Uniform Additivity in Classical and Quantum Information

TL;DR

This paper introduces the concept of uniform additivity in information theory, providing a unified framework that captures all known additive formulas in classical and quantum contexts, and reveals new additive quantities.

Contribution

It characterizes uniformly additive entropic functions using linear programming, unifies classical and quantum additive formulas, and identifies the completely coherent information as a new additive quantity.

Findings

Uniform additivity captures all known quantum additive formulas.

Classical and quantum uniformly additive functions are formally equivalent.

Identifies the completely coherent information as a new additive quantity.

Abstract

Information theory establishes the fundamental limits on data transmission, storage, and processing. Quantum information theory unites information theoretic ideas with an accurate quantum-mechanical description of reality to give a more accurate and complete theory with new and more powerful possibilities for information processing. The goal of both classical and quantum information theory is to quantify the optimal rates of interconversion of different resources. These rates are usually characterized in terms of entropies. However, nonadditivity of many entropic formulas often makes finding answers to information theoretic questions intractable. In a few auspicious cases, such as the classical capacity of a classical channel, the capacity region of a multiple access channel and the entanglement assisted capacity of a quantum channel, additivity allows a full characterization of optimal rates. Here we present a new mathematical property of entropic formulas, uniform additivity, that is both easily evaluated and rich enough to capture all known quantum additive formulas. We give a complete characterization of uniformly additive functions using the linear programming approach to entropy inequalities. In addition to all known quantum formulas, we find a new and intriguing additive quantity: the completely coherent information. We also uncover a remarkable coincidence---the classical and quantum uniformly additive functions are identical; the tractable answers in classical and quantum information theory are formally equivalent. Our techniques pave the way for a deeper understanding of the tractability of information theory, from classical multi-user problems like broadcast channels to the evaluation of quantum channel capacities.