  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 contribution, 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 approach 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.
