Optimized quantum f-divergences and data processing

Abstract

The quantum relative entropy is a measure of the distinguishability of two quantum states, and it is a unifying concept in quantum information theory: many information measures such as entropy, conditional entropy, mutual information, and entanglement measures can be realized from it. As such, there has been broad interest in generalizing the notion to further understand its most basic properties, one of which is the data processing inequality. The quantum f-divergence of Petz is one generalization of the quantum relative entropy, and it also leads to other relative entropies, such as the Petz-Renyi relative entropies. In this paper, I introduce the optimized quantum f-divergence as a related generalization of quantum relative entropy. I prove that it satisfies the data processing inequality, and the method of proof relies upon the operator Jensen inequality, similar to Petz's original approach. Interestingly, the sandwiched Renyi relative entropies are particular examples of the optimized f-divergence. Thus, one benefit of this paper is that there is now a single, unified approach for establishing the data processing inequality for both the Petz-Renyi and sandwiched Renyi relative entropies, for the full range of parameters for which it is known to hold. This paper discusses other aspects of the optimized f-divergence, such as the classical case, the classical-quantum case, and how to construct optimized f-information measures.

36 pages, accepted for publication in the special issue "Shannon's Information Theory 70 years on: applications in classical and quantum physics" for Journal of Physics A

References in corpus (2)

• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Error Exponent in Asymmetric Quantum Hypothesis Testing and Its Application to Classical-Quantum Channel coding

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• Entropy of a quantum channel
• Constraining Quantum Fields using Modular Theory
• Geometric distinguishability measures limit quantum channel estimation and discrimination
• Entanglement and secret-key-agreement capacities of bipartite quantum interactions and read-only memory devices
• Inequalities for quantum divergences and the Audenaert-Datta conjecture
• On contraction coefficients, partial orders and approximation of capacities for quantum channels
• Quantum Rényi and $f$-divergences from integral representations
• Quantum Pufferfish Privacy: A Flexible Privacy Framework for Quantum Systems
• Quantum Neural Estimation of Entropies
• Recoverability for optimized quantum $f$-divergences
• Bounding the forward classical capacity of bipartite quantum channels
• Integral formula for quantum relative entropy implies data processing inequality
• Fidelity-Based Smooth Min-Relative Entropy: Properties and Applications
• Monotonic multi-state quantum $f$-divergences
• An invitation to the sample complexity of quantum hypothesis testing
• Proof of Renyi QNEC for free fermions
• Quantifying the unextendibility of entanglement
• Not even 6 dB: Gaussian quantum illumination in thermal background
• A smallest computable entanglement monotone
• Revisiting the equality conditions of the data processing inequality for the sandwiched Rényi divergence
• Evolved Quantum Boltzmann Machines
• On distinguishability distillation and dilution exponents
• Reverse-type Data Processing Inequality
• Cooperation and dependencies in multipartite systems