Min- and Max-Entropy in Infinite Dimensions

TL;DR

This paper extends the concepts of min- and max-entropy to infinite-dimensional quantum systems, preserving key properties and operational meanings, and establishes an asymptotic equipartition property in this broader context.

Contribution

It introduces infinite-dimensional min- and max-entropies, proves their key properties, and generalizes the asymptotic equipartition property to infinite-dimensional quantum systems.

Findings

Min- and max-entropies retain their properties in infinite dimensions.

Operational interpretations of these entropies are preserved.

An infinite-dimensional quantum asymptotic equipartition property is established.

Abstract

We consider an extension of the conditional min- and max-entropies to infinite-dimensional separable Hilbert spaces. We show that these satisfy characterizing properties known from the finite-dimensional case, and retain information-theoretic operational interpretations, e.g., the min-entropy as maximum achievable quantum correlation, and the max-entropy as decoupling accuracy. We furthermore generalize the smoothed versions of these entropies and prove an infinite-dimensional quantum asymptotic equipartition property. To facilitate these generalizations we show that the min- and max-entropy can be expressed in terms of convergent sequences of finite-dimensional min- and max-entropies, which provides a convenient technique to extend proofs from the finite to the infinite-dimensional setting.