Recent technological advances allow to build revolutionary devices for information processing by making use of quantum mechanics. Most strikingly, quantum communication allows to transmit information with physical security guarantees and quantum algorithms solve problems that are otherwise computationally unfeasible. Having such quantum technologies in sight, it becomes imperative to investigate what can be achieved in practice but also what the ultimate limitations are. Since both questions are fundamentally concerned with processing information, it is natural to seek an information theoretic approach to find an answer.
Entropy inequalities are among the most important tools in the theory of classical and quantum information processing. They have been successfully applied to bound the information flow in classical information networks and to understand the inner workings of deep neural networks. However, in the quantum case our understanding of such inequalities is currently lacking.
Most importantly, the quantum information community is missing a developed theory regarding conditioning on quantum systems. This fact hinders us in fully judging the potential impact of modern quantum technologies, since one is not able to effectively determine the properties of the underlying operations. I propose to set up a conceptually novel information-theoretic framework to quantify practical advantages of quantum-based devices for network communication and machine learning, by mitigating the quantum conditioning problem. I will develop entropic inequalities that describe the change of entropy under specific operations, in particular, quantum gates and channels that can be part of an encoding or decoding circuit for quantum communication, a quantum internet or act as nodes in a quantum neural network. I will develop a framework that allows to prove such entropy inequalities, on a unified basis and employ them to better understand these important technologies of the future.
