Privacy Amplification and Decoupling Without Smoothing

We prove an achievability result for privacy amplification and decoupling in terms of the sandwiched Rényi entropy of order
$\alpha \in (1,2]$
; this extends previous results which worked for
$\alpha =2$
. The fact that this proof works for
$\alpha $
close to 1 means that we can bypass the smooth min-entropy in the many applications where the bound comes from the fully quantum AEP or entropy accumulation, and carry out the whole proof using the Rényi entropy, thereby easily obtaining an error exponent for the final task. This effectively replaces smoothing, which is a difficult high-dimensional optimization problem, by an optimization problem over a single real parameter
$\alpha $
.