Compositional frameworks for supermaps and causality - Wilson, M (2023)
Quantum supermaps are transformations of quantum processes, and have found many applications in quantum foundations and quantum information theory in the past two decades, particularly in the study of causality. Whilst the concept of a supermap is a simple and intuitive one, the current state-of-the-art formalisations of supermaps cannot be applied to arbitrary Hilbert spaces or Operational Probabilistic Theories (OPTs). We review the standard approaches to defining supermaps in quantum theory, wherever possible highlighting the background compositional principles at play using diagrammatic languages and referring to their algebraic formalisation in the field of category theory. The core argument of this thesis is that a more principled and general approach to defining quantum supermaps exists, using a definition of locally-applicable transformation, which can be applied to any symmetric monoidal category. As a consequence this approach can be applied to all quantum processes on general quantum degrees of freedom and to all transformations in OPTs. We identify key compositional features for entire theories of supermaps and show that the supermaps of those theories are always operationally described by locally-applicable transformations. Two tests for a good construction of supermaps on symmetric monoidal categories are identified, recovery of standard physicists definitions for quantum supermaps when applied to categories of standard quantum processes, and existence of key compositional features. By the end of the thesis we find a way to strengthen locally-applicable transformations to construct the theory of polyslots, which passes both tests. Applications of this new general framework for the study of quantum causality and quantum information theory are identified as future potential research directions.