Classical Coding and the Cauchy-Schwarz Inequality

In classical coding, a single quantum state is encoded into classical information. Decoding this classical information in order to regain the original quantum state is known to be impossible. However, one can attempt to construct a state which comes as close as possible. We give bounds on the smallest possible trace distance between the original and the decoded state which can be reached. We give two approaches to the problem: one starting from Keyl and Werner's no-cloning theorem, and one starting from an operator-valued Cauchy-Schwarz inequality