Salman Beigi

Divergences lie at the core of information-theoretic applications. A recently introduced family of f-divergences, defined via an integral representation, has exhibited remarkable properties --- for instance, for the study of contraction coefficients. However, many familiar properties of their classical analogous have remained elusive. In this work, we develop alternative representations of the quantum f-divergences by leveraging the recently established quantum layer-cake theorem. These new formulations enable us to establish several key properties, including monotonicity and connections to other divergences. As our main application, we show how these representations unify and streamline various proofs in quantum hypothesis testing, yielding tighter achievability bounds through conceptually simple arguments that apply across different error regimes.

Abstract A correlation measure of a bipartite or multipartite source is said to satisfy the tensorization property if evaluating it on independent copies of the source, we get the same quantity as we evaluate it on a single copy. Such measures of correlation naturally appear in the study of some source coding and channel coding problems, as well as quantum nonlocality. This tutorial begins with some motivating examples that elucidate the importance of the tensorization property. Then two main measures of correlation with the tensorization property, namely the maximal correlation and the hypercontractivity ribbon are introduced. Finally some applications of these correlation measures, particularly in the study of quantum nonlocality in various settings are discussed.