A min-entropy uncertainty relation for finite size cryptography

TL;DR

This paper introduces a new min-entropy uncertainty relation applicable to small block sizes, enhancing the practicality of quantum cryptography protocols by enabling security proofs with finite resources.

Contribution

It presents a novel uncertainty relation in terms of smooth min-entropy that is effective for small block lengths, unlike previous relations requiring large blocks.

Findings

Established tight uncertainty relations for Renyi entropies.

Demonstrated the relation's applicability to small block lengths.

Facilitated practical quantum cryptography implementations.

Abstract

Apart from their foundational significance, entropic uncertainty relations play a central role in proving the security of quantum cryptographic protocols. Of particular interest are thereby relations in terms of the smooth min-entropy for BB84 and six-state encodings. Previously, strong uncertainty relations were obtained which are valid in the limit of large block lengths. Here, we prove a new 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 Renyi entropies that may be of independent interest.