Privacy Amplification and Decoupling Without Smoothing
Frédéric Dupuis
Abstract
We prove an achievability result for privacy amplification and decoupling in terms of the sandwiched Rényi entropy of order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha \in (1,2]$ </tex-math></inline-formula> ; this extends previous results which worked for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha =2$ </tex-math></inline-formula> . The fact that this proof works for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> .