Recovery and the Data Processing Inequality for quasi-entropies

Abstract

We prove number of quantitative stability bounds for the cases of equality in Petz's monotonicity theorem for quasi-relative entropies defined in terms of an operator monotone decreasing functions. Included in our results is a bound in terms of the Petz recovery map, but we obtain more general results. The present treatment is entirely elementary and developed in the context of finite dimensional von Neumann algebras where the results are already non-trivial and of interest in quantum information theory.

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• Equality conditions of Data Processing Inequality for $\alpha$-$z$ Rényi relative entropies
• On quantum quasi-relative entropy
• Revisiting the equality conditions of the data processing inequality for the sandwiched Rényi divergence
• Measure of genuine coherence based of quasi-relative entropy
• Conditional Independence of 1D Gibbs States with Applications to Efficient Learning