Saturating Quantum Relative Entropy Inequalities

This dissertation describes the progress made towards understanding several quantum entropies and their mathematical properties. Here, we shall focus on various quantum relative entropies, that measure distinguishability between two quantum states (or two entities). In particular, we will focus all details towards understanding the widely used data processing inequality for quantum relative entropy. The data processing inequality describes how knowledge about quantum states (that describe a quantum system) cannot increase whenever the states undergo some noisy action (or a local operation). In chapter 3, we give a detailed explanation of the existing framework for the Umegaki relative entropy, the α− R´enyi relative entropy, the α− sandwiched R´enyi relative entropy, and a more general family of quantum relative entropies, namely the α − z R´enyi relative entropy. Chapter 4 is dedicated to the mathematical framework and tools we use to contribute towards saturating the data processing inequality of the α − z R´enyi relative entropy. In particular, we focus on jointly concave and jointly convex trace functionals. In Chapter 5, we describe and prove our main results with regards to saturating the data processing inequality for the α − z R´enyi relative entropy, under certain parameters, α and z. We prove necessary and algebraically sufficient conditions to saturate the data processing inequality for the α−z R´enyi relative entropy whenever 1 < α ≤ 2 and α/2 ≤ z ≤ α, provided that z > 1. Moreover, these conditions coincide whenever α = z.