Partially smoothed information measures

Smooth entropies are a tool for quantifying resource trade-offs in (quantum) information theory and cryptography.

In typical bi- and multi-partite problems, however, some of the sub-systems are often left unchanged and this is not reflected by the standard smoothing of information measures over a ball of close states.

We propose to smooth instead only over a ball of close states which also have some of the reduced states on the relevant sub-systems fixed.

This partial smoothing of information measures naturally allows to give more refined characterizations of various information-theoretic problems in the one-shot setting.

In particular, we immediately get asymptotic second-order characterizations for tasks such as privacy amplification against classical side information or classical state splitting.

For quantum problems like state merging the general resource trade-off is tightly characterized by partially smoothed information measures as well.