Quantum Data Processing Inequalities and their Reverse

Abstract

Any reasonable measure of quantum information must satisfy a data processing inequality, that is, it must not increase under the action of a quantum channel. The same is, therefore, true for measures of distinguishability of quantum states. In this thesis, we study two families of distinguishability measures that are particularly interesting: the Riemannian metric (more precisely, the corresponding semi-norm) and the standard quantum f-divergences (sometimes referred to as just standard f-divergences). However, rather than focusing on the information lost, we ask about the information preserved - namely, a reverse data processing inequality. As is established in this thesis, an exact reverse data processing inequality for all states acted on by a specific channel is not possible for these measures if the output dimension of the quantum channel is no greater than the input dimension (which includes several important channels). Instead, we settle for a reverse data processing inequality on a restricted set of input states, or oftentimes it suffices to only compare the loss of information incurred via two given quantum channels in general. This thesis demonstrates cases of a restricted reverse data processing inequality for these measures and initiates a study of the similarities between the Riemannian metrics and standard quantum f-divergences in this context.