New Entropies Define Efficient Information Theory and Quantum System Limits.

The quantification of information, a cornerstone of modern science and technology, increasingly demands measures that reflect not only complexity but also the practical limitations of its manipulation. Researchers are now developing a more nuanced information theory for the quantum realm, addressing the efficiency with which information can be processed and utilised. This work, detailed in a new article entitled ‘Fully Quantum Computational Entropies’, introduces a pair of novel quantum entropies – minimum and maximum – designed to characterise these limitations. Noam Avidan, Thomas A. Hahn, Joseph M. Renes, and Rotem Arnon, collaborating across the Weizmann Institute of Science and ETH Zurich, establish fundamental properties for these entropies, including their behaviour under data processing and a crucial chain rule, and demonstrate their relevance to practical quantum information tasks such as entanglement distillation.
Zurich, Switzerland – Researchers unveil a novel information-theoretic framework designed to address limitations inherent in existing approaches by explicitly incorporating efficiency as a central principle, offering a fresh perspective on systems operating under resource constraints. This work establishes a foundation for quantifying the costs and feasibility of information manipulation, thereby opening new avenues for exploration in the field.

The researchers define both min-entropy and max-entropy, specifically designed to quantify information manipulation costs and feasibility, addressing a recognised gap in current theory. Min-entropy functions as a direct analogue to both classical unpredictability entropy – a measure of how difficult it is to predict a random variable – and the established quantum min-entropy, providing a consistent framework for analysing information uncertainty and associated resource costs. Crucially, min-entropy quantifies the maximum probability of obtaining a particular outcome, providing a lower bound on the information content.

Essential properties for this min-entropy, including adherence to the data processing inequality – which states that no operation can increase information – and a well-defined chain rule – allowing for the calculation of entropy for composite systems – ensure its mathematical coherence and broad applicability across diverse information-theoretic scenarios. These properties facilitate consistent analysis of information transformations and their associated resource costs, enabling a deeper understanding of efficient information processing in quantum systems.

Complementing the min-entropy, the max-entropy gains operational significance through an alternative formulation, revealing its capacity to capture the efficiency of entanglement distillation under practical constraints. Entanglement distillation is a crucial process for quantum communication and computation, reducing the effects of noise on quantum states, and the max-entropy provides a precise measure of how effectively this can be achieved using limited local operations. This connection to a practical quantum information task highlights the operational relevance of the newly defined entropy and demonstrates its potential for real-world applications.

Researchers support these theoretical developments with detailed appendices, employing the probabilistic method to demonstrate the existence of hard-to-invert permutations, ensuring the mathematical validity of the proposed entropies and their associated properties. These rigorous proofs enhance the credibility of the framework and provide a clear understanding of its underlying principles, with detailed explanations and consistent notation.

Future work will focus on exploring the implications of this framework for specific quantum technologies and developing practical algorithms for estimating the defined entropies. The team also plans to investigate the relationship between these entropies and other measures of quantum information, such as entanglement and coherence. This research represents a significant step towards a more complete and applicable information theory, paving the way for advancements in quantum communication, computation, and cryptography.