The Smooth Entropy Formalism for von Neumann Algebras

TL;DR

This paper extends the smooth entropy formalism to infinite-dimensional quantum systems modeled by von Neumann algebras, maintaining key properties and operational interpretations, with applications to quantum cryptography.

Contribution

It generalizes the smooth entropy framework from finite-dimensional systems to von Neumann algebras, enabling new analysis of quantum cryptographic protocols in infinite dimensions.

Findings

Recovery of characterizing properties of smooth min- and max-entropy in von Neumann algebras

Generalization of entropic uncertainty relations with quantum side information

Feasibility of privacy amplification and data compression with infinite-dimensional quantum side information

Abstract

We discuss information-theoretic concepts on infinite-dimensional quantum systems. In particular, we lift the smooth entropy formalism as introduced by Renner and collaborators for finite-dimensional systems to von Neumann algebras. For the smooth conditional min- and max-entropy we recover similar characterizing properties and information-theoretic operational interpretations as in the finite-dimensional case. We generalize the entropic uncertainty relation with quantum side information of Tomamichel and Renner and discuss applications to quantum cryptography. In particular, we prove the possibility to perform privacy amplification and classical data compression with quantum side information modeled by a von Neumann algebra.