Quantum Information Theory
Contextuality describes systems whose behaviour precludes any explanation in terms of a classical notion of reality wherein all properties can be ascribed values simultaneously. The primary examples of such systems arise within quantum mechanics. Contextuality subsumes nonlocality as a special case. It is significant not just for the theoretical foundations of quantum theory but also as a potential resource for achieving advantage in a variety of information-theoretic tasks, e.g. computation. Here, we study contextuality both abstractly, by considering high-level, theory-independent mathematical frameworks, and concretely, as resources in particular tasks. This work lies at the intersection of many fields including mathematical logic, quantum physics, programming languages, and quantum computation.
Entanglement is a fundamental feature of quantum physics, and is a useful ingredient in quantum technologies such as quantum cryptography and metrology. Due in part to these applications, and because entanglement is considered to be a distinguishing quality of non-classical behaviour, a rich mathematical theory of entanglement has been developed in recent years which attempts to understand how entanglement in a physical system can be detected and quantified. We investigate the mathematical structure of entangled states and develop techniques to characterise their non-classical properties.
Discrete mathematical structures in quantum theory
Finite relational structures and homomorphisms between them are pervasive notions in logic, computer science and combinatorics. Many relevant questions in finite model theory, constraint satisfaction and graph theory can be phrased in terms of (existence or number of) homomorphisms between finite structures. Developing a quantum theory about these fundamental structures can serve to delineate the scope of quantum advantage and understand the use of quantum resources in an abstract setting. We follow the tradition of using category theory to combine concepts across disciplines that are not yet related, with the hope of provide sensible definitions at a formal level and then use them to single out fundamental aspects along with potential new applications.
Quantum Shannon Theory
The advent of quantum theory offers new and interesting ways of transmitting, extracting and utilising information. Quantum Shannon theory aims to understand both the theoretical limits of these processes, such as what is the maximum rate of transmission using a quantum channel? How efficiently can quantum information be compressed without error? How much information can we learn from a single quantum measurement? Broadly speaking, it is a generalisation of classical information theory.