Error Exponents for Quantum Packing Problems via An Operator Layer Cake Theorem

TL;DR

This paper establishes a new one-shot bound for classical-quantum channel coding, deriving the optimal error exponent and extending to various quantum packing problems, using an innovative operator layer cake theorem.

Contribution

It introduces an operator layer cake theorem and applies it to derive the optimal error exponent for quantum packing problems, extending previous bounds to infinite-dimensional spaces.

Findings

Proves a one-shot random coding bound for classical-quantum channels.

Derives the optimal error exponent for rates above the critical rate.

Shows the equivalence of pretty-good measurement to a randomized Holevo-Helstrom measurement.

Abstract

In this work, we prove a one-shot random coding bound for classical-quantum channel coding, a problem conjectured by Burnashev and Holevo in 1998. By choosing the optimal input distribution, the bound implies the optimal error exponent (i.e., the reliability function) of classical-quantum channels for rates above the critical rate, even in infinite-dimensional Hilbert spaces. Our result extends to various quantum packing-type problems, including classical communication over any fully quantum channel with or without entanglement-assistance, constant composition codes, and classical data compression with quantum side information via fixed-length or variable-length coding. Our technical ingredient is to establish an operator layer cake theorem - the directional derivative of an operator logarithm admits an integral representation of certain projections. This shows that a kind of pretty-good measurement is equivalent to a randomized Holevo-Helstrom measurement, which provides an operational explanation of why the pretty-good measurement is pretty good.