Min-entropy uncertainty relation for finite-size cryptography
Nelly H. Y. Ng, Mario Berta, Stephanie Wehner · Physical Review A · 2012
Apart from their foundational significance, entropic uncertainty relations play a central role in proving the security of quantum cryptographic protocols. Of particular interest are therefore relations in terms of the smooth min-entropy for Bennett-Brassard 1984 (BB84) and six-state encodings. The smooth min-entropy ${H}_{\mathrm{min}}^{\ensuremath{\epsilon}}(X/B)$ quantifies the negative logarithm of the probability for an attacker $B$ to guess $X$, except with a small failure probability $\ensuremath{\epsilon}$. Previously, strong uncertainty relations were obtained which are valid in the limit of large block lengths. Here, we prove an alternative uncertainty relation in terms of the smooth min-entropy that is only marginally less strong but has the crucial property that it can be applied to rather small block lengths. This paves the way for a practical implementation of many cryptographic protocols. As part of our proof we show tight uncertainty relations for a family of R\'enyi entropies that may be of independent interest.