Processing Information with Small Quantum Devices

One of the predominant challenges when engineering future quantum information processors is that complex quantum states are notoriously hard to prepare, maintain and control. Hence, there will be severe limitations on the size of quantum computers for the foreseeable future, and it is natural to ask: Can even a small quantum device offer significant advantages over classical information processing?

I will discuss recent progress on such questions, with an emphasis on
the novel mathematical techniques that are required to tackle them.
First, in the context of channel coding, I will show how we can use results from quantum hypothesis testing to characterize the fundamental limits of information transmission. Second, in the context of quantum cryptography, I will show how an entropic formulation of Heisenberg's uncertainty principle leads to a concise and general security proof for quantum key distribution.