Moderate deviation expansion for fully quantum tasks

TL;DR

This paper characterizes the finite block length trade-offs between communication cost and error in fully quantum information tasks within the moderate deviation regime, providing exact formulas and technical tools for quantum information theory.

Contribution

It introduces a precise characterization of moderate deviation trade-offs for various quantum communication tasks and develops a tight relation between key quantum information measures.

Findings

Exact trade-off characterizations for quantum source coding, state splitting, and entanglement-assisted channel coding.

Derived a tight relation between partially smoothed max-information and hypothesis testing relative entropy.

Expanded the partially smoothed max-information for i.i.d. states in the moderate deviation regime.

Abstract

The moderate deviation regime is concerned with the finite block length trade-off between communication cost and error for information processing tasks in the asymptotic regime, where the communication cost approaches a capacity-like quantity and the error vanishes at the same time. We find exact characterisations of these trade-offs for a variety of fully quantum communication tasks, including quantum source coding, quantum state splitting, entanglement-assisted quantum channel coding, and entanglement-assisted quantum channel simulation. The main technical tool we derive is a tight relation between the partially smoothed max-information and the hypothesis testing relative entropy. This allows us to obtain the expansion of the partially smoothed max-information for i.i.d. states in the moderate deviation regime.