Information entropy of complex probability

TL;DR

The paper expands probabilistic modeling to complex-valued probabilities and develops a corresponding information-theoretic framework. It defines a complex probability measure and extends Shannon entropy to complex probabilities, yielding a complex entropy whose real part captures magnitude-based uncertainty and whose imaginary part encodes phase information. The work derives key properties and extensions, including maximum entropy, joint entropy, conditional entropy, equilibration, and cross entropy in the complex setting, providing a unified view of information in complex probability spaces. This framework has potential implications for statistical mechanics, signal processing, and information theory, while highlighting interpretational and computational challenges in high-dimensional or highly phase-coherent contexts.

Abstract

Probability theory is fundamental for modeling uncertainty, with traditional probabilities being real and non-negative. Complex probability extends this concept by allowing complex-valued probabilities, opening new avenues for analysis in various fields. This paper explores the information-theoretic aspects of complex probability, focusing on its definition, properties, and applications. We extend Shannon entropy to complex probability and examine key properties, including maximum entropy, joint entropy, conditional entropy, equilibration, and cross entropy. These results offer a framework for understanding entropy in complex probability spaces and have potential applications in fields such as statistical mechanics and information theory.