From uniformity to fluctuations: Quantifying the security cost of imperfect randomness in quantum key distribution
Phys. Rev. A 113, 032612 – Published 11 March, 2026
DOI: https://doi.org/10.1103/nqpb-8h13
Abstract
In this work, we examine the impact of imperfect randomness on the security of quantum key distribution (QKD) protocols with a set of linearly independent states, with particular emphasis on deviations from uniform a priori probabilities in quantum state preparation. Both stationary and nonstationary forms of randomness imperfections are considered. In the stationary case—where the preparation probabilities remain constant throughout the entire sequence—we demonstrate that increasing asymmetry monotonically decreases the source entropy and the Holevo information, thereby reducing the conditional entropy and the extractable secret key length. We also explicitly quantify the impact of preparation bias on the achievable secret key rate in protocols based on linearly independent states. In contrast, the non–independent and identically distributed regime, characterized by signalwise fluctuations in the a priori probabilities, is shown to yield a reduction in conditional entropy proportional to the second moment of the deviation distribution, assuming small perturbations. We further analyze several representative classes of biased distributions, establishing upper and lower bounds on the corresponding conditional entropies. These results highlight the necessity of rigorously accounting for randomness imperfections in practical QKD systems and suggest concrete strategies for adjusting key length estimation under realistic, nonideal conditions.
