Inevitable Negativity: Additivity Commands Negative Quantum Channel Entropy

TL;DR

This paper develops a framework for comparing uncertainty in classical and quantum channels, revealing that quantum channel entropy can be negative, which challenges traditional entropy concepts.

Contribution

It introduces three approaches to channel majorization that converge, providing a new foundation for defining and interpreting quantum channel entropy.

Findings

All entropy functions for quantum channels must be negative.

Three approaches to channel majorization are shown to be equivalent.

The framework clarifies the interpretation of entropy in quantum channels.

Abstract

Quantum channels represent a broad spectrum of operations crucial to quantum information theory, encompassing everything from the transmission of quantum information to the manipulation of various resources. In the domain of states, the concept of majorization serves as a fundamental tool for comparing the uncertainty inherent in both classical and quantum systems. This paper establishes a rigorous framework for assessing the uncertainty in both classical and quantum channels. By employing a specific class of superchannels, we introduce and elucidate three distinct approaches to channel majorization: constructive, axiomatic, and operational. Intriguingly, these methodologies converge to a consistent ordering. This convergence not only provides a robust basis for defining entropy functions for channels but also clarifies the interpretation of entropy in this broader context. Most notably, our findings reveal that any viable entropy function for quantum channels must assume negative values, thereby challenging traditional notions of entropy.