Systematic Construction of Monotone Quantum Relative Entropies and Barycentric Rényi Divergences

Belangrijkste concepten

The authors provide systematic ways to define monotone quantum relative entropies and multi-variate quantum Rényi divergences starting from a set of monotone quantum relative entropies.

Samenvatting

The paper focuses on defining quantum extensions of classical information-theoretic quantities with good mathematical properties, particularly monotonicity and additivity.
Key highlights:

The authors give a general procedure to construct monotone and additive quantum relative entropies from a given one with the same properties. This provides a new one-parameter family of quantum relative entropies interpolating between the Umegaki and Belavkin-Staszewski relative entropies.

The authors use a generalization of a classical variational formula to define multi-variate quantum Rényi quantities corresponding to any finite set of quantum relative entropies and real weights. They show that monotone quantum relative entropies define monotone Rényi quantities.

The authors introduce "barycentric Rényi divergences" as a new class of quantum Rényi divergences. They show that if the generating quantum relative entropies are lower semi-continuous, monotone, and additive, and at least one is strictly larger than the Umegaki relative entropy, then the resulting barycentric Rényi divergences are strictly between the log-Euclidean and maximal Rényi divergences.

The authors analyze the properties of the new quantum relative entropies and Rényi divergences, including monotonicity, additivity, lower semi-continuity, and relations to other known quantities.

Statistieken

The paper does not contain explicit numerical data or statistics. It focuses on the mathematical properties of the defined quantum information-theoretic quantities.

Citaten

"Interestingly, despite its central importance in information theory, only two additive and monotone quantum extensions of the classical relative entropy have been known so far, the Umegaki and the Belavkin-Staszewski relative entropies, which are the minimal and the maximal ones, respectively, with these properties."
"We show that if both Dq0 and Dq1 are lower semi-continuous, monotone, and additive quantum relative entropies, and at least one of them is strictly larger than the Umegaki relative entropy then the resulting barycentric Rényi divergences are strictly between the log-Euclidean and the maximal Rényi divergences, and hence they are different from any previously studied quantum Rényi divergence."

Belangrijkste Inzichten Gedestilleerd Uit

Geometric relative entropies and barycentric Rényi divergences

by Milá... om arxiv.org 04-19-2024

https://arxiv.org/pdf/2207.14282.pdf

Diepere vragen

How can the new quantum relative entropies and Rényi divergences be applied to solve specific problems in quantum information theory and quantum computing

The new quantum relative entropies and Rényi divergences introduced in the paper have various applications in quantum information theory and quantum computing.

State Discrimination: These new quantum divergences can be used to quantify the dissimilarity between quantum states, which is crucial in tasks like state discrimination. By measuring the difference between two quantum states using these divergences, one can optimize strategies for distinguishing between quantum states accurately.

Channel Capacity: In quantum communication, understanding the capacity of quantum channels is essential for efficient information transmission. The new divergences can help in characterizing the capacity of quantum channels and optimizing communication protocols.

Entanglement Manipulation: Quantum entanglement is a key resource in quantum computing. The new divergences can aid in quantifying the amount of entanglement present in a quantum system and studying how it can be manipulated for various quantum computing tasks.

Quantum State Conversion: The divergences can be used to analyze the convertibility of quantum states under different operations. Understanding the conditions under which one quantum state can be transformed into another is crucial in quantum information processing.

Error Correction: Quantum error correction is vital for maintaining the integrity of quantum information. The divergences can play a role in designing error-correcting codes and protocols to protect quantum information from noise and errors.

Overall, the new quantum relative entropies and Rényi divergences provide powerful tools for analyzing and solving a wide range of problems in quantum information theory and quantum computing.

What are the operational interpretations and physical meanings of the new quantum information-theoretic quantities introduced in this work

The new quantum information-theoretic quantities introduced in this work have operational interpretations and physical meanings that are crucial in quantum information processing.

Monotonicity: The monotonicity of these quantities under certain operations reflects the preservation of information content during quantum processes. Monotonicity under quantum operations signifies the non-increase of information content, which is essential for maintaining the integrity of quantum information.

Additivity: The additivity property of these quantities can be linked to the composability of quantum systems. Additivity under tensor products implies that the information content of composite quantum systems is the sum of the information content of individual systems, which is fundamental in quantum computing and communication.

State Convertibility: The quantities can be used to determine the convertibility of quantum states under specific operations. Understanding the conditions for converting one quantum state into another provides insights into the dynamics of quantum systems and the feasibility of quantum protocols.

Quantum Channel Capacity: By quantifying the divergence between quantum states, these quantities can help in determining the capacity of quantum channels for transmitting information reliably. Channel capacity is a key parameter in designing efficient quantum communication systems.

Entanglement Measures: In the context of entanglement theory, these quantities can serve as measures of entanglement between quantum systems. They provide a way to quantify the amount of entanglement present in a quantum state, which is crucial for various quantum information processing tasks.

Overall, the operational interpretations and physical meanings of these new quantum information-theoretic quantities provide valuable insights into the behavior and properties of quantum systems.

Can the techniques used in this paper be extended to define other types of quantum divergences or information measures with desirable properties

The techniques used in this paper to define quantum divergences and information measures with desirable properties can be extended to explore a wide range of related concepts in quantum information theory.

Generalized Divergences: The methodology of constructing quantum divergences from a given set of relative entropies can be applied to define other types of quantum divergences with specific properties. By exploring different combinations and transformations of existing divergences, new information measures can be developed.

Multi-Variate Measures: The approach of defining multi-variate quantum R´enyi quantities based on a set of quantum relative entropies can be extended to define multi-variate measures beyond R´enyi divergences. This can lead to the development of new information-theoretic tools for analyzing complex quantum systems.

Operational Interpretations: The techniques used in this paper can be further explored to establish operational interpretations for a broader class of quantum information measures. By connecting mathematical properties with operational significance, these measures can find applications in various quantum information processing tasks.

Error Correction and Quantum Communication: Extending the techniques to define divergences and measures that are relevant to error correction codes and quantum communication protocols can enhance the efficiency and reliability of quantum information processing systems.

By building upon the methods and principles presented in this work, researchers can continue to innovate and develop new quantum information measures with diverse applications in quantum computing and quantum communication.