Quantum Renyi relative entropies and their use

The past decade of research in quantum information theory has witnessed  extraordinary progress in understanding communication over quantum channels, due in large part to quantum generalizations of the classical Renyi relative entropy. One generalization is known as the sandwiched Renyi relative entropy and finds its use in characterizing asymptotic behavior in quantum hypothesis testing. It has also found use in establishing strong converse theorems (fundamental communication capacity limitations) for a variety of quantum communication tasks. Another generalization is known as the geometric Renyi relative entropy and finds its use in establishing strong converse theorems for feedback assisted protocols, which apply to quantum key distribution and distributed quantum computing scenarios. Finally, a generalization now known as the Petz--Renyi relative entropy plays a critical role for statements of achievability in quantum communication. In this talk, I will review these quantum generalizations of the classical Renyi relative entropy, discuss their relevant information-theoretic properties, and the applications mentioned above.