A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks

TL;DR

This paper develops a hierarchy of information measures for quantum information tasks, enabling tight second-order asymptotics and finite block length bounds for data compression and randomness extraction with quantum side information.

Contribution

It introduces a hierarchy of information quantities that connect one-shot entropies with asymptotic spectral entropies, improving analysis of quantum information tasks.

Findings

Derives tight second order asymptotics for quantum tasks.

Establishes a hierarchy linking one-shot and spectral entropies.

Provides bounds on operational quantities for finite block lengths.

Abstract

We consider two fundamental tasks in quantum information theory, data compression with quantum side information as well as randomness extraction against quantum side information. We characterize these tasks for general sources using so-called one-shot entropies. We show that these characterizations - in contrast to earlier results - enable us to derive tight second order asymptotics for these tasks in the i.i.d. limit. More generally, our derivation establishes a hierarchy of information quantities that can be used to investigate information theoretic tasks in the quantum domain: The one-shot entropies most accurately describe an operational quantity, yet they tend to be difficult to calculate for large systems. We show that they asymptotically agree up to logarithmic terms with entropies related to the quantum and classical information spectrum, which are easier to calculate in the i.i.d. limit. Our techniques also naturally yields bounds on operational quantities for finite block lengths.