged between Alice and Bob. We compute tight numerical lower bounds in terms of the sandwiched Rényi entropy of order alpha, optimized via a two-step Frank-Wolfe algorithm, in the presence of a trusted flipping probability q. We find that trusted noise improves the key rate only for a finite interval of alpha, from the alpha -> 1 limit up to alpha approx 1.4. By optimizing on the value of alpha, we determine finite-size key rates for different values of the QBER, observing enhancement due to trusted noise both in asymptotic and finite-size regimes. Finally, we determine the maximum tolerable QBER as a function of the block size.