Optimized quantum f-divergences and data processing

TL;DR

This paper introduces the optimized quantum f-divergence, a new generalization of quantum relative entropy, proving it satisfies the data processing inequality and unifying the understanding of various quantum relative entropies.

Contribution

The paper presents the optimized quantum f-divergence, establishing its data processing inequality and unifying the treatment of Petz-Renyi and sandwiched Renyi relative entropies.

Findings

Proves the data processing inequality for the optimized quantum f-divergence.

Shows sandwiched Renyi relative entropies are special cases of the new divergence.

Provides a unified framework for quantum relative entropy measures.

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.