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Axiomatic Approach to Measures of Total Correlations
Gabriel L Moraes1, Renato M Angelo1, Ana C S Costa1
1Department of Physics, Federal University of Paraná, P.O. Box 19044, Curitiba 81531-980, PR, Brazil.
Entropy (Basel, Switzerland)
|January 8, 2025
Summary
Quantifying total correlations in quantum mechanics is challenging. This study explores reliable measures, finding quantum mutual information valid despite ordering issues in some systems.
Area of Science:
- Quantum Information Science
- Quantum Mechanics
- Mathematical Physics
Background:
- Correlations are fundamental in science, especially quantum mechanics.
- Quantifying total correlations accurately remains an open challenge.
- Existing measures like quantum mutual information (QMI) have limitations.
Purpose of the Study:
- To discuss the challenge of defining a reliable measure for total correlations.
- To outline essential properties for effective correlation measures.
- To introduce novel correlation measures and evaluate existing ones.
Main Methods:
- Review of existing correlation measures: QMI, p-norm, quantum Pearson coefficient.
- Introduction of new measures using Rényi and Tsallis relative entropies, and Kullback-Leibler divergence.
- Analysis of measure properties and their behavior in two-qubit systems.
Main Results:
- Quantum mutual information, p-norm, and Pearson measure are equivalent for two-qubit systems.
- All three measures exhibit an ordering problem.
- New measures based on relative entropies and KL divergence were introduced.
Conclusions:
- Despite criticisms and ordering problems, QMI is argued to be a valid measure of total correlations.
- The study highlights the ongoing debate and challenges in quantifying quantum correlations.
- Further research into robust correlation measures is warranted.
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