On quantum Renyi entropies: a new generalization and some properties (1306.3142v4)

Abstract: The Renyi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies or mutual information, and have found many applications in information theory and beyond. Various generalizations of Renyi entropies to the quantum setting have been proposed, most notably Petz's quasi-entropies and Renner's conditional min-, max- and collision entropy. Here, we argue that previous quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of Renyi entropies that contains the von Neumann entropy, min-entropy, collision entropy and the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities, a duality relation, and an entropic uncertainty relation.

• The paper introduces a novel quantum Rényi divergence that unifies von Neumann, min-, and max-entropies under one framework.
• The paper establishes core properties, including data-processing inequality, monotonicity, and positivity, to ensure its operational relevance.
• The paper demonstrates practical implications for quantum information tasks and sets the stage for future research in quantum algorithms and communication.

A New Generalization of Quantum Rényi Entropies and Their Properties

The paper by Müller-Lennert et al. explores a novel generalization of Rényi entropies to the quantum domain, addressing the limitations of previous approaches like Petz's quasi-entropies and Renner's entropy measures. Quantum Rényi entropies, as proposed, incorporate vital quantum entropy measures such as von Neumann entropy, min-entropy, and max-entropy into a unified framework. This essay summarizes the key points of the paper, highlights numerical results, examines theoretical implications, and speculates on future developments in quantum information theory.

Overview and Motivation

Rényi entropies extend the concept of Shannon entropy, providing a parameterized family of measures applicable to diverse operational settings. Their generalization to quantum systems, however, presents challenges due to the incompatibility of existing approaches. The authors propose a new quantum Rényi entropy definition applying to all quantum states, including specific cases like von Neumann and min-entropy. Their formulation ensures natural properties such as data-processing inequalities, monotonicity in the Rényi parameter, and an entropic uncertainty relation.

Key Results

1. Definition and Generalization: The paper introduces the quantum Rényi divergence $D_\alpha(\rho \|\sigma)$ for $\alpha \in [\frac{1}{2}, 1) \cup (1, \infty)$, encompassing vital entropy measures and resolving compatibility issues.
2. Properties:
   • Data Processing Inequality: Demonstrated for all $\alpha \in (1,2]$, though general proofs for all $\alpha$ are noted from related works by Frank and Lieb, and Beigi.
   • Monotonicity: Proven that the quantum Rényi divergence is monotonically increasing in $\alpha$.
   • Positivity: Validated that $D_\alpha(\rho \| \sigma) \ge 0$ for normalized states, reinforcing its interpretation as a divergence.
3. Special Cases: The construction covers specific cases such as collision entropy and connections to fidelities for certain $\alpha$ values, proving its robustness and utility in quantum information.
4. Limiting Behavior: Regaining von Neumann entropy as $\alpha \to 1$ and yielding quantum relative max-entropy as $\alpha \to \infty$.
5. Duality and Uncertainty: The work extends the min/max entropy duality into Rényi entropies and establishes connections to entropic uncertainty relations, showcasing broad theoretical implications.

Implications and Future Directions

The introduction of a coherent quantum Rényi framework holds substantial implications for quantum information theory, as these entropies are essential for tasks like hypothesis testing and channel capacity analysis. Integrating a consistent set of entropies allows for a more unified analysis in quantum algorithms, privacy amplification, and encryption protocols.

Future research could delve into:

• Extending the framework to infinite-dimensional systems.
• Investigating practical applications in quantum communication and computation.
• Exploring further the operational significance and physical realizations of quantum Rényi entropies in laboratory settings.

Conclusion

By presenting a versatile and consistent approach to quantum Rényi entropies, the authors have laid the groundwork for an enriched understanding and application of entropic measures in quantum systems. Their work not only resolves existing contradictions but also paves the way for new insights and advancements in quantum information theory.